{"title":"Stickervellen","description":"\u003cp\u003ePixel-art-natuurkundestickervellen: A5-kiss-cut-vellen met elk vijf of negen stickers, van portretten van natuurkundigen zoals Newton, Curie, Tesla, Feynman en Hawking tot wet-personages voor de wet van Lenz, tsjerenkovlicht, spin en kosmische tijd. Elke productpagina heeft een natuurkundepaneel met de vergelijkingen, symbolen en bronnen.\u003c\/p\u003e\n\u003cp\u003eDe verzendtarieven staan in ons \u003ca href=\"\/nl\/policies\/shipping-policy\"\u003eVerzendbeleid\u003c\/a\u003e.\u003c\/p\u003e","products":[{"product_id":"physicist-portraits-1-sticker-sheet","title":"Physicist Portraits Vol. 1 Pixel Sticker Sheet – 10 Physics Stickers","description":"\u003cp\u003e10 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: Physicist Portraits Vol. 1. Every character below has its own sticker page with the full physics panel.\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/emmy-noether-sticker-symmetry-in-conservation-out-physics-sticker\"\u003eEmmy Noether\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eNoether's theorem says that every continuous symmetry of a system's laws comes with a conserved quantity. Symmetry under shifts in time gives conservation of energy, shifts in space give momentum, and rotations give angular momentum.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eQ = Σ\u003csub\u003ei\u003c\/sub\u003e (∂L\/∂q̇\u003csub\u003ei\u003c\/sub\u003e) δq\u003csub\u003ei\u003c\/sub\u003e\u003c\/strong\u003e\u003cbr\u003eQ: conserved Noether charge (momentum for a shift, angular momentum for a rotation) (kg·m\/s, J·s); L: Lagrangian (J); qᵢ: generalised coordinates (lengths or angles) (m, rad); ̇qᵢ: generalised velocities (m\/s, rad\/s); ε: small symmetry parameter (a length for a shift, an angle for a rotation, a time for a clock shift) (m, rad, s); δ qᵢ: direction of the symmetry shift (its size is carried by ε)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/emilie-du-chatelet-sticker-2x-the-speed-4x-the-dent-physics-sticker\"\u003eÉmilie du Châtelet\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eKinetic energy grows with the square of speed, so twice the speed carries four times the energy. Du Châtelet used experiments with balls dropped into clay, where the dent depth grew with the square of the speed, to argue for this 'vis viva' measure.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eE\u003csub\u003ek\u003c\/sub\u003e = ½mv²\u003c\/strong\u003e\u003cbr\u003eE_k: kinetic energy (J); m: mass (kg); v: impact speed (m\/s); g: gravitational acceleration (m\/s²); h: drop height (m); F: average resisting force of the clay (assumed constant) (N)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/chien-shiung-wu-sticker-she-broke-symmetry-and-records-physics-sticker\"\u003eChien-Shiung Wu\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn Wu's experiment, cobalt-60 nuclei were lined up by a magnetic field, and the electrons from their beta decay came out preferentially opposite to the nuclear spin. A mirror image of that experiment would show the reverse, so the weak force distinguishes left from right.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003e⁶⁰Co → ⁶⁰Ni* + e⁻ + ν̄\u003csub\u003ee\u003c\/sub\u003e\u003c\/strong\u003e\u003cbr\u003eJ: nuclear spin (axial vector) (J·s); pₑ: electron momentum (polar vector) (kg·m\/s); ̂P: parity (mirror) operation\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/galileo-galilei-sticker-telescope-tea-trouble-with-the-church-physics-sticker\"\u003eGalileo Galilei\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eGalileo found that a falling body's distance grows with the square of the time. In equal time intervals the distances covered go in the ratio of the odd numbers 1, 3, 5 and so on, which is the signature of constant acceleration.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ed = ½gt²\u003c\/strong\u003e\u003cbr\u003ed: distance fallen from rest (m); g: gravitational acceleration (m\/s²); t: time (s); d_n: distance covered in the n-th equal time interval (m)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/isaac-newton-sticker-gravity-hits-different-physics-sticker\"\u003eIsaac Newton\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eNewton's second law says force equals the rate of change of momentum. His law of gravitation says every two masses attract with a force proportional to their product and inversely proportional to the square of their distance, and together they explain falling apples and planetary orbits.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003e\u003cb\u003eF\u003c\/b\u003e = d\u003cb\u003ep\u003c\/b\u003e\/dt\u003c\/strong\u003e\u003cbr\u003eF: net force (N); p: momentum (kg·m\/s); G: gravitational constant (m³·kg⁻¹·s⁻²); m₁, m₂: masses (kg); r: separation (m)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/james-clerk-maxwell-sticker-field-goals-achieved-physics-sticker\"\u003eJames Clerk Maxwell\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eMaxwell added one missing term, the displacement current, to the known laws of electricity and magnetism. With it the four equations are consistent and predict waves moving at the speed of light, which showed that light itself is an electromagnetic wave.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003e∇·\u003cb\u003eE\u003c\/b\u003e = ρ\/ε\u003csub\u003e0\u003c\/sub\u003e, ∇·\u003cb\u003eB\u003c\/b\u003e = 0\u003c\/strong\u003e\u003cbr\u003eE: electric field (V\/m); B: magnetic flux density (T); ρ: charge density (C\/m³); J: current density (A\/m²); ε₀, μ₀: vacuum permittivity, permeability (F\/m, N\/A²); c: speed of light (m\/s)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/michael-faraday-sticker-all-quiet-in-here-physics-sticker\"\u003eMichael Faraday\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eGauss's law relates the electric flux through a closed surface to the charge inside it. In a closed conductor at equilibrium the charges sit on the outside, so the field in an empty cavity is zero, which is why Faraday was safe inside his charged cage.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003e∮\u003csub\u003eS\u003c\/sub\u003e \u003cb\u003eE\u003c\/b\u003e·d\u003cb\u003eA\u003c\/b\u003e = Q\u003csub\u003eenc\u003c\/sub\u003e\/ε\u003csub\u003e0\u003c\/sub\u003e\u003c\/strong\u003e\u003cbr\u003eE: electric field (V\/m); dA: outward area element of a closed surface S (m²); Q_(enc): charge enclosed by S (C); ε₀: vacuum permittivity (F\/m)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/nikola-tesla-sticker-ac-slays-dc-pixel-art-physics-sticker\"\u003eNikola Tesla\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003ePower lost in a wire is the current squared times the resistance, so sending power at high voltage and low current wastes much less. Transformers, which only work with alternating current, step the voltage up or down in proportion to the number of turns on each coil.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eV(t) = V\u003csub\u003e0\u003c\/sub\u003e sin(ωt)\u003c\/strong\u003e\u003cbr\u003eV: voltage (V); V₀: peak voltage (V); ω: angular frequency (rad\/s); V_s, V_p: secondary, primary voltage (V); N_s, N_p: turns on secondary, primary; P_(loss): resistive power loss (W)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/ernest-rutherford-sticker-some-came-back-physics-sticker\"\u003eErnest Rutherford\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eRutherford fired alpha particles at gold foil and some bounced almost straight back. His formula, with its steep dependence on 1\/sin⁴(θ\/2), only works if the positive charge sits in a tiny, dense nucleus.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003edσ\/dΩ = (zZe²\/(16πε\u003csub\u003e0\u003c\/sub\u003eE))² \/ sin⁴(θ\/2)\u003c\/strong\u003e\u003cbr\u003edσ\/dΩ: differential cross section per solid angle (m²\/sr); z: charge number of the alpha particle (2); Z: charge number of the target nucleus (gold: 79); e: elementary charge (C); ε₀: vacuum permittivity (F\/m); E: kinetic energy of the alpha particle (J)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/hideki-yukawa-sticker-heavy-ball-short-game-physics-sticker\"\u003eHideki Yukawa\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eYukawa proposed that nuclear forces come from exchanging a massive particle, so they fade out exponentially beyond a range of about ħ\/mc. Turning the observed short range around predicted the mass of the carrier, later identified as the pion.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eV(r) = −(g²\/4π) e\u003csup\u003e−r\/R\u003c\/sup\u003e\/r, R = ħ\/(mc)\u003c\/strong\u003e\u003cbr\u003eV(r): potential energy between two nucleons (J); g: coupling strength (g²\/4π has units of energy times length here) (J^(1\/2)·m^(1\/2)); r: separation (m); R: range of the force (m); λ: Yukawa's range parameter, the inverse of R (he took 5×10¹² per centimetre) (m⁻¹); ħ: reduced Planck constant (J·s)\u003c\/p\u003e","brand":"Printify","offers":[{"title":"White \/ Kiss-Cut \/ 5.83″ × 8.27″","offer_id":50964876787815,"sku":"28254359185110691288","price":19.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/8509343553496974353_2048.jpg?v=1793674458"},{"product_id":"physicist-portraits-2-sticker-sheet","title":"Physicist Portraits Vol. 2 Pixel Sticker Sheet – 10 Physics Stickers","description":"\u003cp\u003e10 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: Physicist Portraits Vol. 2. Every character below has its own sticker page with the full physics panel.\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/christiaan-huygens-sticker-every-point-a-source-physics-sticker\"\u003eChristiaan Huygens\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eHuygens' principle treats every point on a wavefront as a source of small spherical wavelets. After a short time, the new wavefront is the surface that touches all of those wavelets, which explains reflection and refraction.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003er = vΔt\u003c\/strong\u003e\u003cbr\u003er: radius of each secondary wavelet after time \\Delta t (m); v: wave speed in the medium (m\/s); Δ t: time step (s)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/lise-meitner-sticker-split-happens-physics-sticker\"\u003eLise Meitner\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eWhen a uranium nucleus splits, the pieces weigh less than the original, and the missing mass appears as energy. Meitner and Frisch estimated it from the electric repulsion between the two fragments at the moment they part: about 200 MeV per fission.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eQ = (M\u003csub\u003einitial\u003c\/sub\u003e − M\u003csub\u003efinal\u003c\/sub\u003e)c²\u003c\/strong\u003e\u003cbr\u003eQ: energy released (quoted in MeV) (J); M: total rest mass before\/after (kg); c: speed of light (m\/s); E_C: Coulomb repulsion energy of two touching fragments (J); Z₁, Z₂: fragment proton numbers; e: elementary charge (C)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/maria-goeppert-mayer-sticker-a-full-floor-is-magic-physics-sticker\"\u003eMaria Goeppert Mayer\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eProtons and neutrons fill energy shells in the nucleus, much like electrons in an atom. Adding strong spin-orbit coupling moves the shell gaps to the observed magic numbers 2, 8, 20, 28, 50, 82 and 126, where nuclei are unusually stable.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eE\u003csub\u003eN\u003c\/sub\u003e = ħω(N + 3\/2)\u003c\/strong\u003e\u003cbr\u003eE_N: energy of oscillator shell N (J); N: oscillator shell number; ħω: oscillator level spacing (J); g_N: states in shell N per nucleon species (spin included); V_(ℓ s): spin–orbit potential (J); C(r): radial strength of the spin–orbit term (with orbital and spin angular momentum measured in units of ħ) (J)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/marie-curie-sticker-she-literally-glowed-up-physics-sticker\"\u003eMarie Curie\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eEach radioactive nucleus has a fixed chance per second of decaying, so the number left falls exponentially. The half-life is ln 2 divided by that decay constant, and the activity is the decay constant times the number of nuclei.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eN(t) = N\u003csub\u003e0\u003c\/sub\u003e e\u003csup\u003e−λt\u003c\/sup\u003e\u003c\/strong\u003e\u003cbr\u003eN: number of undecayed nuclei; λ: decay constant (s⁻¹); t_(1\/2): half-life (s); A: activity (1 Bq = 1 decay per second) (Bq)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/c-v-raman-sticker-most-pass-free-a-few-pay-a-toll-physics-sticker\"\u003eC. V. Raman\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eMost light scattered by molecules keeps its colour, but a tiny fraction gives up or takes energy from a molecular vibration and comes out shifted. That Raman shift is a fingerprint of the molecule's vibrations.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ehν\u003csub\u003es\u003c\/sub\u003e = hν\u003csub\u003e0\u003c\/sub\u003e − hν\u003csub\u003evib\u003c\/sub\u003e (Stokes)\u003c\/strong\u003e\u003cbr\u003eh: Planck constant (J·s); ν₀: frequency of incident light (Hz); ν_s: frequency of scattered light (Hz); ν_(vib): molecular vibration frequency (Hz); Δν̃: Raman shift in wavenumber (usually quoted per centimetre) (m⁻¹, cm⁻¹); λ₀, λ_s: incident and scattered wavelengths (m)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/max-planck-sticker-he-quantized-the-vibe-physics-sticker\"\u003eMax Planck\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003ePlanck found that the glow of a hot body could only be explained if energy is exchanged in packets of hν. His formula gives the full blackbody spectrum at any temperature and started quantum theory.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eE = hν\u003c\/strong\u003e\u003cbr\u003eE: quantum of energy (J); h: Planck constant (exact since 2019) (J·s); ν: frequency (Hz); B_ν: spectral radiance (W·m⁻²·sr⁻¹·Hz⁻¹); T: temperature (K); k_B: Boltzmann constant (J\/K)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/niels-bohr-sticker-said-nothing-clearly-explained-everything-physics-sticker\"\u003eNiels Bohr\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn Bohr's model the energy levels of hydrogen are minus the Rydberg energy divided by n squared. When an electron drops from a higher level to a lower one it emits a photon carrying exactly the energy difference, which explains hydrogen's sharp spectral lines.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eE\u003csub\u003en\u003c\/sub\u003e = −R\u003csub\u003e∞\u003c\/sub\u003ehc\/n², with R\u003csub\u003e∞\u003c\/sub\u003ehc ≈ 13.6 eV\u003c\/strong\u003e\u003cbr\u003eE_n: energy of level n (infinite-nuclear-mass approximation; hydrogen uses reduced mass; quoted in eV) (J); R_∞ hc: Rydberg energy (CODATA 2022) (J); n: principal quantum number; h: Planck constant (J·s); ν: photon frequency (Hz); Eᵢ, E_f: initial and final level energies (J)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/enrico-fermi-sticker-within-an-order-physics-sticker\"\u003eEnrico Fermi\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eThe Fermi–Dirac distribution gives the chance that a state of energy E is occupied by a fermion. Deep below the chemical potential states are full, far above it they are empty, and exactly at it the chance is one half.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ef(E) = 1\/(e\u003csup\u003e(E−μ)\/(kT)\u003c\/sup\u003e + 1)\u003c\/strong\u003e\u003cbr\u003ef(E): Fermi–Dirac occupation probability; E: single-particle energy (J); μ: chemical potential (J); k: Boltzmann constant (J\/K); T: temperature (K)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/richard-feynman-sticker-quantum-it-s-complicated-physics-sticker\"\u003eRichard Feynman\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eFeynman's rule: to get from a to b, a quantum particle takes every possible path, and each path contributes a phase set by its action. You add all those amplitudes and square the total to get the probability.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eK(b,a) = Σ\u003csub\u003epaths\u003c\/sub\u003e e\u003csup\u003eiS\/ħ\u003c\/sup\u003e\u003c\/strong\u003e\u003cbr\u003eK(b,a): amplitude to go from a to b (propagator), in d spatial dimensions (m^(-d)); S: classical action along a path (J·s); ħ: reduced Planck constant (J·s); φ_k: amplitude for alternative k; P: probability\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/wolfgang-pauli-sticker-one-per-state-physics-sticker\"\u003eWolfgang Pauli\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eTwo identical fermions must have a wavefunction that changes sign when they are swapped. If both tried to occupy the same state, the wavefunction would be zero everywhere, so the arrangement is forbidden: one fermion per state.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eψ(1,2) = [φ\u003csub\u003ea\u003c\/sub\u003e(1)φ\u003csub\u003eb\u003c\/sub\u003e(2) − φ\u003csub\u003ea\u003c\/sub\u003e(2)φ\u003csub\u003eb\u003c\/sub\u003e(1)]\/√2\u003c\/strong\u003e\u003cbr\u003eψ: two-particle antisymmetrised wavefunction; φₐ, φ_b: single-particle orbitals (including spin); a, b: full sets of quantum numbers (n,ℓ,m_ℓ,m_s)\u003c\/p\u003e","brand":"Printify","offers":[{"title":"White \/ Kiss-Cut \/ 5.83″ × 8.27″","offer_id":50964876853351,"sku":"31018306477649342260","price":19.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/405259607989861568_2048.jpg?v=1793674470"},{"product_id":"physicist-portraits-3-sticker-sheet","title":"Physicist Portraits Vol. 3 Pixel Sticker Sheet – 10 Physics Stickers","description":"\u003cp\u003e10 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: Physicist Portraits Vol. 3. Every character below has its own sticker page with the full physics panel.\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/satyendra-nath-bose-sticker-particle-guy-major-stats-physics-sticker\"\u003eSatyendra Nath Bose\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eBose counted photons as truly identical particles and recovered Planck's law. The resulting Bose–Einstein distribution gives the average number of particles in a state of given energy, and it allows many bosons to pile into the same state.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003en̄(ε) = 1\/(e\u003csup\u003e(ε−μ)\/(k\u003csub\u003eB\u003c\/sub\u003eT)\u003c\/sup\u003e − 1)\u003c\/strong\u003e\u003cbr\u003ēn: mean occupation of a single-particle state; ε: state energy (J); μ: chemical potential (J); k_B: Boltzmann constant (J\/K); T: temperature (K)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/cecilia-payne-gaposchkin-sticker-almost-certainly-real-physics-sticker\"\u003eCecilia Payne-Gaposchkin\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eSaha's equation gives how much of each element is ionized in a hot gas at a given temperature and density. Cecilia Payne used it to correct for ionization in stellar spectra, and showed that stars are made mostly of hydrogen and helium.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003en\u003csub\u003ei+1\u003c\/sub\u003en\u003csub\u003ee\u003c\/sub\u003e\/n\u003csub\u003ei\u003c\/sub\u003e = (2\/Λ³)(g\u003csub\u003ei+1\u003c\/sub\u003e\/g\u003csub\u003ei\u003c\/sub\u003e) exp(−χ\u003csub\u003ei\u003c\/sub\u003e\/(k\u003csub\u003eB\u003c\/sub\u003eT))\u003c\/strong\u003e\u003cbr\u003enᵢ, nᵢ₊₁: number densities of ionisation stages i and i+1 (m⁻³); nₑ: electron number density (m⁻³); gᵢ: statistical weight of stage i; χᵢ: ionisation energy from stage i (J); k_B: Boltzmann constant (J\/K); T: temperature (K)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/fritz-zwicky-sticker-too-fast-to-be-lit-physics-sticker\"\u003eFritz Zwicky\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn a gravitationally bound cluster, the galaxies' speeds are set by the total mass. Zwicky used their velocity spread to estimate the Coma cluster's mass and found it far larger than the light suggested.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eσ² ∼ GM\/R\u003c\/strong\u003e\u003cbr\u003eσ: line-of-sight velocity dispersion of galaxies (km\/s); G: gravitational constant (pc M_☉⁻¹ (km\/s)²); M: gravitating mass within radius R (M_☉); R: characteristic cluster radius (pc); M_(vir): virial mass implied by \\sigma and R (M_☉)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/enrico-fermi-sticker-within-an-order-physics-sticker\"\u003eEnrico Fermi\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eThe Fermi–Dirac distribution gives the chance that a state of energy E is occupied by a fermion. Deep below the chemical potential states are full, far above it they are empty, and exactly at it the chance is one half.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ef(E) = 1\/(e\u003csup\u003e(E−μ)\/(kT)\u003c\/sup\u003e + 1)\u003c\/strong\u003e\u003cbr\u003ef(E): Fermi–Dirac occupation probability; E: single-particle energy (J); μ: chemical potential (J); k: Boltzmann constant (J\/K); T: temperature (K)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/jocelyn-bell-burnell-sticker-she-found-pulsars-they-gave-her-silence-physics-sticker\"\u003eJocelyn Bell Burnell\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA pulsar's very regular pulses mean it must be extremely dense to spin that fast without flying apart, which points to a neutron star. Radio pulses at lower frequencies arrive later after passing through interstellar electrons, and that delay measures how many electrons lie along the way.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eρ̄ ≳ 3π\/(GP²)\u003c\/strong\u003e\u003cbr\u003eρ̄: minimum mean density for a body spinning with period P not to fly apart (kg\/m³); G: gravitational constant (m³·kg⁻¹·s⁻²); P: rotation period (s); Δ t: arrival-time delay between frequencies f₁ \u0026lt; f₂ (s); DM: dispersion measure, ∫ nₑ dl (free electrons per area along the line of sight) (m⁻²); f₁, f₂: observing frequencies (Hz)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/subrahmanyan-chandrasekhar-sticker-no-seconds-physics-sticker\"\u003eSubrahmanyan Chandrasekhar\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA white dwarf is held up by electron degeneracy pressure. Chandrasekhar showed that once the electrons become relativistic, this pressure can only support a limited mass, the Chandrasekhar limit. Above that, collapse cannot be stopped.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eM\u003csub\u003eCh\u003c\/sub\u003e ≈ (5.73\/μ\u003csub\u003ee\u003c\/sub\u003e²) M\u003csub\u003e☉\u003c\/sub\u003e\u003c\/strong\u003e\u003cbr\u003eM_(Ch): Chandrasekhar mass limit (idealised: cold, non-rotating, ideal electron gas) (kg); ω₃⁰: constant from the Lane–Emden n = 3 solution (about 2.018); ħ: reduced Planck constant (J·s); c: speed of light (m\/s); G: gravitational constant (m³·kg⁻¹·s⁻²); μₑ: mean molecular weight per electron\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/stephen-hawking-sticker-black-holes-been-there-bent-that-physics-sticker\"\u003eStephen Hawking\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eHawking found that black holes radiate like warm bodies, with a temperature inversely proportional to their mass. Bekenstein's entropy is proportional to the area of the horizon, not its volume, which links gravity, thermodynamics and quantum theory.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eT\u003csub\u003eH\u003c\/sub\u003e = ħc³\/(8πGMk\u003csub\u003eB\u003c\/sub\u003e)\u003c\/strong\u003e\u003cbr\u003eT_H: Hawking temperature (K); M: black-hole mass (kg); S_(BH): black-hole entropy (J\/K); A: horizon area (m²); ħ, c, G, k_B: fundamental constants (J·s, m\/s, m³·kg⁻¹·s⁻², J\/K)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/vera-rubin-sticker-galaxy-rotation-she-saw-the-dark-side-physics-sticker\"\u003eVera Rubin\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIf most of a galaxy's mass were in its bright centre, stars far out would orbit more slowly. Vera Rubin found the speeds stay roughly constant far out, which means the enclosed mass keeps growing with radius: unseen matter.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ev(r)² = GM(r)\/r\u003c\/strong\u003e\u003cbr\u003ev(r): circular orbital speed (m\/s); M(r): enclosed mass (kg); r: radius (m); G: gravitational constant (m³·kg⁻¹·s⁻²)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/ole-romer-sticker-io-isnt-late-light-is-physics-sticker\"\u003eOle Rømer\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIo's eclipses behind Jupiter ran later when Earth was farther from Jupiter, because the light had farther to travel. Rømer used that delay to show that light has a finite speed, crossing the width of Earth's orbit in a matter of minutes.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eΔt = Δd\/c\u003c\/strong\u003e\u003cbr\u003eΔ t: extra light travel time (s); Δ d: extra distance light must travel (m); c: speed of light in vacuum (m\/s); AU: astronomical unit, about the mean Earth–Sun distance (exactly 149 597 870 700 m since 2012) (m)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/ludwig-boltzmann-sticker-there-are-more-ways-to-be-a-mess-physics-sticker\"\u003eLudwig Boltzmann\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eBoltzmann's formula says entropy measures the number of microscopic arrangements W that look the same from outside. Disordered states have overwhelmingly more arrangements than ordered ones, so systems drift toward disorder. Since 2019 the constant k_B has an exact defined value.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eS = k\u003csub\u003eB\u003c\/sub\u003e ln W\u003c\/strong\u003e\u003cbr\u003eS: entropy (J\/K); k_B: Boltzmann constant (J\/K); W: number of microstates consistent with the macrostate; ln: natural logarithm (written 'log' on the tomb)\u003c\/p\u003e","brand":"Printify","offers":[{"title":"White \/ Kiss-Cut \/ 5.83″ × 8.27″","offer_id":50964877115495,"sku":"21651741330824757363","price":19.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/30727314536498660_2048.jpg?v=1793674485"},{"product_id":"classical-physics-1-sticker-sheet","title":"Classical Physics Vol. 1 Pixel Sticker Sheet – 10 Physics Stickers","description":"\u003cp\u003e10 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: Classical Physics Vol. 1. Every character below has its own sticker page with the full physics panel.\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/adiabatix-adiabatic-invariant\"\u003eAdiabatix\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eAt the 1911 Solvay conference Lorentz asked what happens to a pendulum if you shorten its string very slowly. Einstein answered: the energy changes, but the ratio of energy to frequency stays the same. Paul Ehrenfest built this into a principle (1916): for slow changes, the action J stays constant. That is why quantum numbers stay put when conditions change gently.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eJ = ∮p dq\u003c\/strong\u003e\u003cbr\u003eJ: action over one cycle (J·s); p: momentum; q: position coordinate; ∮: integral over one full oscillation\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/teacupa-tea-leaf-paradox\"\u003eTeacupa\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn a stirred cup, the swirl is held in balance by a pressure that rises toward the wall (the surface is higher there). Near the bottom, friction slows the swirl, but the pressure difference set by the surface is still there. So near the floor the inward pressure push wins, and water flows toward the centre, rises up the middle and returns outward near the top. This secondary flow carries the sunken leaves to the centre, where they settle.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003e∂p\/∂r = ρ v\u003csub\u003eθ\u003c\/sub\u003e²\/r\u003c\/strong\u003e\u003cbr\u003ep: pressure (Pa); r: distance from the axis (m); ρ: density of the water (kg\/m³); v\u003csub\u003eθ\u003c\/sub\u003e: swirl (azimuthal) speed (m\/s)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/infrara-herschel-infrared\"\u003eInfrara\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn 1800 William Herschel placed thermometers in the colours of a prism spectrum and found the temperature rose from violet to red. Then he moved one past the red end, where no light could be seen, and it rose further still (“Experiments on the Refrangibility of the Invisible Rays of the Sun”, Phil. Trans. R. Soc. 90, 284). The Sun emits invisible rays beyond the red: infrared.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eλ\u003csub\u003eIR\u003c\/sub\u003e \u0026gt; λ\u003csub\u003ered\u003c\/sub\u003e ≈ 0.7 µm\u003c\/strong\u003e\u003cbr\u003eλ\u003csub\u003eIR\u003c\/sub\u003e: infrared wavelengths, about 0.7 µm to 1 mm; λ\u003csub\u003ered\u003c\/sub\u003e: long-wavelength end of the visible spectrum\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/braessa-spring-paradox\"\u003eBraessa\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eTwo identical springs hang in series, joined by a short string, with the weight at the bottom. Two longer “safety” strings hang slack, one from the support to the lower spring and one from the upper spring to the weight. When the short string is cut, the load is carried by two parallel paths, each a spring plus a long string, so each spring carries half the weight. If the strings are long enough the weight ends up higher than before.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eseries: k\u003csub\u003eeff\u003c\/sub\u003e = k\/2   →   parallel: k\u003csub\u003eeff\u003c\/sub\u003e = 2k\u003c\/strong\u003e\u003cbr\u003ek: stiffness of each spring (N\/m); k\u003csub\u003eeff\u003c\/sub\u003e: effective stiffness of the support\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/bucketta-newtons-bucket\"\u003eBucketta\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn the Scholium of the Principia (1687), Isaac Newton described a bucket of water hung on a twisted rope. When the bucket starts spinning, the water at first stays flat. As the water picks up the rotation, its surface becomes concave, and it stays curved even after the bucket is stopped, while the water keeps turning. Newton argued that the curve shows rotation relative to absolute space, not relative to the bucket.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ez = ω²r²\/(2g)\u003c\/strong\u003e\u003cbr\u003ez: height of the surface above its lowest point, on the axis (m); ω: angular speed of the water (rad\/s); r: distance from the axis (m); g: gravitational acceleration (m\/s²)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/chainfount-chain-fountain\"\u003eChainfount\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003ePile a long metal bead chain in a jar, pull one end over the rim and let it drop to the floor. The chain then pours itself out, and instead of sliding over the rim it leaps up into a self-supporting arch above the jar. It is also known as the Mould effect, after the video that made it famous. John Biggins and Mark Warner (Proc. R. Soc. A 470, 20130689, 2014) predicted and measured that the arch height is proportional to the drop height.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eh\u003csub\u003e2\u003c\/sub\u003e ∝ h\u003csub\u003e1\u003c\/sub\u003e\u003c\/strong\u003e\u003cbr\u003eh\u003csub\u003e1\u003c\/sub\u003e: height of the jar above the floor, the drop height (m); h\u003csub\u003e2\u003c\/sub\u003e: height of the fountain above the jar rim (m)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/coriolix-coriolis-effect-pixel-sticker\"\u003eCoriolix\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn a rotating frame, moving objects appear to be pushed sideways by the Coriolis term, −2Ω × v. Something moving in a straight line as seen from outside traces a curve as seen from the rotating platform.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003e\u003cb\u003ea\u003c\/b\u003e′ = −2\u003cb\u003eΩ\u003c\/b\u003e×\u003cb\u003ev\u003c\/b\u003e′ − \u003cb\u003eΩ\u003c\/b\u003e×(\u003cb\u003eΩ\u003c\/b\u003e×\u003cb\u003er\u003c\/b\u003e′)\u003c\/strong\u003e\u003cbr\u003ea', v', r': acceleration, velocity and position seen on the turning disc (m\/s², m\/s, m); Ω: angular velocity of the disc (anticlockwise) (rad\/s); v: speed of the ball, thrown from the centre (m\/s); t: time since the throw (s); R: radius of the disc (m); Δφ: angle by which the ball misses the flag at the rim (rad)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/dalembertia-zero-drag\"\u003eDalembertia\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn 1752 Jean le Rond d'Alembert showed that for steady, incompressible, inviscid and irrotational (potential) flow, a body moving at constant speed feels zero drag. The streamlines and the pressure are symmetric front to back, so the pressure pushes on the rear as hard as on the front. Yet every real body in air or water feels drag.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eφ = U (r + a³\/2r²) cos θ   ⇒   F\u003csub\u003edrag\u003c\/sub\u003e = 0\u003c\/strong\u003e\u003cbr\u003eφ: velocity potential of steady flow past a sphere (m²\/s); U: speed of the uniform flow far away (m\/s); a: sphere radius (m); r, θ: distance from the centre and angle from the flow direction; F\u003csub\u003edrag\u003c\/sub\u003e: net force along the flow (N)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/janssena-silo-pressure\"\u003eJanssena\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn a tall column of water, pressure at the bottom keeps rising with depth. In a silo full of grain it does not. The grains push sideways on the walls, and friction with the walls carries part of the weight. Below a few silo diameters, the pressure at the bottom levels off. H. A. Janssen measured this with grain in model silos (“Versuche über Getreidedruck in Silozellen”, Zeitschrift des Vereines Deutscher Ingenieure 39, 1045, 1895).\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ep(z) = p\u003csub\u003e∞\u003c\/sub\u003e(1 − e\u003csup\u003e−z\/λ\u003c\/sup\u003e), λ = R\/(2μK)\u003c\/strong\u003e\u003cbr\u003ep(z): vertical pressure at depth z (Pa); p\u003csub\u003e∞\u003c\/sub\u003e: saturation pressure, ρgλ (Pa); z: depth below the grain surface (m); λ: saturation length (m); R: silo radius (m); μ: wall friction coefficient; K: ratio of horizontal to vertical stress\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/libratrix-lagrange-points-pixel-sticker\"\u003eLibratrix\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eTwo bodies orbiting each other have five points where a small third body can keep pace with them. For the Sun and Earth, L2 lies a little beyond Earth, and the triangular points L4 and L5 are stable only when one mass greatly outweighs the other.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003er\u003csub\u003eL2\u003c\/sub\u003e ≈ R(M\u003csub\u003e2\u003c\/sub\u003e\/(3M\u003csub\u003e1\u003c\/sub\u003e))\u003csup\u003e1\/3\u003c\/sup\u003e\u003c\/strong\u003e\u003cbr\u003er_(L2): distance from the smaller body to L2 (second line: Sun–Earth L2, beyond Earth) (m); R: distance between the two bodies (m); M₁, M₂: masses of the larger and the smaller body (kg); M₁\/M₂: mass ratio; above about 24.96, L4 and L5 are stable (Routh)\u003c\/p\u003e","brand":"Printify","offers":[{"title":"White \/ Kiss-Cut \/ 5.83″ × 8.27″","offer_id":50964877443175,"sku":"33317769571062772207","price":19.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/3764902297399688554_2048.jpg?v=1793674501"},{"product_id":"classical-physics-2-sticker-sheet","title":"Classical Physics Vol. 2 Pixel Sticker Sheet – 10 Physics Stickers","description":"\u003cp\u003e10 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: Classical Physics Vol. 2. Every character below has its own sticker page with the full physics panel.\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/magnara-magnus-effect-pixel-sticker\"\u003eMagnara\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA spinning ball moving through air is pushed sideways, toward the side where its surface moves with the airflow. That Magnus force, proportional to spin crossed with velocity, makes spinning balls curve.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003e\u003cb\u003eF\u003c\/b\u003e\u003csub\u003eM\u003c\/sub\u003e = α \u003cb\u003eω\u003c\/b\u003e×\u003cb\u003ev\u003c\/b\u003e\u003c\/strong\u003e\u003cbr\u003eF_M: Magnus force on the spinning ball (N); α: empirical coefficient (depends on size, spin regime, fluid) (kg); ω: spin angular-velocity vector (rad\/s); v: centre-of-mass velocity through the air (m\/s); m: mass of the ball (kg); b: linear drag coefficient (kg\/s)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/meldea-melde-experiment\"\u003eMeldea\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eAround 1859 Franz Melde tied a string to the prong of a tuning fork and pulled it taut (Ann. Phys. 185, 193, 1860). At the right tensions the string broke into steady standing waves, with nodes that stood still and loops that blurred. Change the tension and the number of loops jumps.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ef = n√(T\/μ)\/(2L)\u003c\/strong\u003e\u003cbr\u003ef: frequency of the string vibration (Hz); n: number of loops; T: tension (N); μ: mass per unit length (kg\/m); L: length of the string (m)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/pendulux-foucault-pendulum-pixel-sticker\"\u003ePendulux\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA pendulum keeps swinging in a fixed plane while the Earth turns beneath it. The swing plane appears to rotate at the Earth's rotation rate times the sine of the latitude: once a sidereal day at the poles, and not at all on the equator.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eω\u003csub\u003eprec\u003c\/sub\u003e = Ω\u003csub\u003e⊕\u003c\/sub\u003e sin φ\u003c\/strong\u003e\u003cbr\u003eω_(prec): rate at which the swing plane turns relative to the floor (clockwise seen from above in the Northern Hemisphere) (°\/h); Ω_(oplus): Earth's rotation rate relative to the stars: one turn per sidereal day of 23.93 h (°\/h); φ: latitude (°)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/spinwell-angular-momentum-pixel-sticker\"\u003eSpinwell\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eAngular momentum is moment of inertia times angular speed, and without an outside torque it stays constant. Pull your arms in and your moment of inertia drops, so you spin faster.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eL = Iω\u003c\/strong\u003e\u003cbr\u003eL: angular momentum about the spin axis (kg m² \/ s); I: moment of inertia about that axis (kg m²); ω: angular speed (rad\/s); τ: external torque about the axis (N m)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/terminara-terminal-velocity-pixel-sticker\"\u003eTerminara\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA falling object speeds up until air drag, which grows with the square of speed, balances its weight. From then on it falls at a steady terminal velocity, approaching it along a tanh curve.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003em dv\/dt = mg − ½ρC\u003csub\u003ed\u003c\/sub\u003eAv²\u003c\/strong\u003e\u003cbr\u003em: mass of the skydiver (here 80 kg) (kg); v: falling speed (m\/s); t: time since leaving the plane, falling from rest (s); g: gravitational acceleration (m\/s²); ρ: air density (here 1.225 kg\/m^3, sea-level standard) (kg\/m³); C_d A: drag coefficient times frontal area (here 0.45 m^2, belly-down) (m²)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/barkhausa-barkhausen-noise\"\u003eBarkhausa\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eMagnetise a piece of iron slowly and smoothly and its magnetisation does not change smoothly. It grows in many tiny, sudden jumps as the walls between magnetic domains snag on defects and then snap free. Heinrich Barkhausen heard this in 1919 by winding a coil around the iron and amplifying the induced pulses into a telephone receiver: a crackle (“Zwei mit Hilfe der neuen Verstärker entdeckte Erscheinungen”, Physikalische Zeitschrift 20, 401, 1919). Each jump induces a small voltage pulse in the coil.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eV = −N dΦ\/dt\u003c\/strong\u003e\u003cbr\u003eV: voltage induced in the pick-up coil (V); N: number of turns of the coil; Φ: magnetic flux through the coil (Wb); t: time (s)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/displacera-displacement-current\"\u003eDisplacera\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eAmpère's law broke down at a charging capacitor: current flows in the wire but not across the gap. James Clerk Maxwell fixed it in 1861–65 by adding a term for the changing electric field in the gap, which makes a magnetic field just as a real current does (“A Dynamical Theory of the Electromagnetic Field”, Phil. Trans. 155, 459, 1865). With it, the equations predicted electromagnetic waves travelling at the speed of light.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eJ\u003csub\u003ed\u003c\/sub\u003e = ε\u003csub\u003e0\u003c\/sub\u003e ∂E\/∂t\u003c\/strong\u003e\u003cbr\u003eJ\u003csub\u003ed\u003c\/sub\u003e: displacement current density (A\/m²); ε\u003csub\u003e0\u003c\/sub\u003e: permittivity of free space (F\/m); ∂E\/∂t: rate of change of the electric field (V m⁻¹ s⁻¹)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/faradra-faraday-rotation\"\u003eFaradra\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn September 1845 Michael Faraday sent polarised light through a block of heavy lead-borate glass placed between the poles of a strong electromagnet, with the field along the beam. When the magnet was switched on, the plane of polarisation turned. It was the first experimental link between light and magnetism. He reported it in the Nineteenth Series of his Experimental Researches in Electricity (Phil. Trans. R. Soc. 136, 1, 1846). The rotation angle grows with the field and with the path length.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eβ = VBd\u003c\/strong\u003e\u003cbr\u003eβ: rotation angle of the plane of polarisation (rad); V: Verdet constant of the material (rad T⁻¹ m⁻¹); B: magnetic flux density along the beam (tesla, T); d: path length through the material (m)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/galvana-mirror-galvanometer\"\u003eGalvana\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eWilliam Thomson, later Lord Kelvin, developed the mirror galvanometer around the 1858 transatlantic telegraph cable, whose signals were too faint for ordinary needles. A tiny mirror carrying small magnets hangs inside a coil; a current twists it slightly, and a beam of light reflected from it sweeps across a scale. By the law of reflection, turning a mirror by an angle θ turns the reflected beam by twice that angle.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ebeam deflection = 2θ\u003c\/strong\u003e\u003cbr\u003eθ: rotation angle of the mirror (rad); 2θ: rotation angle of the reflected beam (rad)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/inertix-tolman-stewart\"\u003eInertix\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eSpin a coil of wire fast and stop it suddenly: the free charges inside keep going for a moment, like passengers in a braking bus, and a pulse of current flows. Richard Tolman and Thomas Stewart measured it in 1916 (Phys. Rev. 8, 97). From the charge that flowed they got the carriers' charge-to-mass ratio, close to that of the electron, showing that current in metals is carried by electrons.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eQ = (m\/e) v\u003csub\u003e0\u003c\/sub\u003e L \/ R\u003c\/strong\u003e\u003cbr\u003eQ: charge that flows after stopping (C); m\/e: mass-to-charge ratio of the carriers (kg\/C); v\u003csub\u003e0\u003c\/sub\u003e: rim speed of the wire before stopping (m\/s); L: length of the wire (m); R: resistance of the coil (Ω)\u003c\/p\u003e","brand":"Printify","offers":[{"title":"White \/ Kiss-Cut \/ 5.83″ × 8.27″","offer_id":50966844276839,"sku":"28301990384207583364","price":19.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/5152942607117677361_2048.jpg?v=1793676395"},{"product_id":"classical-physics-4-sticker-sheet","title":"Classical Physics Vol. 4 Pixel Sticker Sheet – 10 Physics Stickers","description":"\u003cp\u003e10 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: Classical Physics Vol. 4. Every character below has its own sticker page with the full physics panel.\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/saltara-hydraulic-jump-pixel-sticker\"\u003eSaltara\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA fast, shallow flow can suddenly jump to a slow, deep one. Momentum conservation across the jump gives the ratio of the two depths in terms of the incoming Froude number, which compares flow speed with the speed of shallow-water waves.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eFr = v\/√(gh)\u003c\/strong\u003e\u003cbr\u003eFr: Froude number: flow speed over the speed of shallow-water waves; v: depth-averaged flow speed (m\/s); g: gravitational acceleration (m\/s²); h: depth of the flow (m); h₁, h₂: depths just before and just after a planar jump in a straight channel (m); Fr₁: Froude number of the incoming flow\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/vortessa-karman-vortex-street-pixel-sticker\"\u003eVortessa\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eFlow past a cylinder sheds vortices from alternate sides, at a frequency set by the flow speed divided by the cylinder's diameter, times the Strouhal number. Von Kármán showed that the staggered vortex street is stable only for a particular spacing ratio.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ef = St·U\/D, St ≈ 0.2\u003c\/strong\u003e\u003cbr\u003ef: shedding frequency: full left-right cycles per second (Hz); St: Strouhal number, about 0.2 to 0.21 for Re from about 300 to 200,000 (lower near onset); U: speed of the oncoming flow (m\/s); D: diameter of the cylinder (m); Re: Reynolds number; ν: kinematic viscosity of the fluid (m²\/s)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/weissenberga-rod-climbing\"\u003eWeissenberga\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eStir water with a spinning rod and the surface dips into a vortex, because inertia flings the liquid outward. Spin a rod in a polymer solution, such as dough or a dilute polyacrylamide solution, and the liquid climbs up the rod instead. Stretched along the circular streamlines, the long polymer chains act like stretched rubber bands. The resulting tension around the streamlines squeezes the fluid inward, towards the rod, and the only way it can go is up. Karl Weissenberg described this behaviour in “A continuum theory of rheological phenomena” (Nature 159, 310, 1947).\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eN\u003csub\u003e1\u003c\/sub\u003e = τ\u003csub\u003e11\u003c\/sub\u003e − τ\u003csub\u003e22\u003c\/sub\u003e \u0026gt; 0\u003c\/strong\u003e\u003cbr\u003eN\u003csub\u003e1\u003c\/sub\u003e: first normal stress difference (Pa); τ\u003csub\u003e11\u003c\/sub\u003e: normal stress along the flow direction (Pa); τ\u003csub\u003e22\u003c\/sub\u003e: normal stress across the flow, in the direction of the velocity gradient (Pa)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/winetear-tears-of-wine\"\u003eWinetear\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eSwirl a glass of strong wine and a thin film climbs the wall above the liquid, gathers into a ridge and falls back as drops: tears, or legs. Alcohol evaporates from the film faster than water, so the film becomes more watery, and water has a higher surface tension than alcohol. The difference in surface tension pulls liquid up the glass from the bulk below, where there is more alcohol. James Thomson, Lord Kelvin's brother, described it in 1855 (“On certain curious motions observable at the surfaces of wine and other alcoholic liquors”, Phil. Mag. 10, 330). A flow driven like this is now called a Marangoni flow.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eτ = ∇γ\u003c\/strong\u003e\u003cbr\u003eτ: shear stress on the liquid surface (Pa); ∇γ: gradient of surface tension along the surface (N\/m²)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/langmuira-langmuir-probe\"\u003eLangmuira\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eStick a small electrode into a plasma and sweep its voltage. Irving Langmuir and Harold Mott-Smith worked out what the current tells you (“The Theory of Collectors in Gaseous Discharges”, Phys. Rev. 28, 727, 1926, after their 1924 papers). Below the plasma potential the probe repels most electrons, and only those with enough energy to climb the barrier arrive. For a Maxwellian electron gas that electron current rises exponentially with probe voltage, so the slope of ln I gives the electron temperature. Above the plasma potential the current saturates.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eI\u003csub\u003ee\u003c\/sub\u003e ∝ exp(e(V − V\u003csub\u003ep\u003c\/sub\u003e)\/(kT\u003csub\u003ee\u003c\/sub\u003e))\u003c\/strong\u003e\u003cbr\u003eI\u003csub\u003ee\u003c\/sub\u003e: electron current to the probe (A); V: probe voltage (V); V\u003csub\u003ep\u003c\/sub\u003e: plasma potential (V); k: Boltzmann constant (J\/K); T\u003csub\u003ee\u003c\/sub\u003e: electron temperature (K)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/candelor-candela-683\"\u003eCandelor\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eThe candela, the SI unit of luminous intensity, links watts of light to how bright that light looks to a standard human eye. It is defined by fixing the luminous efficacy of monochromatic radiation at 540 × 10¹² Hz (a green of about 555 nm) at exactly 683 lm\/W. For other wavelengths the watts are weighted by the CIE photopic sensitivity curve V(λ).\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eΦ\u003csub\u003ev\u003c\/sub\u003e = K\u003csub\u003em\u003c\/sub\u003e ∫ V(λ) Φ\u003csub\u003ee,λ\u003c\/sub\u003e dλ,   K\u003csub\u003ecd\u003c\/sub\u003e = 683 lm\/W\u003c\/strong\u003e\u003cbr\u003eΦ\u003csub\u003ev\u003c\/sub\u003e: luminous flux (lumen, lm); K\u003csub\u003ecd\u003c\/sub\u003e: luminous efficacy of 540 THz radiation, 683 lm\/W exactly (defining constant of the 2019 SI); K\u003csub\u003em\u003c\/sub\u003e: maximum photopic efficacy, ≈ 683 lm\/W; V(λ): CIE photopic luminous efficiency function, peak 1 near 555 nm; Φ\u003csub\u003ee,λ\u003c\/sub\u003e: spectral radiant flux (W\/nm); λ: wavelength (nm)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/halotra-22-degree-halo\"\u003eHalotra\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eThe real halo is made by millions of tiny hexagonal ice crystals in thin, high cirrus cloud, not by water droplets. Light entering one side face of a crystal and leaving by another passes through what is in effect a 60° prism. Every such prism has a minimum angle of deviation, and light piles up near it, so a bright ring appears at that angle from the Sun.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eD\u003csub\u003emin\u003c\/sub\u003e = 2 arcsin(n sin 30°) − 60°\u003c\/strong\u003e\u003cbr\u003eD\u003csub\u003emin\u003c\/sub\u003e: minimum deviation, the halo radius (degrees); n: refractive index of ice, about 1.31 for visible light; 30°: half the 60° prism angle between alternate faces\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/nicolette-nicol-prism\"\u003eNicolette\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn 1828 William Nicol of Edinburgh cut a calcite rhomb along a diagonal and cemented the halves back together with Canada balsam (Edinb. New Phil. J. 6, 83). Calcite splits light into an ordinary ray (n\u003csub\u003eo\u003c\/sub\u003e ≈ 1.658) and an extraordinary ray (n\u003csub\u003ee\u003c\/sub\u003e ≈ 1.486, along the slowest direction). Balsam sits between them at about 1.55. The ordinary ray, meeting the lower-index balsam at a steep angle, is totally internally reflected out of the side; the extraordinary ray passes. What leaves the end is linearly polarised.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003en\u003csub\u003eo\u003c\/sub\u003e \u0026gt; n\u003csub\u003ebalsam\u003c\/sub\u003e \u0026gt; n\u003csub\u003ee\u003c\/sub\u003e\u003c\/strong\u003e\u003cbr\u003en\u003csub\u003eo\u003c\/sub\u003e: ordinary refractive index of calcite, ≈ 1.658; n\u003csub\u003ebalsam\u003c\/sub\u003e: Canada balsam, ≈ 1.55; n\u003csub\u003ee\u003c\/sub\u003e: extraordinary index, ≈ 1.486 (sodium light)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/perota-fabry-perot\"\u003ePerota\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eTwo parallel, partly reflecting mirrors trap light that bounces back and forth. Only wavelengths that fit a whole number of half-waves into the gap build up and pass through; the rest cancel out. The transmission is a comb of sharp peaks, spaced by the free spectral range.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eFSR = c\/(2nL)\u003c\/strong\u003e\u003cbr\u003eFSR: free spectral range, the frequency spacing between peaks (Hz); c: speed of light (m\/s); n: refractive index between the mirrors; L: mirror spacing (m)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/tweezra-optical-tweezers\"\u003eTweezra\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eFocus a laser beam tightly through a microscope lens and a small transparent bead is pulled towards the brightest point and held there. Light bends as it passes through the bead, and the change in its momentum pushes back on the bead. For a bead that bends light more than its surroundings, the net gradient force points up the intensity gradient, towards the focus. Arthur Ashkin accelerated and trapped particles with radiation pressure in 1970 (Phys. Rev. Lett. 24, 156), and with colleagues demonstrated the single-beam gradient trap, the optical tweezers, in 1986 (Opt. Lett. 11, 288).\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eF\u003csub\u003egrad\u003c\/sub\u003e ∝ ∇I\u003c\/strong\u003e\u003cbr\u003eF\u003csub\u003egrad\u003c\/sub\u003e: gradient force on the bead (N); ∇I: gradient of the light intensity (W\/m³)\u003c\/p\u003e","brand":"Printify","offers":[{"title":"White \/ Kiss-Cut \/ 5.83″ × 8.27″","offer_id":50966852501607,"sku":"14474346663117771923","price":19.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/18390166916276149494_2048.jpg?v=1793676402"},{"product_id":"quantum-1-sticker-sheet","title":"Quantum Physics Vol. 1 Pixel Sticker Sheet – 12 Physics Stickers","description":"\u003cp\u003e12 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: Quantum Physics Vol. 1. Every character below has its own sticker page with the full physics panel.\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/berryphase-geometric-phase\"\u003eBerryphase\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eCarry a quantum system slowly round a closed loop of its settings and it returns to its starting state, apart from a phase. Michael Berry showed in 1984 that part of this phase depends only on the geometry of the loop, not on how long it takes (Proc. R. Soc. A 392, 45). For a spin-½ in the state aligned with a magnetic field (m = +½), with the field direction swept slowly round a closed loop, the geometric phase is minus half the solid angle that loop encloses: γ = −mΩ = −Ω\/2. The anti-aligned state (m = −½) picks up +Ω\/2.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eγ = −Ω\/2\u003c\/strong\u003e\u003cbr\u003eγ: geometric (Berry) phase of the aligned state (rad); Ω: solid angle enclosed by the loop traced by the field direction, seen from the origin (sr); m: spin projection along the field in units of ħ (dimensionless; +½ aligned, −½ anti-aligned)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/dehaasa-einstein-de-haas\"\u003eDehaasa\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eMagnetise an iron rod and it starts to turn. In 1915 Albert Einstein and Wander de Haas hung an iron cylinder on a fine fibre inside a coil (Verh. Dtsch. Phys. Ges. 17, 152). Reversing the magnetisation flips the tiny magnets inside the iron, and their angular momentum flips with them. Angular momentum is conserved, so the whole rod twists the other way. The ratio of magnetic moment to angular momentum sets the g-factor.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eL = 2mμ\/(ge)\u003c\/strong\u003e\u003cbr\u003eL: angular momentum of the carriers (J·s); μ: magnetic moment (A·m²); m: electron mass (kg); e: elementary charge (C); g: gyromagnetic g-factor\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/homestaka-solar-neutrino-problem\"\u003eHomestaka\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eRaymond Davis Jr.'s experiment in the Homestake gold mine, South Dakota, 1478 m underground, used a tank of about 615 tonnes of perchloroethylene (tetrachloroethylene, C₂Cl₄). A solar electron neutrino can turn a chlorine-37 nucleus into argon-37 (threshold 0.814 MeV), and the few argon atoms were extracted and counted. From 1970 to 1994 it measured about a third of the rate predicted by John Bahcall's standard solar model.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003emeasured 2.56 ± 0.16 ± 0.16 SNU   vs   predicted 7.6\u003csup\u003e+1.3\u003c\/sup\u003e\u003csub\u003e−1.1\u003c\/sub\u003e SNU\u003c\/strong\u003e\u003cbr\u003eSNU: solar neutrino unit: 10⁻³⁶ captures per target atom per second; 2.56: final Homestake result (Cleveland et al., ApJ 496, 505, 1998), statistical and systematic errors; 7.6: chlorine prediction of the BP2000 standard solar model (Bahcall, Pinsonneault and Basu, 2001)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/helicita-goldhaber-neutrino-helicity\"\u003eHelicita\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn 1958 Maurice Goldhaber, Lee Grodzins and Andrew Sunyar measured the handedness of the neutrino (“Helicity of Neutrinos”, Phys. Rev. 109, 1015). Europium-152m captures an electron and emits a neutrino; the daughter nucleus recoils and then emits a gamma ray. Selecting gammas emitted against the recoil and measuring their circular polarisation with magnetised iron tied the neutrino's spin to its motion. Neutrinos turned out left-handed: spin opposite to motion.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eh = −1\u003c\/strong\u003e\u003cbr\u003eh: helicity, the spin component along the direction of motion, in units of the maximum\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/expulsieur-meissner-effect-pixel-sticker\"\u003eExpulsieur\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA superconductor expels magnetic field from its interior, which a merely perfect conductor would not do. The field only penetrates a thin layer, dying away exponentially over the London penetration depth.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eB\u003csub\u003einside\u003c\/sub\u003e = 0\u003c\/strong\u003e\u003cbr\u003eB_(inside): magnetic field in the bulk of the superconductor (applied field below the critical field) (T); B(x): field at depth x below the surface (T); B₀: field at the surface (T); x: depth below the surface (m); λ_L: London penetration depth (m); m: electron mass (kg)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/landaura-conductance-quantum\"\u003eLandaura\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn a narrow ballistic conductor, such as a quantum point contact in a two-dimensional electron gas, electrons travel in a few transverse modes, like lanes on a road. Each fully open lane carries one conductance quantum, counting both spin states. Widening the constriction opens one lane at a time, so the conductance rises in a staircase. This was first seen in 1988 in GaAs point contacts (van Wees et al.; Wharam et al.).\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eG = (2e²\/h) Σ\u003csub\u003en\u003c\/sub\u003e T\u003csub\u003en\u003c\/sub\u003e,   G\u003csub\u003e0\u003c\/sub\u003e = 2e²\/h ≈ 7.748 091 729 × 10⁻⁵ S\u003c\/strong\u003e\u003cbr\u003eG: conductance (siemens, S); e: elementary charge, exact in the 2019 SI; h: Planck constant, exact in the 2019 SI; T\u003csub\u003en\u003c\/sub\u003e: transmission probability of mode n (0 to 1); 2: spin degeneracy, two spin states per lane; G\u003csub\u003e0\u003c\/sub\u003e: conductance quantum, exact (CODATA 2022)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/squidra-dc-squid\"\u003eSquidra\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA dc SQUID is a superconducting loop broken by two Josephson junctions. The current waves through the two junctions interfere, so the voltage across the device swings periodically as the magnetic flux through the loop changes. One full period is one flux quantum. R. C. Jaklevic, J. Lambe, A. H. Silver and J. E. Mercereau first showed this interference (“Quantum Interference Effects in Josephson Tunneling”, Phys. Rev. Lett. 12, 159, 1964).\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eΦ\u003csub\u003e0\u003c\/sub\u003e = h\/2e ≈ 2.067833848 × 10⁻¹⁵ Wb\u003c\/strong\u003e\u003cbr\u003eΦ\u003csub\u003e0\u003c\/sub\u003e: magnetic flux quantum (weber, Wb); h: Planck constant; 2e: charge of a Cooper pair\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/stairohm-quantum-hall-effect-pixel-sticker\"\u003eStairohm\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn a thin conductor at low temperature and strong magnetic field, the Hall resistance locks onto exact plateaus at h\/e² divided by a whole number. These plateaus are so precise and reproducible that they now serve as the standard of electrical resistance.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eR\u003csub\u003exy\u003c\/sub\u003e = h\/(νe²) = R\u003csub\u003eK\u003c\/sub\u003e\/ν\u003c\/strong\u003e\u003cbr\u003eR_(xy): Hall resistance: transverse voltage divided by current (Ω); h: Planck constant (exact in the SI since 2019) (J·s); e: elementary charge (exact in the SI since 2019) (C); ν: filling factor: an integer on each plateau; R_K: von Klitzing constant, h\/e² (Ω); Rₓₓ: longitudinal resistance along the sample (Ω)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/tesora-transition-edge-sensor\"\u003eTesora\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA transition-edge sensor is a thin superconducting film held right at the edge of its superconducting transition, where its resistance rises from zero to normal within a tiny temperature range. Absorb a photon or particle and the film warms very slightly, its resistance jumps, and the current through it changes measurably. K. D. Irwin showed how to hold it stable on the edge with voltage bias and electrothermal feedback (Appl. Phys. Lett. 66, 1998, 1995). The steepness is measured by a logarithmic sensitivity.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eα = d ln R \/ d ln T\u003c\/strong\u003e\u003cbr\u003eα: logarithmic temperature sensitivity, dimensionless; R: film resistance (Ω); T: film temperature (K)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/echoa-spin-echo\"\u003eEchoa\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn a magnetic field, nuclear spins precess at slightly different rates because the field is not perfectly uniform, so after a pulse knocks them over they fan out and the signal fades. Erwin Hahn found in 1950 (“Spin Echoes”, Phys. Rev. 80, 580) that a second pulse makes them run back together, and a signal reappears: the spin echo. In the version by H. Y. Carr and E. M. Purcell (Phys. Rev. 94, 630, 1954), a π\/2 pulse is followed after a time τ by a π pulse that flips the fan over. The fast spins are now behind the slow ones, catch up, and all line up again at 2τ.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003et\u003csub\u003eecho\u003c\/sub\u003e = 2τ\u003c\/strong\u003e\u003cbr\u003eτ: time between the first and second pulse (s); t\u003csub\u003eecho\u003c\/sub\u003e: time of the echo after the first pulse (s)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/purcella-purcell-effect\"\u003ePurcella\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eAn excited atom's spontaneous emission rate is not fixed: it depends on the electromagnetic modes around it. Put the atom in a small, high-quality resonant cavity tuned to its transition and it can emit much faster than in free space. E. M. Purcell pointed this out for nuclear magnetic moments at radio frequencies in a short abstract (“Spontaneous Emission Probabilities at Radio Frequencies”, Phys. Rev. 69, 681, 1946). The enhancement is the Purcell factor.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eF = 3Q(λ\/n)³ \/ (4π²V)\u003c\/strong\u003e\u003cbr\u003eF: Purcell factor, emission rate in the cavity over the free-space rate; Q: quality factor of the cavity; λ: free-space wavelength of the transition (m); n: refractive index of the cavity medium; V: mode volume of the cavity (m³)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/barnetta-barnett-effect\"\u003eBarnetta\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eSpin an unmagnetised iron rod and it becomes magnetised. Samuel Barnett showed this in 1915 (Phys. Rev. 6, 239): rotation acts on the electrons' angular momentum like a magnetic field, so their spins tend to line up. It is the reverse of the Einstein–de Haas effect, and gave g ≈ 2 for iron.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eM ∝ Ω\u003c\/strong\u003e\u003cbr\u003eM: magnetisation (A\/m); Ω: angular speed of rotation (rad\/s)\u003c\/p\u003e","brand":"Printify","offers":[{"title":"White \/ Kiss-Cut \/ 5.83″ × 8.27″","offer_id":50966860726375,"sku":"18755123285379818490","price":19.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/13385295286637407814_2048.jpg?v=1793676408"},{"product_id":"quantum-2-sticker-sheet","title":"Quantum Physics Vol. 2 Pixel Sticker Sheet – 12 Physics Stickers","description":"\u003cp\u003e12 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: Quantum Physics Vol. 2. Every character below has its own sticker page with the full physics panel.\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/corpuscle-jj-thomson\"\u003eCorpuscle\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn 1897 J. J. Thomson crossed electric and magnetic fields over a cathode-ray beam (“Cathode Rays”, Phil. Mag. 44, 293). He tuned them so the beam ran straight, which gives its speed, v = E\/B; then the deflection from one field alone gives the mass-to-charge ratio. The result was about a thousand times smaller than for the hydrogen ion, whatever the gas or electrode metal. He called the particles corpuscles: we call them electrons.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ev = E\/B\u003c\/strong\u003e\u003cbr\u003ev: beam speed (m\/s); E: electric field (V\/m); B: magnetic field (T); m\/e: mass-to-charge ratio of the particle (kg\/C)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/dynodra-photomultiplier\"\u003eDynodra\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA photomultiplier tube turns one photon into a measurable pulse. The photon frees an electron from the photocathode. It is then accelerated onto the first dynode, where it knocks out several secondary electrons. Each of those is accelerated onto the next dynode, and so on. With δ secondaries per electron at each of N stages, the gain multiplies.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eG = δ\u003csup\u003eN\u003c\/sup\u003e, so ln G = N ln δ\u003c\/strong\u003e\u003cbr\u003eG: overall electron gain; δ: secondary-emission ratio per dynode; N: number of dynodes\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/magnetonix-bohr-magneton\"\u003eMagnetonix\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eAn electron with one unit of orbital angular momentum ħ has a magnetic moment of one Bohr magneton. This is the natural unit for atomic and electron magnetism.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eμ\u003csub\u003eB\u003c\/sub\u003e = eħ\/(2m\u003csub\u003ee\u003c\/sub\u003e) = 9.274 010 0657(29) × 10⁻²⁴ J\/T\u003c\/strong\u003e\u003cbr\u003eμ\u003csub\u003eB\u003c\/sub\u003e: Bohr magneton (J\/T); e: elementary charge (exact); ħ: reduced Planck constant (exact); m\u003csub\u003ee\u003c\/sub\u003e: electron mass (measured)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/pumpette-single-electron-pump\"\u003ePumpette\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA single-electron pump is a nanoscale island (a quantum dot or metal island) cut off by tunnel barriers. Its charging energy is large enough that electrons can only cross one at a time (Coulomb blockade). Driving the gates at frequency f moves one electron per cycle, so the current is set by counting.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eI = ef\u003c\/strong\u003e\u003cbr\u003eI: current (A); e: elementary charge, 1.602 176 634 × 10⁻¹⁹ C exactly; f: pumping frequency (Hz)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/quantix-flux-quantum\"\u003eQuantix\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn a superconducting ring the electrons pair up into Cooper pairs and share a single macroscopic wavefunction. The phase must come back to itself around the loop, so the fluxoid (the flux, plus a small supercurrent term) is quantised in whole multiples of Φ\u003csub\u003e0\u003c\/sub\u003e. In a type-II superconductor the field enters as Abrikosov vortices, each carrying one Φ\u003csub\u003e0\u003c\/sub\u003e, arranged in a triangular lattice.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eΦ\u003csub\u003e0\u003c\/sub\u003e = h\/2e ≈ 2.067 833 848 × 10⁻¹⁵ Wb\u003c\/strong\u003e\u003cbr\u003eΦ\u003csub\u003e0\u003c\/sub\u003e: magnetic flux quantum (Wb); h: Planck constant (exact); 2e: charge of a Cooper pair\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/splitterra-cockcroft-walton\"\u003eSplitterra\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn April 1932 at the Cavendish Laboratory, John Cockcroft and Ernest Walton fired protons from a high-voltage accelerator at lithium. Scintillation screens flashed with pairs of alpha particles flying apart (Proc. R. Soc. A 137, 229, 1932). It was the first nucleus split by artificially accelerated particles, and the energy released matched E = mc².\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ep + ⁷Li → 2α + 17.3 MeV\u003c\/strong\u003e\u003cbr\u003ep: proton; ⁷Li: lithium-7 nucleus; α: alpha particle, a helium-4 nucleus\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/feeblea-taylor-feeble-light\"\u003eFeeblea\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn 1909 a young G. I. Taylor photographed the shadow of a needle with light so dim that, in modern terms, rarely more than one photon was in the box at a time (Proc. Camb. Phil. Soc. 15, 114). His longest exposure ran about 2000 hours, roughly three months. The diffraction fringes were just as sharp as with bright light.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003et\u003csub\u003eexposure\u003c\/sub\u003e ≈ 2000 h\u003c\/strong\u003e\u003cbr\u003et\u003csub\u003eexposure\u003c\/sub\u003e: exposure time of the faintest photograph\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/cptera-cpt-theorem\"\u003eCptera\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eFlip all charges (C), mirror space (P) and run time backwards (T), all at once, and any local, Lorentz-invariant quantum field theory looks exactly the same. This CPT theorem was worked out by Julian Schwinger (1951), Gerhart Lüders (1954) and Wolfgang Pauli (1955). It forces a particle and its antiparticle to have exactly equal mass and lifetime.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eCPT: C × P × T is an exact symmetry\u003c\/strong\u003e\u003cbr\u003eC: charge conjugation, particle ↔ antiparticle; P: parity, mirror reflection of space; T: time reversal\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/dicka-superradiance\"\u003eDicka\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eN excited atoms that radiate independently give N times the light of one. But if they are packed close together, within about a wavelength, or otherwise coupled to the same light field, they can lock into phase and radiate as one giant dipole. Then the peak intensity grows as N², and the light comes out in a short, bright burst. Robert Dicke predicted this in “Coherence in Spontaneous Radiation Processes” (Phys. Rev. 93, 99, 1954).\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eI\u003csub\u003epeak\u003c\/sub\u003e ∝ N²\u003c\/strong\u003e\u003cbr\u003eI\u003csub\u003epeak\u003c\/sub\u003e: peak emitted intensity (W); N: number of emitters radiating in phase\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/hanleva-hanle-effect\"\u003eHanleva\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eShine polarised light at the mercury resonance line (253.7 nm) into mercury vapour and the atoms re-emit it as polarised resonance fluorescence. Add a weak magnetic field and the polarisation drops. Wilhelm Hanle reported this in “Über magnetische Beeinflussung der Polarisation der Resonanzfluoreszenz” (Z. Phys. 30, 93, 1924). The excited atoms behave like little oscillating dipoles that precess around the field at the Larmor frequency. If they turn through a large angle before they emit, the emitted light comes out with a scrambled direction of polarisation.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eΔB ≈ ħ \/ (g μ\u003csub\u003eB\u003c\/sub\u003e τ)\u003c\/strong\u003e\u003cbr\u003eΔB: field scale over which the polarisation falls (tesla, T); ħ: reduced Planck constant; g: Landé g-factor of the excited state; μ\u003csub\u003eB\u003c\/sub\u003e: Bohr magneton; τ: lifetime of the excited state (s)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/starka-stark-effect\"\u003eStarka\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003ePut atoms in a strong electric field and their spectral lines split and shift. Johannes Stark observed this in hydrogen in 1913 and published it in full in Annalen der Physik (“Beobachtungen über den Effekt des elektrischen Feldes auf Spektrallinien”, Ann. Phys. 348, 965, 1914). Antonino Lo Surdo saw it independently the same year. Hydrogen is special: states with the same n but different shapes have the same energy, so the field mixes them and the shift is proportional to the field itself, giving a symmetric pattern of components.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eΔE = (3\/2) n (n\u003csub\u003e1\u003c\/sub\u003e − n\u003csub\u003e2\u003c\/sub\u003e) e a\u003csub\u003e0\u003c\/sub\u003e F\u003c\/strong\u003e\u003cbr\u003eΔE: energy shift of the level (J); n: principal quantum number; n\u003csub\u003e1\u003c\/sub\u003e, n\u003csub\u003e2\u003c\/sub\u003e: parabolic quantum numbers; e: elementary charge (C); a\u003csub\u003e0\u003c\/sub\u003e: Bohr radius (m); F: electric field strength (V\/m)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/augermeit-auger-meitner\"\u003eAugermeit\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eRemove an electron from an atom's innermost K shell and an electron from the L shell drops into the hole. The energy released can leave as an X-ray photon. Or it can pass directly to another electron, which is ejected from the atom with a characteristic energy: an Auger electron. No photon is emitted and reabsorbed. For a KLL transition the Auger electron's energy is roughly the K binding energy minus two L binding energies, which makes it a fingerprint of the element. That is the basis of Auger electron spectroscopy.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eE\u003csub\u003eKLL\u003c\/sub\u003e ≈ E\u003csub\u003eK\u003c\/sub\u003e − E\u003csub\u003eL\u003c\/sub\u003e − E\u003csub\u003eL\u003c\/sub\u003e\u003c\/strong\u003e\u003cbr\u003eE\u003csub\u003eKLL\u003c\/sub\u003e: kinetic energy of the ejected Auger electron (eV); E\u003csub\u003eK\u003c\/sub\u003e: binding energy of the K-shell electron (eV); E\u003csub\u003eL\u003c\/sub\u003e: binding energies of the two L-shell electrons involved (eV)\u003c\/p\u003e","brand":"Printify","offers":[{"title":"White \/ Kiss-Cut \/ 5.83″ × 8.27″","offer_id":50966868951143,"sku":"47631973255872779782","price":19.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/858166197133163503_2048.jpg?v=1793676414"},{"product_id":"quantum-3-sticker-sheet","title":"Quantum Physics Vol. 3 Pixel Sticker Sheet – 11 Physics Stickers","description":"\u003cp\u003e11 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: Quantum Physics Vol. 3. Every character below has its own sticker page with the full physics panel.\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/condensa-bose-einstein-condensate-pixel-sticker\"\u003eCondensa\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eBelow a critical temperature, a gas of bosons starts collecting in its lowest quantum state. The fraction in that state grows as the temperature drops, following 1 − (T\/Tc)^(3\/2), and the critical temperature depends on the gas's density and particle mass.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eN\u003csub\u003e0\u003c\/sub\u003e\/N = 1 − (T\/T\u003csub\u003ec\u003c\/sub\u003e)\u003csup\u003e3\/2\u003c\/sup\u003e (T ≤ T\u003csub\u003ec\u003c\/sub\u003e)\u003c\/strong\u003e\u003cbr\u003eN₀: number of bosons in the ground state; N: total number of bosons; T: temperature (K); T_c: critical temperature for condensation (K); n: number density (m⁻³); m: boson mass (kg)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/fogplate-becquerel\"\u003eFogplate\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eHenri Becquerel suspected that uranium salts, made to glow by sunlight, might give off X-rays. Paris stayed cloudy in late February 1896, so he put plates wrapped in black paper, with uranium salt on top, in a drawer. On 1 March he developed them anyway and found strong images: the uranium emitted penetrating rays with no sunlight at all (Comptes rendus 122, 501, 1896). It was radioactivity.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003euranium → rays, no light needed\u003c\/strong\u003e\u003cbr\u003euranium salt: potassium uranyl sulfate in his first tests; plate: photographic plate wrapped in black paper\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/bohra-bohr-radius\"\u003eBohra\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn hydrogen's ground state (1s), the radial probability density r²|ψ|² peaks at exactly a\u003csub\u003e0\u003c\/sub\u003e (with the proton treated as infinitely heavy). This is the atomic unit of length.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ea\u003csub\u003e0\u003c\/sub\u003e = 4πε\u003csub\u003e0\u003c\/sub\u003eħ²\/(m\u003csub\u003ee\u003c\/sub\u003ee²) = 5.291 772 105 44(82) × 10⁻¹¹ m ≈ 52.9 pm\u003c\/strong\u003e\u003cbr\u003ea\u003csub\u003e0\u003c\/sub\u003e: Bohr radius (m); ε\u003csub\u003e0\u003c\/sub\u003e: vacuum permittivity (measured since 2019); ħ: reduced Planck constant (exact); m\u003csub\u003ee\u003c\/sub\u003e: electron mass (measured); e: elementary charge (exact)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/massratio-proton-electron\"\u003eMassratio\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eThe proton is about 1836 times heavier than the electron. This dimensionless ratio sets why atoms have a tiny heavy nucleus with a light electron cloud, why nuclei barely move in chemistry (the Born–Oppenheimer approximation), and the ratio of vibrational to electronic energies in molecules. It is measured with Penning-trap mass comparisons and precision spectroscopy of molecular hydrogen ions such as HD⁺.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eμ = m\u003csub\u003ep\u003c\/sub\u003e\/m\u003csub\u003ee\u003c\/sub\u003e = 1836.152 673 426(32)\u003c\/strong\u003e\u003cbr\u003eμ: proton-to-electron mass ratio (dimensionless); m\u003csub\u003ep\u003c\/sub\u003e: proton rest mass, measured; m\u003csub\u003ee\u003c\/sub\u003e: electron rest mass, measured; (32): standard uncertainty in the last two digits, about 1.7 × 10⁻¹¹ relative\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/vevette-higgs-field-pixel-sticker\"\u003eVevette\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eThe Higgs field's potential has its lowest point away from zero, so the field takes a nonzero value v of about 246 GeV everywhere. Particles that interact with it gain masses proportional to that value and their coupling strength.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eV(φ) = −μ²|φ|² + λ|φ|⁴\u003c\/strong\u003e\u003cbr\u003eV: Higgs potential (energy density; natural units) (GeV⁴); φ: Higgs field; on the art the horizontal axis is the field's size (natural units) (GeV); μ²: parameter of the potential, positive (GeV²); λ: self-coupling of the field, positive; ⟨φ⟩: vacuum expectation value: where the field settles, in the valley (GeV); v: electroweak scale, about 246.22 GeV (GeV)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/weinberga-weak-mixing-angle\"\u003eWeinberga\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn the electroweak theory (Glashow, Weinberg, Salam), the neutral gauge fields B and W³ mix. Rotating them by the weak mixing angle θ\u003csub\u003eW\u003c\/sub\u003e gives the massless photon and the massive Z boson. The angle also links the couplings: e = g sin θ\u003csub\u003eW\u003c\/sub\u003e.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003esin²θ\u003csub\u003eW\u003c\/sub\u003e ≈ 0.231 (MS-bar at M\u003csub\u003eZ\u003c\/sub\u003e)\u003c\/strong\u003e\u003cbr\u003eθ\u003csub\u003eW\u003c\/sub\u003e: weak mixing (Weinberg) angle, about 28.7°; MS-bar: modified minimal-subtraction renormalisation scheme; M\u003csub\u003eZ\u003c\/sub\u003e: Z boson mass scale, about 91.19 GeV\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/kapitzadirac-electron-diffraction-light\"\u003eKapitzadirac\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eTwo laser beams travelling in opposite directions form a standing wave, with bright and dark bands half a wavelength apart. Pyotr Kapitza and Paul Dirac predicted in 1933 (“The reflection of electrons from standing light waves”, Proc. Camb. Phil. Soc. 29, 297) that electrons crossing such a wave would be diffracted, as if by a grating. The electrons are pushed away from the bright bands by the ponderomotive force, and the regular pattern of bands acts as a periodic grating.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ed = λ \/ 2\u003c\/strong\u003e\u003cbr\u003ed: period of the light grating (m); λ: wavelength of the laser light (m)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/casimira-casimir-effect-pixel-sticker\"\u003eCasimira\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003ePut two uncharged, perfectly conducting parallel plates a small distance a apart in vacuum and they attract. Between the plates only electromagnetic modes that fit the boundary conditions are allowed; outside, every mode is. Summing the zero-point energies ½ħω of the modes, and subtracting the free-space value, leaves a finite energy that falls as the plates approach. Hendrik Casimir derived it in 1948.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eP = π²ħc \/ (240 a⁴)\u003c\/strong\u003e\u003cbr\u003eP: attractive pressure, force per unit plate area (Pa); ħ: reduced Planck constant, h\/2π = 1.054 571 817 × 10⁻³⁴ J s; c: speed of light, 299 792 458 m s⁻¹; a: gap between the plates (m); π²\/240: numerical factor from summing the allowed modes\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/delaychoice-wheeler-delayed-choice\"\u003eDelaychoice\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn a Mach–Zehnder interferometer, a single photon meets a first beam splitter and then two paths. If a second beam splitter recombines the paths, the photon shows interference: which detector clicks depends on the phase difference. If the second splitter is removed, each detector clicks half the time, as if the photon took one path. John Wheeler proposed in 1978 making this choice after the photon has passed the first splitter. Jacques et al. (Science 315, 966, 2007) did it with single photons and a random choice that was space-like separated from the photon's entry.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ewith splitter: P\u003csub\u003e1\u003c\/sub\u003e = cos²(φ\/2), P\u003csub\u003e2\u003c\/sub\u003e = sin²(φ\/2);   without: P\u003csub\u003e1\u003c\/sub\u003e = P\u003csub\u003e2\u003c\/sub\u003e = ½\u003c\/strong\u003e\u003cbr\u003eP\u003csub\u003e1\u003c\/sub\u003e, P\u003csub\u003e2\u003c\/sub\u003e: probabilities that detector 1 or 2 clicks; φ: phase difference between the two arms (rad)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/bifurcor-stern-gerlach-experiment-pixel-sticker\"\u003eBifurcor\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA magnetic field that changes with height pushes on an atom's magnetic moment. Because an electron's spin can only point up or down, silver atoms in the Stern–Gerlach magnet split into exactly two beams instead of smearing out.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eF\u003csub\u003ez\u003c\/sub\u003e = μ\u003csub\u003ez\u003c\/sub\u003e ∂B\u003csub\u003ez\u003c\/sub\u003e\/∂z\u003c\/strong\u003e\u003cbr\u003eF_z: force on the atom along the field gradient (N); μ_z: z-component of the atom's magnetic moment (for silver, set by its one outer electron) (J\/T); B_z: z-component of the magnetic field (T); z: position along the gradient, up the page (m); g_s: electron spin g-factor, about 2.0023; μ_B: Bohr magneton (J\/T)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/revolvix-spinor-rotation-pixel-sticker\"\u003eRevolvix\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eRotating a spin-½ particle by one full turn multiplies its wavefunction by −1; only two full turns bring it back to its original state. Neutron interferometers measured this sign change directly in 1975.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eR(2π)ψ = −ψ\u003c\/strong\u003e\u003cbr\u003eR(θ): operator that rotates a spin-1\/2 state by angle θ about the axis n; ψ: two-component spinor state of a spin-1\/2 particle; n̂: unit vector along the rotation axis; σ: the three Pauli matrices (σx, σy, σz); 2π, 4π: one full turn (360°) and two full turns (720°) (rad)\u003c\/p\u003e","brand":"Printify","offers":[{"title":"White \/ Kiss-Cut \/ 5.83″ × 8.27″","offer_id":50966877175911,"sku":"25657300833391065990","price":19.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/1183536305024002566_2048.jpg?v=1793676420"},{"product_id":"forces-fields-1-sticker-sheet","title":"Forces \u0026 Fields Vol. 1 Pixel Sticker Sheet – 7 Physics Stickers","description":"\u003cp\u003e7 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: Forces \u0026amp; Fields Vol. 1. Every character below has its own sticker page with the full physics panel.\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/azurwake-cherenkov-radiation-pixel-sticker\"\u003eAzurwake\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA charged particle moving through a medium faster than light travels in that medium emits light in a cone, like a sonic boom. The cone angle depends on the particle's speed and the refractive index, and Frank and Tamm's formula gives how many photons come out per unit length.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ecos θ = 1\/(nβ), β \u0026gt; 1\/n\u003c\/strong\u003e\u003cbr\u003eθ: angle between the emitted light and the particle's path (the wavefront cone has half-angle 90° − θ) (rad); n: refractive index of the medium, about 1.33 for water; β: particle speed as a fraction of c; N: number of photons emitted; x: distance travelled by the particle (m); λ: wavelength of the emitted light (m)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/bremsella-bremsstrahlung-pixel-sticker\"\u003eBremsella\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eAny accelerating charge radiates. Larmor's formula says the power grows with the square of the charge and the square of its acceleration, which is why electrons that slow down sharply in matter give off braking radiation.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eP = q²a²\/(6πε\u003csub\u003e0\u003c\/sub\u003ec³)\u003c\/strong\u003e\u003cbr\u003eP: power radiated by the accelerating charge (non-relativistic Larmor formula) (W); q: charge of the particle (C); a: magnitude of its acceleration (m\/s²); ε₀: vacuum permittivity (F\/m); c: speed of light (m\/s)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/capacitwin-two-capacitor-paradox\"\u003eCapacitwin\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eCharge one capacitor C to voltage V and join it to an identical empty one. Charge is conserved, so each ends at V\/2 with half the charge. But the stored energy is now Q²\/4C instead of Q²\/2C: half the energy is gone. This holds however small the wire resistance is.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eΔE = ½ · C\u003csub\u003e1\u003c\/sub\u003eC\u003csub\u003e2\u003c\/sub\u003e\/(C\u003csub\u003e1\u003c\/sub\u003e + C\u003csub\u003e2\u003c\/sub\u003e) · (V\u003csub\u003e1\u003c\/sub\u003e − V\u003csub\u003e2\u003c\/sub\u003e)²\u003c\/strong\u003e\u003cbr\u003eΔE: energy lost when the capacitors are joined (J); C\u003csub\u003e1\u003c\/sub\u003e, C\u003csub\u003e2\u003c\/sub\u003e: capacitances (F); V\u003csub\u003e1\u003c\/sub\u003e, V\u003csub\u003e2\u003c\/sub\u003e: starting voltages (V)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/diagonix-maluss-law-pixel-sticker\"\u003eDiagonix\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA polarizer passes the fraction cos²θ of already-polarized light. Two crossed polarizers block everything, but slip a third one between them at 45° and one-eighth of the original unpolarized light gets through.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eI = I\u003csub\u003e0\u003c\/sub\u003e cos²θ\u003c\/strong\u003e\u003cbr\u003eI: intensity after one ideal polariser (W\/m²); I₀: intensity arriving (unpolarised for the three-polariser line) (W\/m²); θ: angle between the light's polarisation and the polariser's transmission axis (rad); I_(out): intensity after the vertical, 45° and horizontal polarisers in turn (W\/m²)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/gyrolla-cyclotron-motion-pixel-sticker\"\u003eGyrolla\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA magnetic field bends a charged particle into a circle without changing its speed. Faster particles make bigger circles, but each lap takes the same time, so the cyclotron frequency depends only on charge, mass and field. That is what makes the cyclotron accelerator work.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003er = mv\/(qB)\u003c\/strong\u003e\u003cbr\u003eq: charge of the particle (C); v: speed perpendicular to the field (m\/s); B: magnetic flux density (T); m: mass of the particle (kg); r: radius of the circle (m); f_c: cyclotron frequency, laps per second (non-relativistic; at high energy it falls by 1\/γ) (Hz)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/kibbla-kibble-balance\"\u003eKibbla\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA Kibble balance runs in two modes. In weighing mode, a current I in a coil sitting in a radial magnetic field pushes up with force BLI, balancing the weight mg. In moving mode, the same coil is moved at speed v and a voltage U = BLv is induced. The hard-to-measure product BL cancels between the two, leaving a match between mechanical and electrical power.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003emgv = UI\u003c\/strong\u003e\u003cbr\u003em: test mass (kg); g: local gravitational acceleration (m\/s²); v: coil velocity (m\/s); U: induced voltage (V); I: coil current (A)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/lenzor-lenzs-law-pixel-sticker\"\u003eLenzor\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA changing magnetic flux induces a voltage, and the induced current always flows so as to oppose the change. A magnet dropped through a copper pipe therefore drives eddy currents that brake it, so it drifts down at a steady terminal speed.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eℰ = −dΦ\u003csub\u003eB\u003c\/sub\u003e\/dt\u003c\/strong\u003e\u003cbr\u003emathcalE: EMF induced around one ring of the pipe (V); Φ_B: magnetic flux through that ring (Wb); t: time (s); m: mass of the falling magnet (kg); g: gravitational acceleration (m\/s²); F_(eddy): upward drag from the eddy currents; it grows with the magnet's speed (N)\u003c\/p\u003e","brand":"Printify","offers":[{"title":"White \/ Kiss-Cut \/ 5.83″ × 8.27″","offer_id":50966885400679,"sku":"29186501020521845643","price":19.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/139833868696833101_2048.jpg?v=1793676426"},{"product_id":"forces-fields-2-sticker-sheet","title":"Forces \u0026 Fields Vol. 2 Pixel Sticker Sheet – 7 Physics Stickers","description":"\u003cp\u003e7 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: Forces \u0026amp; Fields Vol. 2. Every character below has its own sticker page with the full physics panel.\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/minkowskara-photon-momentum\"\u003eMinkowskara\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eHow much momentum does light carry inside a medium of refractive index n? Hermann Minkowski (1908) gave a momentum that grows with n; Max Abraham (1909–1910) gave one that shrinks as 1\/n. Both follow from consistent electrodynamics, and experiments have seemed to support each in different setups.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ep\u003csub\u003eMinkowski\u003c\/sub\u003e = nħω\/c,   p\u003csub\u003eAbraham\u003c\/sub\u003e = ħω\/(nc)\u003c\/strong\u003e\u003cbr\u003ep: momentum of one photon in the medium; n: refractive index of the medium; ħ: reduced Planck constant (exact in the 2019 SI); ω: angular frequency of the light; c: speed of light in vacuum (exact)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/penninga-penning-trap\"\u003ePenninga\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eEarnshaw's theorem says no arrangement of static electric charges alone can hold a charge in stable equilibrium. A Penning trap gets around it by adding a strong, uniform magnetic field to an electric quadrupole field. A trapped particle then has three motions: a fast modified cyclotron circle (ω\u003csub\u003e+\u003c\/sub\u003e), an axial bounce along the field (ω\u003csub\u003ez\u003c\/sub\u003e), and a slow magnetron drift (ω\u003csub\u003e−\u003c\/sub\u003e). L. S. Brown and G. Gabrielse showed that, even in an imperfect trap, the three combine to give the true cyclotron frequency (“Geonium theory”, Rev. Mod. Phys. 58, 233, 1986).\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eω\u003csub\u003ec\u003c\/sub\u003e² = ω\u003csub\u003e+\u003c\/sub\u003e² + ω\u003csub\u003ez\u003c\/sub\u003e² + ω\u003csub\u003e−\u003c\/sub\u003e²\u003c\/strong\u003e\u003cbr\u003eω\u003csub\u003ec\u003c\/sub\u003e: free-space cyclotron frequency, qB\/m (rad\/s); ω\u003csub\u003e+\u003c\/sub\u003e: modified cyclotron frequency (rad\/s); ω\u003csub\u003ez\u003c\/sub\u003e: axial frequency (rad\/s); ω\u003csub\u003e−\u003c\/sub\u003e: magnetron frequency (rad\/s)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/refracta-fermats-principle-pixel-sticker\"\u003eRefracta\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eLight takes the path between two points that takes the least time. That single rule gives Snell's law of refraction: light bends toward the slower medium, so that the sine of the angle divided by the speed is the same on both sides.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003esin θ\u003csub\u003e1\u003c\/sub\u003e\/v\u003csub\u003e1\u003c\/sub\u003e = sin θ\u003csub\u003e2\u003c\/sub\u003e\/v\u003csub\u003e2\u003c\/sub\u003e\u003c\/strong\u003e\u003cbr\u003eθ₁: angle of the path from the normal on the sand (light: in the first medium) (rad); θ₂: angle of the path from the normal in the water (light: in the second medium) (rad); v₁, v₂: speeds on each side (m\/s); n₁, n₂: refractive indices; c: speed of light in vacuum (m\/s); ds: element of path length (m)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/scatterina-rayleigh-scattering-pixel-sticker\"\u003eScatterina\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eSmall particles and molecules scatter short wavelengths much more strongly, in proportion to one over wavelength to the fourth power. Blue light is scattered about 5.9 times more than red, which is why the sky looks blue.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eI\u003csub\u003es\u003c\/sub\u003e ∝ 1\/λ⁴\u003c\/strong\u003e\u003cbr\u003eI_s: power scattered by a molecule much smaller than the wavelength (W); λ: wavelength of the light (m)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/thomsonia-thomson-cross-section\"\u003eThomsonia\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eWhen light meets a free electron at low energy, its electric field shakes the electron, which re-radiates at the same frequency. This elastic scattering has a fixed total cross section, the effective target area of one electron. It is J. J. Thomson's classical result, and it is the low-energy limit of quantum electrodynamics.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eσ\u003csub\u003eT\u003c\/sub\u003e = (8π\/3) r\u003csub\u003ee\u003c\/sub\u003e² = 6.652 458 7051(62) × 10⁻²⁹ m²\u003c\/strong\u003e\u003cbr\u003eσ\u003csub\u003eT\u003c\/sub\u003e: Thomson cross section; r\u003csub\u003ee\u003c\/sub\u003e: classical electron radius, e²\/(4πε\u003csub\u003e0\u003c\/sub\u003em\u003csub\u003ee\u003c\/sub\u003ec²) = 2.817 940 3205(13) × 10⁻¹⁵ m; e: elementary charge (exact); ε\u003csub\u003e0\u003c\/sub\u003e: vacuum permittivity (measured); m\u003csub\u003ee\u003c\/sub\u003e: electron mass (measured); c: speed of light (exact)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/gravitas-big-g\"\u003eGravitas\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eG sets the strength of gravity between any two masses. It is measured with torsion balances (Cavendish, 1798) and other lab methods, where the pull between kilogram-scale masses is about 10⁻⁷ N or less.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eG = 6.674 30(15) × 10⁻¹¹ m³ kg⁻¹ s⁻²\u003c\/strong\u003e\u003cbr\u003eG: Newtonian constant of gravitation; (15): standard uncertainty in the last two digits, 2.2 × 10⁻⁵ relative\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/moderon-neutron-moderation-pixel-sticker\"\u003eModeron\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA neutron loses the largest share of its energy when it hits a nucleus of about its own mass. The average logarithmic energy loss per collision tells you how many collisions a moderator needs to slow fission neutrons down to thermal energies.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eα = ((A − 1)\/(A + 1))²\u003c\/strong\u003e\u003cbr\u003eA: mass of the target nucleus in neutron masses (hydrogen: 1); α: smallest possible ratio of neutron energy after to before one collision (a head-on hit); ξ: average loss of ln E per collision (hydrogen: 1); n_(coll): average number of collisions to slow a neutron to thermal; E₀: starting neutron energy, about 2 MeV from fission (MeV); E_(th): thermal neutron energy, about 0.025 eV (eV)\u003c\/p\u003e","brand":"Printify","offers":[{"title":"White \/ Kiss-Cut \/ 5.83″ × 8.27″","offer_id":50966893625447,"sku":"14999103256598222747","price":19.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/3249009372056948214_2048.jpg?v=1793676432"},{"product_id":"thermodynamics-sticker-sheet","title":"Thermodynamics Pixel Sticker Sheet – 12 Physics Stickers","description":"\u003cp\u003e12 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: Thermodynamics. Every character below has its own sticker page with the full physics panel.\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/dilutra-dilution-refrigerator\"\u003eDilutra\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eBelow about 0.87 K a mixture of helium-3 and helium-4 separates into a ³He-rich phase floating on a dilute phase. Even at absolute zero the dilute phase keeps a few percent of ³He dissolved. Ifft, Edwards and Sarwinski measured this limit at 6.37 ± 0.05% (Phys. Rev. Lett. 19, 831, 1967); later work gives about 6.4–6.6% depending on pressure. Pumping ³He out of the dilute phase pulls more across the boundary, and that ‘evaporation’ into the dilute phase absorbs heat. It works continuously, down to a few millikelvin.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ex\u003csub\u003e3\u003c\/sub\u003e(T → 0) ≈ 6.4–6.6%\u003c\/strong\u003e\u003cbr\u003ex\u003csub\u003e3\u003c\/sub\u003e: fraction of ³He atoms in the dilute (⁴He-rich) phase\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/boltzbrain-boltzmann-brain\"\u003eBoltzbrain\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn the 1890s Ludwig Boltzmann suggested that our ordered world might be a rare fluctuation in a universe otherwise in thermal equilibrium. The difficulty is that a small fluctuation is far more likely than a big one. A single brain, complete with false memories, needs a much smaller drop in entropy than a whole ordered universe, so in such a picture most observers would be lone “Boltzmann brains”. Andreas Albrecht and Lorenzo Sorbo (Phys. Rev. D 70, 063528, 2004) discussed it as the so-called “Boltzmann's brain” paradox.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eP ∝ exp(ΔS\/k\u003csub\u003eB\u003c\/sub\u003e)\u003c\/strong\u003e\u003cbr\u003eP: relative probability of a fluctuation; ΔS: entropy change of the fluctuation, negative for a fluctuation into order (J\/K); k\u003csub\u003eB\u003c\/sub\u003e: Boltzmann constant, 1.380 649 × 10⁻²³ J\/K exactly in the 2019 SI\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/nyquista-johnson-noise\"\u003eNyquista\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eAny resistor at a temperature above absolute zero hisses with voltage noise, because its electrons jiggle thermally. John B. Johnson measured it at Bell Labs and Harry Nyquist explained it, both in 1928 (Phys. Rev. 32, 97 and 32, 110). The noise power per unit bandwidth depends only on temperature and resistance, not on the material.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003e⟨V²⟩ = 4k\u003csub\u003eB\u003c\/sub\u003eTRΔf\u003c\/strong\u003e\u003cbr\u003e⟨V²⟩: mean-square noise voltage (V²); k\u003csub\u003eB\u003c\/sub\u003e: Boltzmann constant, exactly 1.380 649 × 10⁻²³ J\/K since 2019; T: temperature (K); R: resistance (Ω); Δf: measurement bandwidth (Hz)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/seebecka-seebeck-effect\"\u003eSeebecka\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn 1821 Thomas Johann Seebeck joined bismuth and copper in a loop and heated one junction. A compass needle beside it swung. A temperature difference between the junctions drives a voltage, and in a closed loop a current: thermoelectricity. He reported it to the Prussian Academy (Abh. Akad. Wiss. Berlin, 1822–23).\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eV = S ΔT\u003c\/strong\u003e\u003cbr\u003eV: thermoelectric voltage (V); S: Seebeck coefficient difference of the two metals (V\/K); ΔT: temperature difference between the junctions (K)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/invertia-negative-absolute-temperature-pixel-sticker\"\u003eInvertia\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eTemperature is defined by how entropy changes when you add energy. In a system with a top energy level, like nuclear spins, putting more particles in the upper state than the lower one makes that slope negative: a negative absolute temperature, which is actually 'hotter than infinity'.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003e1\/T = ∂S\/∂E \u0026lt; 0\u003c\/strong\u003e\u003cbr\u003eT: absolute temperature, in the standard (Boltzmann) definition (K); S: entropy (Boltzmann form, from the number of states at energy E) (J\/K); E: total energy of the spin system (J); N_↑, N_↓: number of particles in the upper and the lower level; Δ E: energy gap between the two levels (J); k_B: Boltzmann constant (J\/K)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/paddlewheel-joule\"\u003ePaddlewheel\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eJames Prescott Joule let weights fall on pulleys, turning a paddle wheel inside an insulated vessel of water. Friction of the paddles warmed the water a tiny amount. Dividing the work done by the falling weights by the heat produced, he reported in 1850 (“On the Mechanical Equivalent of Heat”, Phil. Trans. R. Soc. 140, 61) that 772 foot-pounds of work raise one pound of water by 1 °F.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eW = J Q\u003c\/strong\u003e\u003cbr\u003eW: mechanical work (ft·lbf); J: mechanical equivalent of heat (ft·lbf per BTU); Q: heat produced (BTU)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/radiometra-crookes-radiometer\"\u003eRadiometra\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA Crookes radiometer has four vanes, each black on one side and silver on the other, inside a bulb pumped down to a partial vacuum. In light, the vanes spin with the dark sides retreating. If light pressure drove it, the reflective sides would be pushed harder (a reflected photon gives twice the push of an absorbed one) and it would spin the other way. Instead, the dark side gets warmer, and the thin gas does the work. James Clerk Maxwell (Phil. Trans. R. Soc. 170, 231, 1879) and Osborne Reynolds, in his study of thermal transpiration (Phil. Trans. R. Soc. 170, 727, 1879), explained it: gas creeps around each vane's edge from the cold side to the hot side, and the reaction pushes the vane towards its cooler, silver side.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003edark side warmer ⇒ vane moves dark side trailing\u003c\/strong\u003e\u003cbr\u003edark side: absorbs light and heats up; silver side: reflects light and stays cooler; gas: thin enough for molecules to travel far between collisions, but not a hard vacuum\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/ratchetta-feynman-ratchet\"\u003eRatchetta\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eThe ratchet and pawl, analysed by Smoluchowski (1912) and made famous in the Feynman Lectures (Vol. I, ch. 46), is a vane in a gas joined by an axle to a ratchet wheel with a spring-loaded pawl. Molecules hitting the vane seem to push the wheel forward, while the pawl blocks it going back. But the pawl is also at a temperature and also jiggles. When both are at the same temperature, the pawl lifts and lets the wheel slip back as often as the vane pushes it forward.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003erate forward ∝ e\u003csup\u003e−ε\/k\u003csub\u003eB\u003c\/sub\u003eT\u003csub\u003e1\u003c\/sub\u003e\u003c\/sup\u003e,   rate back ∝ e\u003csup\u003e−ε\/k\u003csub\u003eB\u003c\/sub\u003eT\u003csub\u003e2\u003c\/sub\u003e\u003c\/sup\u003e  ⇒  T\u003csub\u003e1\u003c\/sub\u003e = T\u003csub\u003e2\u003c\/sub\u003e gives ⟨W⟩ = 0\u003c\/strong\u003e\u003cbr\u003eε: energy needed to lift the pawl over one tooth (J); k\u003csub\u003eB\u003c\/sub\u003e: Boltzmann constant, exact in the 2019 SI; T\u003csub\u003e1\u003c\/sub\u003e: temperature of the gas around the vane (K); T\u003csub\u003e2\u003c\/sub\u003e: temperature of the ratchet and pawl (K); ⟨W⟩: average work delivered per unit time\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/ruchardta-ruchardt-experiment\"\u003eRuchardta\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn 1929 Eduard Rüchardt measured the heat-capacity ratio of a gas with a simple trick (Phys. Z. 30, 58). A ball fits snugly in the neck of a large flask. Push it and it bounces up and down on the springy gas below. The squeezes are quick, so no heat flows and the gas acts adiabatically. Timing the oscillation gives γ.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eγ = 4π²mV\/(A²pT²)\u003c\/strong\u003e\u003cbr\u003eγ: ratio of heat capacities C\u003csub\u003ep\u003c\/sub\u003e\/C\u003csub\u003ev\u003c\/sub\u003e; m: mass of the ball (kg); V: volume of gas (m³); A: cross-section of the neck (m²); p: gas pressure under the ball (Pa); T: oscillation period (s)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/sortex-maxwells-demon-pixel-sticker\"\u003eSortex\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eLandauer argued that erasing one bit of information must release at least k_B T ln 2 of heat. Because a Maxwell's demon has to erase its records eventually, this cost is what saves the second law of thermodynamics.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eQ\u003csub\u003eerase\u003c\/sub\u003e ≥ k\u003csub\u003eB\u003c\/sub\u003eT ln 2\u003c\/strong\u003e\u003cbr\u003eQ_(erase): heat released to the surroundings when one bit of memory is erased (a lower bound, not a fixed cost) (J); k_B: Boltzmann constant (J\/K); T: temperature of the surroundings that absorb the heat (K); ln 2: natural logarithm of 2: one bit is a choice between two states\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/argonix-rayleigh-ramsay\"\u003eArgonix\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eLord Rayleigh weighed nitrogen with great care and found that nitrogen taken from air was about half a percent denser than nitrogen made from ammonia, a gap far larger than his errors. With William Ramsay he removed all the nitrogen and oxygen from air and found a stubborn leftover gas, about 1% of the atmosphere, with a new spectrum and no chemistry at all. They named it argon, ‘the lazy one’ (“Argon, a New Constituent of the Atmosphere”, Phil. Trans. A 186, 187, 1895).\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eρ\u003csub\u003eair N₂\u003c\/sub\u003e ≈ 1.005 ρ\u003csub\u003echem N₂\u003c\/sub\u003e\u003c\/strong\u003e\u003cbr\u003eρ\u003csub\u003eair N₂\u003c\/sub\u003e: density of ‘nitrogen’ from air, which still contains argon; ρ\u003csub\u003echem N₂\u003c\/sub\u003e: density of nitrogen from chemical sources\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/echobel-acoustic-gas-thermometer\"\u003eEchobel\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn acoustic gas thermometry, the resonant frequencies of a gas-filled spherical cavity of known size give the speed of sound. For a dilute monatomic gas the speed of sound fixes k\u003csub\u003eB\u003c\/sub\u003eT per atom. Measurements of this kind at the triple point of water (NPL, LNE, INRiM and others) gave the most precise values of the Boltzmann constant before the 2019 redefinition.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ec² = γk\u003csub\u003eB\u003c\/sub\u003eT\/m\u003c\/strong\u003e\u003cbr\u003ec: speed of sound in the zero-pressure limit (m\/s); γ: heat-capacity ratio, 5\/3 for an ideal monatomic gas; k\u003csub\u003eB\u003c\/sub\u003e: Boltzmann constant, 1.380 649 × 10⁻²³ J\/K exactly; T: thermodynamic temperature (K); m: mass of one gas atom (kg)\u003c\/p\u003e","brand":"Printify","offers":[{"title":"White \/ Kiss-Cut \/ 5.83″ × 8.27″","offer_id":50966901850215,"sku":"29053796791254063365","price":19.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/7530443038865545052_2048.jpg?v=1793676438"},{"product_id":"relativity-cosmos-sticker-sheet","title":"Relativity \u0026 Cosmos Pixel Sticker Sheet – 11 Physics Stickers","description":"\u003cp\u003e11 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: Relativity \u0026amp; Cosmos. Every character below has its own sticker page with the full physics panel.\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/magnetoconda-moving-magnet\"\u003eMagnetoconda\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eMove a magnet past a loop of wire and a current flows. Hold the magnet still and move the wire instead, at the same relative speed, and exactly the same current flows. Yet the textbook stories differ. If the magnet moves, a changing magnetic field makes an electric field that drives the charges. If the wire moves, there is no electric field in the magnet's frame, and the magnetic force on the moving charges does the job. A rod of length L moving at speed v across a field B gets the same emf either way.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eε = vBL\u003c\/strong\u003e\u003cbr\u003eε: induced emf (volt, V); v: relative speed of rod and magnet, perpendicular to the field (m\/s); B: magnetic flux density (tesla, T); L: length of the rod across the field (m)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/aberra-bradley-aberration\"\u003eAberra\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eFrom 1725 James Bradley tracked the star Gamma Draconis and found it shifted through the year by about 20 arcseconds, but in a pattern that did not fit parallax. In 1729 he explained it (Phil. Trans. R. Soc. 35, 637): Earth is moving, so the telescope must be tilted slightly forward to catch starlight, like slanting an umbrella when you walk in rain. The tilt is set by Earth’s orbital speed compared with the speed of light.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eθ ≈ v\/c ≈ 20.5″\u003c\/strong\u003e\u003cbr\u003eθ: aberration angle (rad, here ≈ 20.5 arcseconds); v: Earth’s orbital speed, ≈ 30 km\/s; c: speed of light (m\/s)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/faintsol-faint-young-sun\"\u003eFaintsol\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eAs the Sun fuses hydrogen into helium, its core gets denser and hotter, so it slowly brightens. Standard solar models give about 70% of today's luminosity some 4.5 billion years ago. With today's atmosphere and albedo, that would leave the early Earth's average temperature below freezing for roughly its first two billion years. But the geological record shows liquid water on Earth more than 3.8 billion years ago. Carl Sagan and George Mullen stated this paradox in 1972.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eL(t) ≈ [1 + (2\/5)(1 − t\/t\u003csub\u003e⊙\u003c\/sub\u003e)]⁻¹ L\u003csub\u003e⊙\u003c\/sub\u003e\u003c\/strong\u003e\u003cbr\u003eL(t): solar luminosity at age t; t: age of the Sun at that time; t\u003csub\u003e⊙\u003c\/sub\u003e: present age of the Sun, about 4.57 billion years; L\u003csub\u003e⊙\u003c\/sub\u003e: present solar luminosity, 3.828 × 10²⁶ W (IAU nominal value)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/mirrorix-ligo-test-mass\"\u003eMirrorix\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eLIGO measures gravitational waves by comparing the lengths of two 4 km arms with laser light. At the end of each arm hangs a test mass: a 40 kg fused-silica mirror 34 cm across, hung from a four-stage pendulum to isolate it from ground vibration (Advanced LIGO, Class. Quantum Grav. 32, 074001, 2015). On 14 September 2015 a wave from two merging black holes, GW150914, stretched and squeezed the arms with a peak strain of 1.0 × 10⁻²¹ (Phys. Rev. Lett. 116, 061102, 2016).\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eh ≈ ΔL\/L\u003c\/strong\u003e\u003cbr\u003eh: gravitational-wave strain, dimensionless; ΔL: change in the difference of the arm lengths (m); L: arm length, 4 km\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/skyrider-hess-balloon\"\u003eSkyrider\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn 1912 Victor Hess carried electroscopes up in balloons to 5 350 m (Phys. Z. 13, 1084). Their discharge rate, a measure of ionising radiation, first dipped slightly, then climbed steeply with height. The radiation was coming from above, not from the ground. A flight during a near-total solar eclipse and at night showed no drop, so the Sun was not the source. He shared the 1936 Nobel Prize for cosmic rays.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eionisation rate rises with altitude\u003c\/strong\u003e\u003cbr\u003ealtitude: height above sea level, up to 5 350 m in 1912; ionisation rate: ions formed per cm³ per second in the electroscope\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/primordix-big-bang-nucleosynthesis-pixel-sticker\"\u003ePrimordix\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn the first minutes after the Big Bang, the ratio of neutrons to protons froze out at a small fixed value. Nearly all those neutrons ended up in helium-4, which predicts the fraction of ordinary mass that came out as helium, matching observations.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eY\u003csub\u003ep\u003c\/sub\u003e ≃ 2(n\/p)\/(1 + n\/p)\u003c\/strong\u003e\u003cbr\u003eY_p: primordial helium-4 fraction by mass (observed 0.245 ± 0.003); n\/p: neutron-to-proton ratio; Q: rest-energy difference between neutron and proton (MeV); k_B: Boltzmann constant (eV\/K); T_f: temperature when the weak reactions freeze out (thermal energy about 0.7–0.8 MeV) (K); m_n, m_p: neutron and proton masses (kg)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/relicta-cosmic-microwave-background-pixel-sticker\"\u003eRelicta\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eThe cosmic microwave background is light left over from when the universe became transparent. It has a nearly perfect blackbody spectrum, and its temperature has fallen in proportion to the stretching of space, to about 2.7 K today.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eT(z) = T\u003csub\u003e0\u003c\/sub\u003e(1 + z)\u003c\/strong\u003e\u003cbr\u003eT(z): temperature of the background radiation at redshift z (K); z: redshift (last scattering: about 1090); T₀: temperature today (Fixsen 2009: 2.72548 ± 0.00057 K) (K); B_ν: blackbody spectral radiance per unit frequency (W·m⁻²·Hz⁻¹·sr⁻¹); ν: frequency (Hz); h: Planck constant (J·s)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/orbitick-gps-clock-correction-pixel-sticker\"\u003eOrbitick\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eGPS clocks feel weaker gravity than clocks on the ground, so they run fast, and their orbital speed makes them run slightly slow. The net effect would build up timing errors quickly, so the satellite clocks are deliberately set to tick slightly slower before launch.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eΔf\/f ≈ ΔΦ\/c² − v²\/(2c²)\u003c\/strong\u003e\u003cbr\u003eΔ f\/f: fractional rate of a GPS satellite clock relative to a clock on the geoid (positive: it runs fast); ΔΦ: gravitational potential at the satellite minus the effective potential on the geoid, which includes Earth's rotation term (J\/kg); v: orbital speed of the satellite (m\/s); c: speed of light in vacuum (m\/s); f₀: factory-specified frequency of the clock's 10.23 MHz reference before launch (Hz)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/troutona-trouton-noble\"\u003eTroutona\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIf Earth moved through an ether, a charged capacitor hanging on a fine wire should feel a torque turning its plates to face the motion. In 1903 Frederick Trouton and H. R. Noble looked for it (Phil. Trans. A 202, 165, published 1904). They found none: the torque was zero.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003etorque = 0\u003c\/strong\u003e\u003cbr\u003etorque: turning effect on the suspended capacitor (N·m)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/mirrorspin-foucault-speed-of-light\"\u003eMirrorspin\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn 1862 Léon Foucault measured the speed of light in a laboratory with a rapidly rotating mirror (Comptes rendus 55, 501). Light reflects off the spinning mirror to a distant fixed mirror and back. In the round-trip time the rotating mirror turns a tiny angle, so the returning spot lands slightly displaced. From the displacement, the path length and the spin rate he got c = 298 000 ± 500 km\/s, close to the modern value.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eΔα = 2ωD\/c\u003c\/strong\u003e\u003cbr\u003eΔα: angle the mirror turns during the round trip (rad); the returning beam is deflected by 2Δα; ω: angular speed of the mirror (rad\/s); D: distance to the fixed mirror (m); c: speed of light (m\/s)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/muonaut-muon-time-dilation-pixel-sticker\"\u003eMuonaut\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eMuons made high in the atmosphere live only about two microseconds, too short to cross the atmosphere at face value. Moving near the speed of light, their clocks run slow by the factor γ, so many reach the ground; in the muon's frame, the atmosphere is shorter instead.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eΔt = γτ, L = L\u003csub\u003e0\u003c\/sub\u003e\/γ\u003c\/strong\u003e\u003cbr\u003eΔ t: muon lifetime as timed by clocks on the ground (s); τ: mean lifetime in the muon's own rest frame (PDG 2024: 2.1969811 μs; the half-life is shorter, 1.52 μs) (s); γ: Lorentz factor; L₀: thickness of the atmosphere in the ground frame (m); L: the same thickness measured in the muon's frame (m); v: muon speed (m\/s)\u003c\/p\u003e","brand":"Printify","offers":[{"title":"White \/ Kiss-Cut \/ 5.83″ × 8.27″","offer_id":50966910074983,"sku":"18908236297653881170","price":19.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/16573198963890037437_2048.jpg?v=1793676444"},{"product_id":"women-in-physics-sticker-sheet","title":"Women in Physics Pixel Sticker Sheet – 9 Physics Stickers","description":"\u003cp\u003e9 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: Women in Physics. Every character below has its own sticker page with the full physics panel.\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/lise-meitner-sticker-split-happens-physics-sticker\"\u003eLise Meitner\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eWhen a uranium nucleus splits, the pieces weigh less than the original, and the missing mass appears as energy. Meitner and Frisch estimated it from the electric repulsion between the two fragments at the moment they part: about 200 MeV per fission.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eQ = (M\u003csub\u003einitial\u003c\/sub\u003e − M\u003csub\u003efinal\u003c\/sub\u003e)c²\u003c\/strong\u003e\u003cbr\u003eQ: energy released (quoted in MeV) (J); M: total rest mass before\/after (kg); c: speed of light (m\/s); E_C: Coulomb repulsion energy of two touching fragments (J); Z₁, Z₂: fragment proton numbers; e: elementary charge (C)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/marie-curie-sticker-she-literally-glowed-up-physics-sticker\"\u003eMarie Curie\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eEach radioactive nucleus has a fixed chance per second of decaying, so the number left falls exponentially. The half-life is ln 2 divided by that decay constant, and the activity is the decay constant times the number of nuclei.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eN(t) = N\u003csub\u003e0\u003c\/sub\u003e e\u003csup\u003e−λt\u003c\/sup\u003e\u003c\/strong\u003e\u003cbr\u003eN: number of undecayed nuclei; λ: decay constant (s⁻¹); t_(1\/2): half-life (s); A: activity (1 Bq = 1 decay per second) (Bq)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/chien-shiung-wu-sticker-she-broke-symmetry-and-records-physics-sticker\"\u003eChien-Shiung Wu\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn Wu's experiment, cobalt-60 nuclei were lined up by a magnetic field, and the electrons from their beta decay came out preferentially opposite to the nuclear spin. A mirror image of that experiment would show the reverse, so the weak force distinguishes left from right.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003e⁶⁰Co → ⁶⁰Ni* + e⁻ + ν̄\u003csub\u003ee\u003c\/sub\u003e\u003c\/strong\u003e\u003cbr\u003eJ: nuclear spin (axial vector) (J·s); pₑ: electron momentum (polar vector) (kg·m\/s); ̂P: parity (mirror) operation\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/jocelyn-bell-burnell-sticker-she-found-pulsars-they-gave-her-silence-physics-sticker\"\u003eJocelyn Bell Burnell\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA pulsar's very regular pulses mean it must be extremely dense to spin that fast without flying apart, which points to a neutron star. Radio pulses at lower frequencies arrive later after passing through interstellar electrons, and that delay measures how many electrons lie along the way.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eρ̄ ≳ 3π\/(GP²)\u003c\/strong\u003e\u003cbr\u003eρ̄: minimum mean density for a body spinning with period P not to fly apart (kg\/m³); G: gravitational constant (m³·kg⁻¹·s⁻²); P: rotation period (s); Δ t: arrival-time delay between frequencies f₁ \u0026lt; f₂ (s); DM: dispersion measure, ∫ nₑ dl (free electrons per area along the line of sight) (m⁻²); f₁, f₂: observing frequencies (Hz)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/vera-rubin-sticker-galaxy-rotation-she-saw-the-dark-side-physics-sticker\"\u003eVera Rubin\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIf most of a galaxy's mass were in its bright centre, stars far out would orbit more slowly. Vera Rubin found the speeds stay roughly constant far out, which means the enclosed mass keeps growing with radius: unseen matter.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ev(r)² = GM(r)\/r\u003c\/strong\u003e\u003cbr\u003ev(r): circular orbital speed (m\/s); M(r): enclosed mass (kg); r: radius (m); G: gravitational constant (m³·kg⁻¹·s⁻²)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/emmy-noether-sticker-symmetry-in-conservation-out-physics-sticker\"\u003eEmmy Noether\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eNoether's theorem says that every continuous symmetry of a system's laws comes with a conserved quantity. Symmetry under shifts in time gives conservation of energy, shifts in space give momentum, and rotations give angular momentum.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eQ = Σ\u003csub\u003ei\u003c\/sub\u003e (∂L\/∂q̇\u003csub\u003ei\u003c\/sub\u003e) δq\u003csub\u003ei\u003c\/sub\u003e\u003c\/strong\u003e\u003cbr\u003eQ: conserved Noether charge (momentum for a shift, angular momentum for a rotation) (kg·m\/s, J·s); L: Lagrangian (J); qᵢ: generalised coordinates (lengths or angles) (m, rad); ̇qᵢ: generalised velocities (m\/s, rad\/s); ε: small symmetry parameter (a length for a shift, an angle for a rotation, a time for a clock shift) (m, rad, s); δ qᵢ: direction of the symmetry shift (its size is carried by ε)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/cecilia-payne-gaposchkin-sticker-almost-certainly-real-physics-sticker\"\u003eCecilia Payne-Gaposchkin\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eSaha's equation gives how much of each element is ionized in a hot gas at a given temperature and density. Cecilia Payne used it to correct for ionization in stellar spectra, and showed that stars are made mostly of hydrogen and helium.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003en\u003csub\u003ei+1\u003c\/sub\u003en\u003csub\u003ee\u003c\/sub\u003e\/n\u003csub\u003ei\u003c\/sub\u003e = (2\/Λ³)(g\u003csub\u003ei+1\u003c\/sub\u003e\/g\u003csub\u003ei\u003c\/sub\u003e) exp(−χ\u003csub\u003ei\u003c\/sub\u003e\/(k\u003csub\u003eB\u003c\/sub\u003eT))\u003c\/strong\u003e\u003cbr\u003enᵢ, nᵢ₊₁: number densities of ionisation stages i and i+1 (m⁻³); nₑ: electron number density (m⁻³); gᵢ: statistical weight of stage i; χᵢ: ionisation energy from stage i (J); k_B: Boltzmann constant (J\/K); T: temperature (K)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/emilie-du-chatelet-sticker-2x-the-speed-4x-the-dent-physics-sticker\"\u003eÉmilie du Châtelet\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eKinetic energy grows with the square of speed, so twice the speed carries four times the energy. Du Châtelet used experiments with balls dropped into clay, where the dent depth grew with the square of the speed, to argue for this 'vis viva' measure.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eE\u003csub\u003ek\u003c\/sub\u003e = ½mv²\u003c\/strong\u003e\u003cbr\u003eE_k: kinetic energy (J); m: mass (kg); v: impact speed (m\/s); g: gravitational acceleration (m\/s²); h: drop height (m); F: average resisting force of the clay (assumed constant) (N)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/maria-goeppert-mayer-sticker-a-full-floor-is-magic-physics-sticker\"\u003eMaria Goeppert Mayer\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eProtons and neutrons fill energy shells in the nucleus, much like electrons in an atom. Adding strong spin-orbit coupling moves the shell gaps to the observed magic numbers 2, 8, 20, 28, 50, 82 and 126, where nuclei are unusually stable.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eE\u003csub\u003eN\u003c\/sub\u003e = ħω(N + 3\/2)\u003c\/strong\u003e\u003cbr\u003eE_N: energy of oscillator shell N (J); N: oscillator shell number; ħω: oscillator level spacing (J); g_N: states in shell N per nucleon species (spin included); V_(ℓ s): spin–orbit potential (J); C(r): radial strength of the spin–orbit term (with orbital and spin angular momentum measured in units of ħ) (J)\u003c\/p\u003e","brand":"Printify","offers":[{"title":"White \/ Kiss-Cut \/ 5.83″ × 8.27″","offer_id":50966918299751,"sku":"79825821422549706662","price":19.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/12124856556730486760_2048.jpg?v=1793676451"},{"product_id":"paradoxes-puzzles-sticker-sheet","title":"Paradoxes \u0026 Puzzles Pixel Sticker Sheet – 12 Physics Stickers","description":"\u003cp\u003e12 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: Paradoxes \u0026amp; Puzzles. Every character below has its own sticker page with the full physics panel.\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/braessa-spring-paradox\"\u003eBraessa\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eTwo identical springs hang in series, joined by a short string, with the weight at the bottom. Two longer “safety” strings hang slack, one from the support to the lower spring and one from the upper spring to the weight. When the short string is cut, the load is carried by two parallel paths, each a spring plus a long string, so each spring carries half the weight. If the strings are long enough the weight ends up higher than before.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eseries: k\u003csub\u003eeff\u003c\/sub\u003e = k\/2   →   parallel: k\u003csub\u003eeff\u003c\/sub\u003e = 2k\u003c\/strong\u003e\u003cbr\u003ek: stiffness of each spring (N\/m); k\u003csub\u003eeff\u003c\/sub\u003e: effective stiffness of the support\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/dalembertia-zero-drag\"\u003eDalembertia\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn 1752 Jean le Rond d'Alembert showed that for steady, incompressible, inviscid and irrotational (potential) flow, a body moving at constant speed feels zero drag. The streamlines and the pressure are symmetric front to back, so the pressure pushes on the rear as hard as on the front. Yet every real body in air or water feels drag.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eφ = U (r + a³\/2r²) cos θ   ⇒   F\u003csub\u003edrag\u003c\/sub\u003e = 0\u003c\/strong\u003e\u003cbr\u003eφ: velocity potential of steady flow past a sphere (m²\/s); U: speed of the uniform flow far away (m\/s); a: sphere radius (m); r, θ: distance from the centre and angle from the flow direction; F\u003csub\u003edrag\u003c\/sub\u003e: net force along the flow (N)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/teacupa-tea-leaf-paradox\"\u003eTeacupa\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn a stirred cup, the swirl is held in balance by a pressure that rises toward the wall (the surface is higher there). Near the bottom, friction slows the swirl, but the pressure difference set by the surface is still there. So near the floor the inward pressure push wins, and water flows toward the centre, rises up the middle and returns outward near the top. This secondary flow carries the sunken leaves to the centre, where they settle.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003e∂p\/∂r = ρ v\u003csub\u003eθ\u003c\/sub\u003e²\/r\u003c\/strong\u003e\u003cbr\u003ep: pressure (Pa); r: distance from the axis (m); ρ: density of the water (kg\/m³); v\u003csub\u003eθ\u003c\/sub\u003e: swirl (azimuthal) speed (m\/s)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/faintsol-faint-young-sun\"\u003eFaintsol\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eAs the Sun fuses hydrogen into helium, its core gets denser and hotter, so it slowly brightens. Standard solar models give about 70% of today's luminosity some 4.5 billion years ago. With today's atmosphere and albedo, that would leave the early Earth's average temperature below freezing for roughly its first two billion years. But the geological record shows liquid water on Earth more than 3.8 billion years ago. Carl Sagan and George Mullen stated this paradox in 1972.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eL(t) ≈ [1 + (2\/5)(1 − t\/t\u003csub\u003e⊙\u003c\/sub\u003e)]⁻¹ L\u003csub\u003e⊙\u003c\/sub\u003e\u003c\/strong\u003e\u003cbr\u003eL(t): solar luminosity at age t; t: age of the Sun at that time; t\u003csub\u003e⊙\u003c\/sub\u003e: present age of the Sun, about 4.57 billion years; L\u003csub\u003e⊙\u003c\/sub\u003e: present solar luminosity, 3.828 × 10²⁶ W (IAU nominal value)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/minkowskara-photon-momentum\"\u003eMinkowskara\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eHow much momentum does light carry inside a medium of refractive index n? Hermann Minkowski (1908) gave a momentum that grows with n; Max Abraham (1909–1910) gave one that shrinks as 1\/n. Both follow from consistent electrodynamics, and experiments have seemed to support each in different setups.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ep\u003csub\u003eMinkowski\u003c\/sub\u003e = nħω\/c,   p\u003csub\u003eAbraham\u003c\/sub\u003e = ħω\/(nc)\u003c\/strong\u003e\u003cbr\u003ep: momentum of one photon in the medium; n: refractive index of the medium; ħ: reduced Planck constant (exact in the 2019 SI); ω: angular frequency of the light; c: speed of light in vacuum (exact)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/homestaka-solar-neutrino-problem\"\u003eHomestaka\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eRaymond Davis Jr.'s experiment in the Homestake gold mine, South Dakota, 1478 m underground, used a tank of about 615 tonnes of perchloroethylene (tetrachloroethylene, C₂Cl₄). A solar electron neutrino can turn a chlorine-37 nucleus into argon-37 (threshold 0.814 MeV), and the few argon atoms were extracted and counted. From 1970 to 1994 it measured about a third of the rate predicted by John Bahcall's standard solar model.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003emeasured 2.56 ± 0.16 ± 0.16 SNU   vs   predicted 7.6\u003csup\u003e+1.3\u003c\/sup\u003e\u003csub\u003e−1.1\u003c\/sub\u003e SNU\u003c\/strong\u003e\u003cbr\u003eSNU: solar neutrino unit: 10⁻³⁶ captures per target atom per second; 2.56: final Homestake result (Cleveland et al., ApJ 496, 505, 1998), statistical and systematic errors; 7.6: chlorine prediction of the BP2000 standard solar model (Bahcall, Pinsonneault and Basu, 2001)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/bucketta-newtons-bucket\"\u003eBucketta\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn the Scholium of the Principia (1687), Isaac Newton described a bucket of water hung on a twisted rope. When the bucket starts spinning, the water at first stays flat. As the water picks up the rotation, its surface becomes concave, and it stays curved even after the bucket is stopped, while the water keeps turning. Newton argued that the curve shows rotation relative to absolute space, not relative to the bucket.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ez = ω²r²\/(2g)\u003c\/strong\u003e\u003cbr\u003ez: height of the surface above its lowest point, on the axis (m); ω: angular speed of the water (rad\/s); r: distance from the axis (m); g: gravitational acceleration (m\/s²)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/capacitwin-two-capacitor-paradox\"\u003eCapacitwin\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eCharge one capacitor C to voltage V and join it to an identical empty one. Charge is conserved, so each ends at V\/2 with half the charge. But the stored energy is now Q²\/4C instead of Q²\/2C: half the energy is gone. This holds however small the wire resistance is.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eΔE = ½ · C\u003csub\u003e1\u003c\/sub\u003eC\u003csub\u003e2\u003c\/sub\u003e\/(C\u003csub\u003e1\u003c\/sub\u003e + C\u003csub\u003e2\u003c\/sub\u003e) · (V\u003csub\u003e1\u003c\/sub\u003e − V\u003csub\u003e2\u003c\/sub\u003e)²\u003c\/strong\u003e\u003cbr\u003eΔE: energy lost when the capacitors are joined (J); C\u003csub\u003e1\u003c\/sub\u003e, C\u003csub\u003e2\u003c\/sub\u003e: capacitances (F); V\u003csub\u003e1\u003c\/sub\u003e, V\u003csub\u003e2\u003c\/sub\u003e: starting voltages (V)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/boltzbrain-boltzmann-brain\"\u003eBoltzbrain\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn the 1890s Ludwig Boltzmann suggested that our ordered world might be a rare fluctuation in a universe otherwise in thermal equilibrium. The difficulty is that a small fluctuation is far more likely than a big one. A single brain, complete with false memories, needs a much smaller drop in entropy than a whole ordered universe, so in such a picture most observers would be lone “Boltzmann brains”. Andreas Albrecht and Lorenzo Sorbo (Phys. Rev. D 70, 063528, 2004) discussed it as the so-called “Boltzmann's brain” paradox.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eP ∝ exp(ΔS\/k\u003csub\u003eB\u003c\/sub\u003e)\u003c\/strong\u003e\u003cbr\u003eP: relative probability of a fluctuation; ΔS: entropy change of the fluctuation, negative for a fluctuation into order (J\/K); k\u003csub\u003eB\u003c\/sub\u003e: Boltzmann constant, 1.380 649 × 10⁻²³ J\/K exactly in the 2019 SI\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/delaychoice-wheeler-delayed-choice\"\u003eDelaychoice\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn a Mach–Zehnder interferometer, a single photon meets a first beam splitter and then two paths. If a second beam splitter recombines the paths, the photon shows interference: which detector clicks depends on the phase difference. If the second splitter is removed, each detector clicks half the time, as if the photon took one path. John Wheeler proposed in 1978 making this choice after the photon has passed the first splitter. Jacques et al. (Science 315, 966, 2007) did it with single photons and a random choice that was space-like separated from the photon's entry.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ewith splitter: P\u003csub\u003e1\u003c\/sub\u003e = cos²(φ\/2), P\u003csub\u003e2\u003c\/sub\u003e = sin²(φ\/2);   without: P\u003csub\u003e1\u003c\/sub\u003e = P\u003csub\u003e2\u003c\/sub\u003e = ½\u003c\/strong\u003e\u003cbr\u003eP\u003csub\u003e1\u003c\/sub\u003e, P\u003csub\u003e2\u003c\/sub\u003e: probabilities that detector 1 or 2 clicks; φ: phase difference between the two arms (rad)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/magnetoconda-moving-magnet\"\u003eMagnetoconda\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eMove a magnet past a loop of wire and a current flows. Hold the magnet still and move the wire instead, at the same relative speed, and exactly the same current flows. Yet the textbook stories differ. If the magnet moves, a changing magnetic field makes an electric field that drives the charges. If the wire moves, there is no electric field in the magnet's frame, and the magnetic force on the moving charges does the job. A rod of length L moving at speed v across a field B gets the same emf either way.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eε = vBL\u003c\/strong\u003e\u003cbr\u003eε: induced emf (volt, V); v: relative speed of rod and magnet, perpendicular to the field (m\/s); B: magnetic flux density (tesla, T); L: length of the rod across the field (m)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/ratchetta-feynman-ratchet\"\u003eRatchetta\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eThe ratchet and pawl, analysed by Smoluchowski (1912) and made famous in the Feynman Lectures (Vol. I, ch. 46), is a vane in a gas joined by an axle to a ratchet wheel with a spring-loaded pawl. Molecules hitting the vane seem to push the wheel forward, while the pawl blocks it going back. But the pawl is also at a temperature and also jiggles. When both are at the same temperature, the pawl lifts and lets the wheel slip back as often as the vane pushes it forward.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003erate forward ∝ e\u003csup\u003e−ε\/k\u003csub\u003eB\u003c\/sub\u003eT\u003csub\u003e1\u003c\/sub\u003e\u003c\/sup\u003e,   rate back ∝ e\u003csup\u003e−ε\/k\u003csub\u003eB\u003c\/sub\u003eT\u003csub\u003e2\u003c\/sub\u003e\u003c\/sup\u003e  ⇒  T\u003csub\u003e1\u003c\/sub\u003e = T\u003csub\u003e2\u003c\/sub\u003e gives ⟨W⟩ = 0\u003c\/strong\u003e\u003cbr\u003eε: energy needed to lift the pawl over one tooth (J); k\u003csub\u003eB\u003c\/sub\u003e: Boltzmann constant, exact in the 2019 SI; T\u003csub\u003e1\u003c\/sub\u003e: temperature of the gas around the vane (K); T\u003csub\u003e2\u003c\/sub\u003e: temperature of the ratchet and pawl (K); ⟨W⟩: average work delivered per unit time\u003c\/p\u003e","brand":"Printify","offers":[{"title":"White \/ Kiss-Cut \/ 5.83″ × 8.27″","offer_id":50966926524519,"sku":"23426739714267012811","price":19.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/13532698264728282778_2048.jpg?v=1793676457"},{"product_id":"constants-units-sticker-sheet","title":"Constants \u0026 Units Pixel Sticker Sheet – 12 Physics Stickers","description":"\u003cp\u003e12 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: Constants \u0026amp; Units. Every character below has its own sticker page with the full physics panel.\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/kibbla-kibble-balance\"\u003eKibbla\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA Kibble balance runs in two modes. In weighing mode, a current I in a coil sitting in a radial magnetic field pushes up with force BLI, balancing the weight mg. In moving mode, the same coil is moved at speed v and a voltage U = BLv is induced. The hard-to-measure product BL cancels between the two, leaving a match between mechanical and electrical power.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003emgv = UI\u003c\/strong\u003e\u003cbr\u003em: test mass (kg); g: local gravitational acceleration (m\/s²); v: coil velocity (m\/s); U: induced voltage (V); I: coil current (A)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/pumpette-single-electron-pump\"\u003ePumpette\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA single-electron pump is a nanoscale island (a quantum dot or metal island) cut off by tunnel barriers. Its charging energy is large enough that electrons can only cross one at a time (Coulomb blockade). Driving the gates at frequency f moves one electron per cycle, so the current is set by counting.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eI = ef\u003c\/strong\u003e\u003cbr\u003eI: current (A); e: elementary charge, 1.602 176 634 × 10⁻¹⁹ C exactly; f: pumping frequency (Hz)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/echobel-acoustic-gas-thermometer\"\u003eEchobel\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn acoustic gas thermometry, the resonant frequencies of a gas-filled spherical cavity of known size give the speed of sound. For a dilute monatomic gas the speed of sound fixes k\u003csub\u003eB\u003c\/sub\u003eT per atom. Measurements of this kind at the triple point of water (NPL, LNE, INRiM and others) gave the most precise values of the Boltzmann constant before the 2019 redefinition.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ec² = γk\u003csub\u003eB\u003c\/sub\u003eT\/m\u003c\/strong\u003e\u003cbr\u003ec: speed of sound in the zero-pressure limit (m\/s); γ: heat-capacity ratio, 5\/3 for an ideal monatomic gas; k\u003csub\u003eB\u003c\/sub\u003e: Boltzmann constant, 1.380 649 × 10⁻²³ J\/K exactly; T: thermodynamic temperature (K); m: mass of one gas atom (kg)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/quantix-flux-quantum\"\u003eQuantix\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn a superconducting ring the electrons pair up into Cooper pairs and share a single macroscopic wavefunction. The phase must come back to itself around the loop, so the fluxoid (the flux, plus a small supercurrent term) is quantised in whole multiples of Φ\u003csub\u003e0\u003c\/sub\u003e. In a type-II superconductor the field enters as Abrikosov vortices, each carrying one Φ\u003csub\u003e0\u003c\/sub\u003e, arranged in a triangular lattice.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eΦ\u003csub\u003e0\u003c\/sub\u003e = h\/2e ≈ 2.067 833 848 × 10⁻¹⁵ Wb\u003c\/strong\u003e\u003cbr\u003eΦ\u003csub\u003e0\u003c\/sub\u003e: magnetic flux quantum (Wb); h: Planck constant (exact); 2e: charge of a Cooper pair\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/magnetonix-bohr-magneton\"\u003eMagnetonix\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eAn electron with one unit of orbital angular momentum ħ has a magnetic moment of one Bohr magneton. This is the natural unit for atomic and electron magnetism.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eμ\u003csub\u003eB\u003c\/sub\u003e = eħ\/(2m\u003csub\u003ee\u003c\/sub\u003e) = 9.274 010 0657(29) × 10⁻²⁴ J\/T\u003c\/strong\u003e\u003cbr\u003eμ\u003csub\u003eB\u003c\/sub\u003e: Bohr magneton (J\/T); e: elementary charge (exact); ħ: reduced Planck constant (exact); m\u003csub\u003ee\u003c\/sub\u003e: electron mass (measured)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/bohra-bohr-radius\"\u003eBohra\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn hydrogen's ground state (1s), the radial probability density r²|ψ|² peaks at exactly a\u003csub\u003e0\u003c\/sub\u003e (with the proton treated as infinitely heavy). This is the atomic unit of length.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ea\u003csub\u003e0\u003c\/sub\u003e = 4πε\u003csub\u003e0\u003c\/sub\u003eħ²\/(m\u003csub\u003ee\u003c\/sub\u003ee²) = 5.291 772 105 44(82) × 10⁻¹¹ m ≈ 52.9 pm\u003c\/strong\u003e\u003cbr\u003ea\u003csub\u003e0\u003c\/sub\u003e: Bohr radius (m); ε\u003csub\u003e0\u003c\/sub\u003e: vacuum permittivity (measured since 2019); ħ: reduced Planck constant (exact); m\u003csub\u003ee\u003c\/sub\u003e: electron mass (measured); e: elementary charge (exact)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/gravitas-big-g\"\u003eGravitas\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eG sets the strength of gravity between any two masses. It is measured with torsion balances (Cavendish, 1798) and other lab methods, where the pull between kilogram-scale masses is about 10⁻⁷ N or less.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eG = 6.674 30(15) × 10⁻¹¹ m³ kg⁻¹ s⁻²\u003c\/strong\u003e\u003cbr\u003eG: Newtonian constant of gravitation; (15): standard uncertainty in the last two digits, 2.2 × 10⁻⁵ relative\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/weinberga-weak-mixing-angle\"\u003eWeinberga\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn the electroweak theory (Glashow, Weinberg, Salam), the neutral gauge fields B and W³ mix. Rotating them by the weak mixing angle θ\u003csub\u003eW\u003c\/sub\u003e gives the massless photon and the massive Z boson. The angle also links the couplings: e = g sin θ\u003csub\u003eW\u003c\/sub\u003e.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003esin²θ\u003csub\u003eW\u003c\/sub\u003e ≈ 0.231 (MS-bar at M\u003csub\u003eZ\u003c\/sub\u003e)\u003c\/strong\u003e\u003cbr\u003eθ\u003csub\u003eW\u003c\/sub\u003e: weak mixing (Weinberg) angle, about 28.7°; MS-bar: modified minimal-subtraction renormalisation scheme; M\u003csub\u003eZ\u003c\/sub\u003e: Z boson mass scale, about 91.19 GeV\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/landaura-conductance-quantum\"\u003eLandaura\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn a narrow ballistic conductor, such as a quantum point contact in a two-dimensional electron gas, electrons travel in a few transverse modes, like lanes on a road. Each fully open lane carries one conductance quantum, counting both spin states. Widening the constriction opens one lane at a time, so the conductance rises in a staircase. This was first seen in 1988 in GaAs point contacts (van Wees et al.; Wharam et al.).\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eG = (2e²\/h) Σ\u003csub\u003en\u003c\/sub\u003e T\u003csub\u003en\u003c\/sub\u003e,   G\u003csub\u003e0\u003c\/sub\u003e = 2e²\/h ≈ 7.748 091 729 × 10⁻⁵ S\u003c\/strong\u003e\u003cbr\u003eG: conductance (siemens, S); e: elementary charge, exact in the 2019 SI; h: Planck constant, exact in the 2019 SI; T\u003csub\u003en\u003c\/sub\u003e: transmission probability of mode n (0 to 1); 2: spin degeneracy, two spin states per lane; G\u003csub\u003e0\u003c\/sub\u003e: conductance quantum, exact (CODATA 2022)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/massratio-proton-electron\"\u003eMassratio\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eThe proton is about 1836 times heavier than the electron. This dimensionless ratio sets why atoms have a tiny heavy nucleus with a light electron cloud, why nuclei barely move in chemistry (the Born–Oppenheimer approximation), and the ratio of vibrational to electronic energies in molecules. It is measured with Penning-trap mass comparisons and precision spectroscopy of molecular hydrogen ions such as HD⁺.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eμ = m\u003csub\u003ep\u003c\/sub\u003e\/m\u003csub\u003ee\u003c\/sub\u003e = 1836.152 673 426(32)\u003c\/strong\u003e\u003cbr\u003eμ: proton-to-electron mass ratio (dimensionless); m\u003csub\u003ep\u003c\/sub\u003e: proton rest mass, measured; m\u003csub\u003ee\u003c\/sub\u003e: electron rest mass, measured; (32): standard uncertainty in the last two digits, about 1.7 × 10⁻¹¹ relative\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/thomsonia-thomson-cross-section\"\u003eThomsonia\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eWhen light meets a free electron at low energy, its electric field shakes the electron, which re-radiates at the same frequency. This elastic scattering has a fixed total cross section, the effective target area of one electron. It is J. J. Thomson's classical result, and it is the low-energy limit of quantum electrodynamics.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eσ\u003csub\u003eT\u003c\/sub\u003e = (8π\/3) r\u003csub\u003ee\u003c\/sub\u003e² = 6.652 458 7051(62) × 10⁻²⁹ m²\u003c\/strong\u003e\u003cbr\u003eσ\u003csub\u003eT\u003c\/sub\u003e: Thomson cross section; r\u003csub\u003ee\u003c\/sub\u003e: classical electron radius, e²\/(4πε\u003csub\u003e0\u003c\/sub\u003em\u003csub\u003ee\u003c\/sub\u003ec²) = 2.817 940 3205(13) × 10⁻¹⁵ m; e: elementary charge (exact); ε\u003csub\u003e0\u003c\/sub\u003e: vacuum permittivity (measured); m\u003csub\u003ee\u003c\/sub\u003e: electron mass (measured); c: speed of light (exact)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/candelor-candela-683\"\u003eCandelor\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eThe candela, the SI unit of luminous intensity, links watts of light to how bright that light looks to a standard human eye. It is defined by fixing the luminous efficacy of monochromatic radiation at 540 × 10¹² Hz (a green of about 555 nm) at exactly 683 lm\/W. For other wavelengths the watts are weighted by the CIE photopic sensitivity curve V(λ).\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eΦ\u003csub\u003ev\u003c\/sub\u003e = K\u003csub\u003em\u003c\/sub\u003e ∫ V(λ) Φ\u003csub\u003ee,λ\u003c\/sub\u003e dλ,   K\u003csub\u003ecd\u003c\/sub\u003e = 683 lm\/W\u003c\/strong\u003e\u003cbr\u003eΦ\u003csub\u003ev\u003c\/sub\u003e: luminous flux (lumen, lm); K\u003csub\u003ecd\u003c\/sub\u003e: luminous efficacy of 540 THz radiation, 683 lm\/W exactly (defining constant of the 2019 SI); K\u003csub\u003em\u003c\/sub\u003e: maximum photopic efficacy, ≈ 683 lm\/W; V(λ): CIE photopic luminous efficiency function, peak 1 near 555 nm; Φ\u003csub\u003ee,λ\u003c\/sub\u003e: spectral radiant flux (W\/nm); λ: wavelength (nm)\u003c\/p\u003e","brand":"Printify","offers":[{"title":"White \/ Kiss-Cut \/ 5.83″ × 8.27″","offer_id":50966934782055,"sku":"25166154548427143505","price":19.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/1506495440873399564_2048.jpg?v=1793676463"},{"product_id":"light-optics-sticker-sheet","title":"Light \u0026 Optics Pixel Sticker Sheet – 12 Physics Stickers","description":"\u003cp\u003e12 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: Light \u0026amp; Optics. Every character below has its own sticker page with the full physics panel.\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/scatterina-rayleigh-scattering-pixel-sticker\"\u003eScatterina\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eSmall particles and molecules scatter short wavelengths much more strongly, in proportion to one over wavelength to the fourth power. Blue light is scattered about 5.9 times more than red, which is why the sky looks blue.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eI\u003csub\u003es\u003c\/sub\u003e ∝ 1\/λ⁴\u003c\/strong\u003e\u003cbr\u003eI_s: power scattered by a molecule much smaller than the wavelength (W); λ: wavelength of the light (m)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/refracta-fermats-principle-pixel-sticker\"\u003eRefracta\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eLight takes the path between two points that takes the least time. That single rule gives Snell's law of refraction: light bends toward the slower medium, so that the sine of the angle divided by the speed is the same on both sides.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003esin θ\u003csub\u003e1\u003c\/sub\u003e\/v\u003csub\u003e1\u003c\/sub\u003e = sin θ\u003csub\u003e2\u003c\/sub\u003e\/v\u003csub\u003e2\u003c\/sub\u003e\u003c\/strong\u003e\u003cbr\u003eθ₁: angle of the path from the normal on the sand (light: in the first medium) (rad); θ₂: angle of the path from the normal in the water (light: in the second medium) (rad); v₁, v₂: speeds on each side (m\/s); n₁, n₂: refractive indices; c: speed of light in vacuum (m\/s); ds: element of path length (m)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/diagonix-maluss-law-pixel-sticker\"\u003eDiagonix\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA polarizer passes the fraction cos²θ of already-polarized light. Two crossed polarizers block everything, but slip a third one between them at 45° and one-eighth of the original unpolarized light gets through.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eI = I\u003csub\u003e0\u003c\/sub\u003e cos²θ\u003c\/strong\u003e\u003cbr\u003eI: intensity after one ideal polariser (W\/m²); I₀: intensity arriving (unpolarised for the three-polariser line) (W\/m²); θ: angle between the light's polarisation and the polariser's transmission axis (rad); I_(out): intensity after the vertical, 45° and horizontal polarisers in turn (W\/m²)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/azurwake-cherenkov-radiation-pixel-sticker\"\u003eAzurwake\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA charged particle moving through a medium faster than light travels in that medium emits light in a cone, like a sonic boom. The cone angle depends on the particle's speed and the refractive index, and Frank and Tamm's formula gives how many photons come out per unit length.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ecos θ = 1\/(nβ), β \u0026gt; 1\/n\u003c\/strong\u003e\u003cbr\u003eθ: angle between the emitted light and the particle's path (the wavefront cone has half-angle 90° − θ) (rad); n: refractive index of the medium, about 1.33 for water; β: particle speed as a fraction of c; N: number of photons emitted; x: distance travelled by the particle (m); λ: wavelength of the emitted light (m)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/halotra-22-degree-halo\"\u003eHalotra\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eThe real halo is made by millions of tiny hexagonal ice crystals in thin, high cirrus cloud, not by water droplets. Light entering one side face of a crystal and leaving by another passes through what is in effect a 60° prism. Every such prism has a minimum angle of deviation, and light piles up near it, so a bright ring appears at that angle from the Sun.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eD\u003csub\u003emin\u003c\/sub\u003e = 2 arcsin(n sin 30°) − 60°\u003c\/strong\u003e\u003cbr\u003eD\u003csub\u003emin\u003c\/sub\u003e: minimum deviation, the halo radius (degrees); n: refractive index of ice, about 1.31 for visible light; 30°: half the 60° prism angle between alternate faces\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/nicolette-nicol-prism\"\u003eNicolette\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn 1828 William Nicol of Edinburgh cut a calcite rhomb along a diagonal and cemented the halves back together with Canada balsam (Edinb. New Phil. J. 6, 83). Calcite splits light into an ordinary ray (n\u003csub\u003eo\u003c\/sub\u003e ≈ 1.658) and an extraordinary ray (n\u003csub\u003ee\u003c\/sub\u003e ≈ 1.486, along the slowest direction). Balsam sits between them at about 1.55. The ordinary ray, meeting the lower-index balsam at a steep angle, is totally internally reflected out of the side; the extraordinary ray passes. What leaves the end is linearly polarised.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003en\u003csub\u003eo\u003c\/sub\u003e \u0026gt; n\u003csub\u003ebalsam\u003c\/sub\u003e \u0026gt; n\u003csub\u003ee\u003c\/sub\u003e\u003c\/strong\u003e\u003cbr\u003en\u003csub\u003eo\u003c\/sub\u003e: ordinary refractive index of calcite, ≈ 1.658; n\u003csub\u003ebalsam\u003c\/sub\u003e: Canada balsam, ≈ 1.55; n\u003csub\u003ee\u003c\/sub\u003e: extraordinary index, ≈ 1.486 (sodium light)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/perota-fabry-perot\"\u003ePerota\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eTwo parallel, partly reflecting mirrors trap light that bounces back and forth. Only wavelengths that fit a whole number of half-waves into the gap build up and pass through; the rest cancel out. The transmission is a comb of sharp peaks, spaced by the free spectral range.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eFSR = c\/(2nL)\u003c\/strong\u003e\u003cbr\u003eFSR: free spectral range, the frequency spacing between peaks (Hz); c: speed of light (m\/s); n: refractive index between the mirrors; L: mirror spacing (m)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/tweezra-optical-tweezers\"\u003eTweezra\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eFocus a laser beam tightly through a microscope lens and a small transparent bead is pulled towards the brightest point and held there. Light bends as it passes through the bead, and the change in its momentum pushes back on the bead. For a bead that bends light more than its surroundings, the net gradient force points up the intensity gradient, towards the focus. Arthur Ashkin accelerated and trapped particles with radiation pressure in 1970 (Phys. Rev. Lett. 24, 156), and with colleagues demonstrated the single-beam gradient trap, the optical tweezers, in 1986 (Opt. Lett. 11, 288).\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eF\u003csub\u003egrad\u003c\/sub\u003e ∝ ∇I\u003c\/strong\u003e\u003cbr\u003eF\u003csub\u003egrad\u003c\/sub\u003e: gradient force on the bead (N); ∇I: gradient of the light intensity (W\/m³)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/pockella-pockels-effect\"\u003ePockella\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eSome crystals without a centre of symmetry change their refractive index in direct proportion to an applied electric field. Friedrich Pockels studied this linear electro-optic effect in 1893. Put such a crystal between crossed polarisers and a voltage can switch light on and off in nanoseconds.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eΔn ∝ E\u003c\/strong\u003e\u003cbr\u003eΔn: change in refractive index; E: applied electric field (V\/m)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/infrara-herschel-infrared\"\u003eInfrara\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn 1800 William Herschel placed thermometers in the colours of a prism spectrum and found the temperature rose from violet to red. Then he moved one past the red end, where no light could be seen, and it rose further still (“Experiments on the Refrangibility of the Invisible Rays of the Sun”, Phil. Trans. R. Soc. 90, 284). The Sun emits invisible rays beyond the red: infrared.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eλ\u003csub\u003eIR\u003c\/sub\u003e \u0026gt; λ\u003csub\u003ered\u003c\/sub\u003e ≈ 0.7 µm\u003c\/strong\u003e\u003cbr\u003eλ\u003csub\u003eIR\u003c\/sub\u003e: infrared wavelengths, about 0.7 µm to 1 mm; λ\u003csub\u003ered\u003c\/sub\u003e: long-wavelength end of the visible spectrum\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/mirrorspin-foucault-speed-of-light\"\u003eMirrorspin\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn 1862 Léon Foucault measured the speed of light in a laboratory with a rapidly rotating mirror (Comptes rendus 55, 501). Light reflects off the spinning mirror to a distant fixed mirror and back. In the round-trip time the rotating mirror turns a tiny angle, so the returning spot lands slightly displaced. From the displacement, the path length and the spin rate he got c = 298 000 ± 500 km\/s, close to the modern value.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eΔα = 2ωD\/c\u003c\/strong\u003e\u003cbr\u003eΔα: angle the mirror turns during the round trip (rad); the returning beam is deflected by 2Δα; ω: angular speed of the mirror (rad\/s); D: distance to the fixed mirror (m); c: speed of light (m\/s)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/christiaan-huygens-sticker-every-point-a-source-physics-sticker\"\u003eChristiaan Huygens\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eHuygens' principle treats every point on a wavefront as a source of small spherical wavelets. After a short time, the new wavefront is the surface that touches all of those wavelets, which explains reflection and refraction.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003er = vΔt\u003c\/strong\u003e\u003cbr\u003er: radius of each secondary wavelet after time \\Delta t (m); v: wave speed in the medium (m\/s); Δ t: time step (s)\u003c\/p\u003e","brand":"Printify","offers":[{"title":"White \/ Kiss-Cut \/ 5.83″ × 8.27″","offer_id":50966943006823,"sku":"20684385716945323770","price":19.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/11619963504391612442_2048.jpg?v=1793676469"},{"product_id":"cosmic-extremes-sticker-sheet","title":"Cosmic Extremes Pixel Sticker Sheet – 11 Physics Stickers","description":"\u003cp\u003e11 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: Cosmic Extremes. Every character below has its own sticker page with the full physics panel.\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/stephen-hawking-sticker-black-holes-been-there-bent-that-physics-sticker\"\u003eStephen Hawking\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eHawking found that black holes radiate like warm bodies, with a temperature inversely proportional to their mass. Bekenstein's entropy is proportional to the area of the horizon, not its volume, which links gravity, thermodynamics and quantum theory.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eT\u003csub\u003eH\u003c\/sub\u003e = ħc³\/(8πGMk\u003csub\u003eB\u003c\/sub\u003e)\u003c\/strong\u003e\u003cbr\u003eT_H: Hawking temperature (K); M: black-hole mass (kg); S_(BH): black-hole entropy (J\/K); A: horizon area (m²); ħ, c, G, k_B: fundamental constants (J·s, m\/s, m³·kg⁻¹·s⁻², J\/K)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/subrahmanyan-chandrasekhar-sticker-no-seconds-physics-sticker\"\u003eSubrahmanyan Chandrasekhar\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA white dwarf is held up by electron degeneracy pressure. Chandrasekhar showed that once the electrons become relativistic, this pressure can only support a limited mass, the Chandrasekhar limit. Above that, collapse cannot be stopped.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eM\u003csub\u003eCh\u003c\/sub\u003e ≈ (5.73\/μ\u003csub\u003ee\u003c\/sub\u003e²) M\u003csub\u003e☉\u003c\/sub\u003e\u003c\/strong\u003e\u003cbr\u003eM_(Ch): Chandrasekhar mass limit (idealised: cold, non-rotating, ideal electron gas) (kg); ω₃⁰: constant from the Lane–Emden n = 3 solution (about 2.018); ħ: reduced Planck constant (J·s); c: speed of light (m\/s); G: gravitational constant (m³·kg⁻¹·s⁻²); μₑ: mean molecular weight per electron\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/fritz-zwicky-sticker-too-fast-to-be-lit-physics-sticker\"\u003eFritz Zwicky\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn a gravitationally bound cluster, the galaxies' speeds are set by the total mass. Zwicky used their velocity spread to estimate the Coma cluster's mass and found it far larger than the light suggested.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eσ² ∼ GM\/R\u003c\/strong\u003e\u003cbr\u003eσ: line-of-sight velocity dispersion of galaxies (km\/s); G: gravitational constant (pc M_☉⁻¹ (km\/s)²); M: gravitating mass within radius R (M_☉); R: characteristic cluster radius (pc); M_(vir): virial mass implied by \\sigma and R (M_☉)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/vera-rubin-sticker-galaxy-rotation-she-saw-the-dark-side-physics-sticker\"\u003eVera Rubin\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIf most of a galaxy's mass were in its bright centre, stars far out would orbit more slowly. Vera Rubin found the speeds stay roughly constant far out, which means the enclosed mass keeps growing with radius: unseen matter.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ev(r)² = GM(r)\/r\u003c\/strong\u003e\u003cbr\u003ev(r): circular orbital speed (m\/s); M(r): enclosed mass (kg); r: radius (m); G: gravitational constant (m³·kg⁻¹·s⁻²)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/jocelyn-bell-burnell-sticker-she-found-pulsars-they-gave-her-silence-physics-sticker\"\u003eJocelyn Bell Burnell\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA pulsar's very regular pulses mean it must be extremely dense to spin that fast without flying apart, which points to a neutron star. Radio pulses at lower frequencies arrive later after passing through interstellar electrons, and that delay measures how many electrons lie along the way.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eρ̄ ≳ 3π\/(GP²)\u003c\/strong\u003e\u003cbr\u003eρ̄: minimum mean density for a body spinning with period P not to fly apart (kg\/m³); G: gravitational constant (m³·kg⁻¹·s⁻²); P: rotation period (s); Δ t: arrival-time delay between frequencies f₁ \u0026lt; f₂ (s); DM: dispersion measure, ∫ nₑ dl (free electrons per area along the line of sight) (m⁻²); f₁, f₂: observing frequencies (Hz)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/relicta-cosmic-microwave-background-pixel-sticker\"\u003eRelicta\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eThe cosmic microwave background is light left over from when the universe became transparent. It has a nearly perfect blackbody spectrum, and its temperature has fallen in proportion to the stretching of space, to about 2.7 K today.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eT(z) = T\u003csub\u003e0\u003c\/sub\u003e(1 + z)\u003c\/strong\u003e\u003cbr\u003eT(z): temperature of the background radiation at redshift z (K); z: redshift (last scattering: about 1090); T₀: temperature today (Fixsen 2009: 2.72548 ± 0.00057 K) (K); B_ν: blackbody spectral radiance per unit frequency (W·m⁻²·Hz⁻¹·sr⁻¹); ν: frequency (Hz); h: Planck constant (J·s)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/primordix-big-bang-nucleosynthesis-pixel-sticker\"\u003ePrimordix\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn the first minutes after the Big Bang, the ratio of neutrons to protons froze out at a small fixed value. Nearly all those neutrons ended up in helium-4, which predicts the fraction of ordinary mass that came out as helium, matching observations.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eY\u003csub\u003ep\u003c\/sub\u003e ≃ 2(n\/p)\/(1 + n\/p)\u003c\/strong\u003e\u003cbr\u003eY_p: primordial helium-4 fraction by mass (observed 0.245 ± 0.003); n\/p: neutron-to-proton ratio; Q: rest-energy difference between neutron and proton (MeV); k_B: Boltzmann constant (eV\/K); T_f: temperature when the weak reactions freeze out (thermal energy about 0.7–0.8 MeV) (K); m_n, m_p: neutron and proton masses (kg)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/skyrider-hess-balloon\"\u003eSkyrider\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn 1912 Victor Hess carried electroscopes up in balloons to 5 350 m (Phys. Z. 13, 1084). Their discharge rate, a measure of ionising radiation, first dipped slightly, then climbed steeply with height. The radiation was coming from above, not from the ground. A flight during a near-total solar eclipse and at night showed no drop, so the Sun was not the source. He shared the 1936 Nobel Prize for cosmic rays.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eionisation rate rises with altitude\u003c\/strong\u003e\u003cbr\u003ealtitude: height above sea level, up to 5 350 m in 1912; ionisation rate: ions formed per cm³ per second in the electroscope\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/mirrorix-ligo-test-mass\"\u003eMirrorix\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eLIGO measures gravitational waves by comparing the lengths of two 4 km arms with laser light. At the end of each arm hangs a test mass: a 40 kg fused-silica mirror 34 cm across, hung from a four-stage pendulum to isolate it from ground vibration (Advanced LIGO, Class. Quantum Grav. 32, 074001, 2015). On 14 September 2015 a wave from two merging black holes, GW150914, stretched and squeezed the arms with a peak strain of 1.0 × 10⁻²¹ (Phys. Rev. Lett. 116, 061102, 2016).\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eh ≈ ΔL\/L\u003c\/strong\u003e\u003cbr\u003eh: gravitational-wave strain, dimensionless; ΔL: change in the difference of the arm lengths (m); L: arm length, 4 km\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/muonaut-muon-time-dilation-pixel-sticker\"\u003eMuonaut\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eMuons made high in the atmosphere live only about two microseconds, too short to cross the atmosphere at face value. Moving near the speed of light, their clocks run slow by the factor γ, so many reach the ground; in the muon's frame, the atmosphere is shorter instead.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eΔt = γτ, L = L\u003csub\u003e0\u003c\/sub\u003e\/γ\u003c\/strong\u003e\u003cbr\u003eΔ t: muon lifetime as timed by clocks on the ground (s); τ: mean lifetime in the muon's own rest frame (PDG 2024: 2.1969811 μs; the half-life is shorter, 1.52 μs) (s); γ: Lorentz factor; L₀: thickness of the atmosphere in the ground frame (m); L: the same thickness measured in the muon's frame (m); v: muon speed (m\/s)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/homestaka-solar-neutrino-problem\"\u003eHomestaka\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eRaymond Davis Jr.'s experiment in the Homestake gold mine, South Dakota, 1478 m underground, used a tank of about 615 tonnes of perchloroethylene (tetrachloroethylene, C₂Cl₄). A solar electron neutrino can turn a chlorine-37 nucleus into argon-37 (threshold 0.814 MeV), and the few argon atoms were extracted and counted. From 1970 to 1994 it measured about a third of the rate predicted by John Bahcall's standard solar model.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003emeasured 2.56 ± 0.16 ± 0.16 SNU   vs   predicted 7.6\u003csup\u003e+1.3\u003c\/sup\u003e\u003csub\u003e−1.1\u003c\/sub\u003e SNU\u003c\/strong\u003e\u003cbr\u003eSNU: solar neutrino unit: 10⁻³⁶ captures per target atom per second; 2.56: final Homestake result (Cleveland et al., ApJ 496, 505, 1998), statistical and systematic errors; 7.6: chlorine prediction of the BP2000 standard solar model (Bahcall, Pinsonneault and Basu, 2001)\u003c\/p\u003e","brand":"Printify","offers":[{"title":"White \/ Kiss-Cut \/ 5.83″ × 8.27″","offer_id":50966951231591,"sku":"31025817950228738687","price":19.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/16428687048937592307_2048.jpg?v=1793676476"},{"product_id":"weird-quantum-sticker-sheet","title":"Weird Quantum Pixel Sticker Sheet – 12 Physics Stickers","description":"\u003cp\u003e12 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: Weird Quantum. Every character below has its own sticker page with the full physics panel.\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/revolvix-spinor-rotation-pixel-sticker\"\u003eRevolvix\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eRotating a spin-½ particle by one full turn multiplies its wavefunction by −1; only two full turns bring it back to its original state. Neutron interferometers measured this sign change directly in 1975.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eR(2π)ψ = −ψ\u003c\/strong\u003e\u003cbr\u003eR(θ): operator that rotates a spin-1\/2 state by angle θ about the axis n; ψ: two-component spinor state of a spin-1\/2 particle; n̂: unit vector along the rotation axis; σ: the three Pauli matrices (σx, σy, σz); 2π, 4π: one full turn (360°) and two full turns (720°) (rad)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/bifurcor-stern-gerlach-experiment-pixel-sticker\"\u003eBifurcor\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA magnetic field that changes with height pushes on an atom's magnetic moment. Because an electron's spin can only point up or down, silver atoms in the Stern–Gerlach magnet split into exactly two beams instead of smearing out.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eF\u003csub\u003ez\u003c\/sub\u003e = μ\u003csub\u003ez\u003c\/sub\u003e ∂B\u003csub\u003ez\u003c\/sub\u003e\/∂z\u003c\/strong\u003e\u003cbr\u003eF_z: force on the atom along the field gradient (N); μ_z: z-component of the atom's magnetic moment (for silver, set by its one outer electron) (J\/T); B_z: z-component of the magnetic field (T); z: position along the gradient, up the page (m); g_s: electron spin g-factor, about 2.0023; μ_B: Bohr magneton (J\/T)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/casimira-casimir-effect-pixel-sticker\"\u003eCasimira\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003ePut two uncharged, perfectly conducting parallel plates a small distance a apart in vacuum and they attract. Between the plates only electromagnetic modes that fit the boundary conditions are allowed; outside, every mode is. Summing the zero-point energies ½ħω of the modes, and subtracting the free-space value, leaves a finite energy that falls as the plates approach. Hendrik Casimir derived it in 1948.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eP = π²ħc \/ (240 a⁴)\u003c\/strong\u003e\u003cbr\u003eP: attractive pressure, force per unit plate area (Pa); ħ: reduced Planck constant, h\/2π = 1.054 571 817 × 10⁻³⁴ J s; c: speed of light, 299 792 458 m s⁻¹; a: gap between the plates (m); π²\/240: numerical factor from summing the allowed modes\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/delaychoice-wheeler-delayed-choice\"\u003eDelaychoice\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn a Mach–Zehnder interferometer, a single photon meets a first beam splitter and then two paths. If a second beam splitter recombines the paths, the photon shows interference: which detector clicks depends on the phase difference. If the second splitter is removed, each detector clicks half the time, as if the photon took one path. John Wheeler proposed in 1978 making this choice after the photon has passed the first splitter. Jacques et al. (Science 315, 966, 2007) did it with single photons and a random choice that was space-like separated from the photon's entry.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ewith splitter: P\u003csub\u003e1\u003c\/sub\u003e = cos²(φ\/2), P\u003csub\u003e2\u003c\/sub\u003e = sin²(φ\/2);   without: P\u003csub\u003e1\u003c\/sub\u003e = P\u003csub\u003e2\u003c\/sub\u003e = ½\u003c\/strong\u003e\u003cbr\u003eP\u003csub\u003e1\u003c\/sub\u003e, P\u003csub\u003e2\u003c\/sub\u003e: probabilities that detector 1 or 2 clicks; φ: phase difference between the two arms (rad)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/kapitzadirac-electron-diffraction-light\"\u003eKapitzadirac\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eTwo laser beams travelling in opposite directions form a standing wave, with bright and dark bands half a wavelength apart. Pyotr Kapitza and Paul Dirac predicted in 1933 (“The reflection of electrons from standing light waves”, Proc. Camb. Phil. Soc. 29, 297) that electrons crossing such a wave would be diffracted, as if by a grating. The electrons are pushed away from the bright bands by the ponderomotive force, and the regular pattern of bands acts as a periodic grating.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ed = λ \/ 2\u003c\/strong\u003e\u003cbr\u003ed: period of the light grating (m); λ: wavelength of the laser light (m)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/feeblea-taylor-feeble-light\"\u003eFeeblea\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn 1909 a young G. I. Taylor photographed the shadow of a needle with light so dim that, in modern terms, rarely more than one photon was in the box at a time (Proc. Camb. Phil. Soc. 15, 114). His longest exposure ran about 2000 hours, roughly three months. The diffraction fringes were just as sharp as with bright light.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003et\u003csub\u003eexposure\u003c\/sub\u003e ≈ 2000 h\u003c\/strong\u003e\u003cbr\u003et\u003csub\u003eexposure\u003c\/sub\u003e: exposure time of the faintest photograph\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/condensa-bose-einstein-condensate-pixel-sticker\"\u003eCondensa\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eBelow a critical temperature, a gas of bosons starts collecting in its lowest quantum state. The fraction in that state grows as the temperature drops, following 1 − (T\/Tc)^(3\/2), and the critical temperature depends on the gas's density and particle mass.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eN\u003csub\u003e0\u003c\/sub\u003e\/N = 1 − (T\/T\u003csub\u003ec\u003c\/sub\u003e)\u003csup\u003e3\/2\u003c\/sup\u003e (T ≤ T\u003csub\u003ec\u003c\/sub\u003e)\u003c\/strong\u003e\u003cbr\u003eN₀: number of bosons in the ground state; N: total number of bosons; T: temperature (K); T_c: critical temperature for condensation (K); n: number density (m⁻³); m: boson mass (kg)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/invertia-negative-absolute-temperature-pixel-sticker\"\u003eInvertia\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eTemperature is defined by how entropy changes when you add energy. In a system with a top energy level, like nuclear spins, putting more particles in the upper state than the lower one makes that slope negative: a negative absolute temperature, which is actually 'hotter than infinity'.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003e1\/T = ∂S\/∂E \u0026lt; 0\u003c\/strong\u003e\u003cbr\u003eT: absolute temperature, in the standard (Boltzmann) definition (K); S: entropy (Boltzmann form, from the number of states at energy E) (J\/K); E: total energy of the spin system (J); N_↑, N_↓: number of particles in the upper and the lower level; Δ E: energy gap between the two levels (J); k_B: Boltzmann constant (J\/K)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/expulsieur-meissner-effect-pixel-sticker\"\u003eExpulsieur\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA superconductor expels magnetic field from its interior, which a merely perfect conductor would not do. The field only penetrates a thin layer, dying away exponentially over the London penetration depth.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eB\u003csub\u003einside\u003c\/sub\u003e = 0\u003c\/strong\u003e\u003cbr\u003eB_(inside): magnetic field in the bulk of the superconductor (applied field below the critical field) (T); B(x): field at depth x below the surface (T); B₀: field at the surface (T); x: depth below the surface (m); λ_L: London penetration depth (m); m: electron mass (kg)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/berryphase-geometric-phase\"\u003eBerryphase\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eCarry a quantum system slowly round a closed loop of its settings and it returns to its starting state, apart from a phase. Michael Berry showed in 1984 that part of this phase depends only on the geometry of the loop, not on how long it takes (Proc. R. Soc. A 392, 45). For a spin-½ in the state aligned with a magnetic field (m = +½), with the field direction swept slowly round a closed loop, the geometric phase is minus half the solid angle that loop encloses: γ = −mΩ = −Ω\/2. The anti-aligned state (m = −½) picks up +Ω\/2.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eγ = −Ω\/2\u003c\/strong\u003e\u003cbr\u003eγ: geometric (Berry) phase of the aligned state (rad); Ω: solid angle enclosed by the loop traced by the field direction, seen from the origin (sr); m: spin projection along the field in units of ħ (dimensionless; +½ aligned, −½ anti-aligned)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/dicka-superradiance\"\u003eDicka\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eN excited atoms that radiate independently give N times the light of one. But if they are packed close together, within about a wavelength, or otherwise coupled to the same light field, they can lock into phase and radiate as one giant dipole. Then the peak intensity grows as N², and the light comes out in a short, bright burst. Robert Dicke predicted this in “Coherence in Spontaneous Radiation Processes” (Phys. Rev. 93, 99, 1954).\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eI\u003csub\u003epeak\u003c\/sub\u003e ∝ N²\u003c\/strong\u003e\u003cbr\u003eI\u003csub\u003epeak\u003c\/sub\u003e: peak emitted intensity (W); N: number of emitters radiating in phase\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/wolfgang-pauli-sticker-one-per-state-physics-sticker\"\u003eWolfgang Pauli\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eTwo identical fermions must have a wavefunction that changes sign when they are swapped. If both tried to occupy the same state, the wavefunction would be zero everywhere, so the arrangement is forbidden: one fermion per state.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eψ(1,2) = [φ\u003csub\u003ea\u003c\/sub\u003e(1)φ\u003csub\u003eb\u003c\/sub\u003e(2) − φ\u003csub\u003ea\u003c\/sub\u003e(2)φ\u003csub\u003eb\u003c\/sub\u003e(1)]\/√2\u003c\/strong\u003e\u003cbr\u003eψ: two-particle antisymmetrised wavefunction; φₐ, φ_b: single-particle orbitals (including spin); a, b: full sets of quantum numbers (n,ℓ,m_ℓ,m_s)\u003c\/p\u003e","brand":"Printify","offers":[{"title":"White \/ Kiss-Cut \/ 5.83″ × 8.27″","offer_id":50966951297127,"sku":"29705037365584945896","price":19.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/7722761340991221139_2048.jpg?v=1793676482"},{"product_id":"discoveries-that-changed-physics-sticker-sheet","title":"Discoveries That Changed Physics Pixel Sticker Sheet – 11 Physics Stickers","description":"\u003cp\u003e11 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: Discoveries That Changed Physics. Every character below has its own sticker page with the full physics panel.\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/fogplate-becquerel\"\u003eFogplate\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eHenri Becquerel suspected that uranium salts, made to glow by sunlight, might give off X-rays. Paris stayed cloudy in late February 1896, so he put plates wrapped in black paper, with uranium salt on top, in a drawer. On 1 March he developed them anyway and found strong images: the uranium emitted penetrating rays with no sunlight at all (Comptes rendus 122, 501, 1896). It was radioactivity.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003euranium → rays, no light needed\u003c\/strong\u003e\u003cbr\u003euranium salt: potassium uranyl sulfate in his first tests; plate: photographic plate wrapped in black paper\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/corpuscle-jj-thomson\"\u003eCorpuscle\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn 1897 J. J. Thomson crossed electric and magnetic fields over a cathode-ray beam (“Cathode Rays”, Phil. Mag. 44, 293). He tuned them so the beam ran straight, which gives its speed, v = E\/B; then the deflection from one field alone gives the mass-to-charge ratio. The result was about a thousand times smaller than for the hydrogen ion, whatever the gas or electrode metal. He called the particles corpuscles: we call them electrons.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ev = E\/B\u003c\/strong\u003e\u003cbr\u003ev: beam speed (m\/s); E: electric field (V\/m); B: magnetic field (T); m\/e: mass-to-charge ratio of the particle (kg\/C)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/ernest-rutherford-sticker-some-came-back-physics-sticker\"\u003eErnest Rutherford\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eRutherford fired alpha particles at gold foil and some bounced almost straight back. His formula, with its steep dependence on 1\/sin⁴(θ\/2), only works if the positive charge sits in a tiny, dense nucleus.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003edσ\/dΩ = (zZe²\/(16πε\u003csub\u003e0\u003c\/sub\u003eE))² \/ sin⁴(θ\/2)\u003c\/strong\u003e\u003cbr\u003edσ\/dΩ: differential cross section per solid angle (m²\/sr); z: charge number of the alpha particle (2); Z: charge number of the target nucleus (gold: 79); e: elementary charge (C); ε₀: vacuum permittivity (F\/m); E: kinetic energy of the alpha particle (J)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/splitterra-cockcroft-walton\"\u003eSplitterra\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn April 1932 at the Cavendish Laboratory, John Cockcroft and Ernest Walton fired protons from a high-voltage accelerator at lithium. Scintillation screens flashed with pairs of alpha particles flying apart (Proc. R. Soc. A 137, 229, 1932). It was the first nucleus split by artificially accelerated particles, and the energy released matched E = mc².\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ep + ⁷Li → 2α + 17.3 MeV\u003c\/strong\u003e\u003cbr\u003ep: proton; ⁷Li: lithium-7 nucleus; α: alpha particle, a helium-4 nucleus\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/argonix-rayleigh-ramsay\"\u003eArgonix\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eLord Rayleigh weighed nitrogen with great care and found that nitrogen taken from air was about half a percent denser than nitrogen made from ammonia, a gap far larger than his errors. With William Ramsay he removed all the nitrogen and oxygen from air and found a stubborn leftover gas, about 1% of the atmosphere, with a new spectrum and no chemistry at all. They named it argon, ‘the lazy one’ (“Argon, a New Constituent of the Atmosphere”, Phil. Trans. A 186, 187, 1895).\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eρ\u003csub\u003eair N₂\u003c\/sub\u003e ≈ 1.005 ρ\u003csub\u003echem N₂\u003c\/sub\u003e\u003c\/strong\u003e\u003cbr\u003eρ\u003csub\u003eair N₂\u003c\/sub\u003e: density of ‘nitrogen’ from air, which still contains argon; ρ\u003csub\u003echem N₂\u003c\/sub\u003e: density of nitrogen from chemical sources\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/infrara-herschel-infrared\"\u003eInfrara\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn 1800 William Herschel placed thermometers in the colours of a prism spectrum and found the temperature rose from violet to red. Then he moved one past the red end, where no light could be seen, and it rose further still (“Experiments on the Refrangibility of the Invisible Rays of the Sun”, Phil. Trans. R. Soc. 90, 284). The Sun emits invisible rays beyond the red: infrared.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eλ\u003csub\u003eIR\u003c\/sub\u003e \u0026gt; λ\u003csub\u003ered\u003c\/sub\u003e ≈ 0.7 µm\u003c\/strong\u003e\u003cbr\u003eλ\u003csub\u003eIR\u003c\/sub\u003e: infrared wavelengths, about 0.7 µm to 1 mm; λ\u003csub\u003ered\u003c\/sub\u003e: long-wavelength end of the visible spectrum\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/skyrider-hess-balloon\"\u003eSkyrider\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn 1912 Victor Hess carried electroscopes up in balloons to 5 350 m (Phys. Z. 13, 1084). Their discharge rate, a measure of ionising radiation, first dipped slightly, then climbed steeply with height. The radiation was coming from above, not from the ground. A flight during a near-total solar eclipse and at night showed no drop, so the Sun was not the source. He shared the 1936 Nobel Prize for cosmic rays.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eionisation rate rises with altitude\u003c\/strong\u003e\u003cbr\u003ealtitude: height above sea level, up to 5 350 m in 1912; ionisation rate: ions formed per cm³ per second in the electroscope\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/aberra-bradley-aberration\"\u003eAberra\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eFrom 1725 James Bradley tracked the star Gamma Draconis and found it shifted through the year by about 20 arcseconds, but in a pattern that did not fit parallax. In 1729 he explained it (Phil. Trans. R. Soc. 35, 637): Earth is moving, so the telescope must be tilted slightly forward to catch starlight, like slanting an umbrella when you walk in rain. The tilt is set by Earth’s orbital speed compared with the speed of light.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eθ ≈ v\/c ≈ 20.5″\u003c\/strong\u003e\u003cbr\u003eθ: aberration angle (rad, here ≈ 20.5 arcseconds); v: Earth’s orbital speed, ≈ 30 km\/s; c: speed of light (m\/s)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/ole-romer-sticker-io-isnt-late-light-is-physics-sticker\"\u003eOle Rømer\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIo's eclipses behind Jupiter ran later when Earth was farther from Jupiter, because the light had farther to travel. Rømer used that delay to show that light has a finite speed, crossing the width of Earth's orbit in a matter of minutes.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eΔt = Δd\/c\u003c\/strong\u003e\u003cbr\u003eΔ t: extra light travel time (s); Δ d: extra distance light must travel (m); c: speed of light in vacuum (m\/s); AU: astronomical unit, about the mean Earth–Sun distance (exactly 149 597 870 700 m since 2012) (m)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/homestaka-solar-neutrino-problem\"\u003eHomestaka\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eRaymond Davis Jr.'s experiment in the Homestake gold mine, South Dakota, 1478 m underground, used a tank of about 615 tonnes of perchloroethylene (tetrachloroethylene, C₂Cl₄). A solar electron neutrino can turn a chlorine-37 nucleus into argon-37 (threshold 0.814 MeV), and the few argon atoms were extracted and counted. From 1970 to 1994 it measured about a third of the rate predicted by John Bahcall's standard solar model.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003emeasured 2.56 ± 0.16 ± 0.16 SNU   vs   predicted 7.6\u003csup\u003e+1.3\u003c\/sup\u003e\u003csub\u003e−1.1\u003c\/sub\u003e SNU\u003c\/strong\u003e\u003cbr\u003eSNU: solar neutrino unit: 10⁻³⁶ captures per target atom per second; 2.56: final Homestake result (Cleveland et al., ApJ 496, 505, 1998), statistical and systematic errors; 7.6: chlorine prediction of the BP2000 standard solar model (Bahcall, Pinsonneault and Basu, 2001)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/galileo-galilei-sticker-telescope-tea-trouble-with-the-church-physics-sticker\"\u003eGalileo Galilei\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eGalileo found that a falling body's distance grows with the square of the time. In equal time intervals the distances covered go in the ratio of the odd numbers 1, 3, 5 and so on, which is the signature of constant acceleration.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ed = ½gt²\u003c\/strong\u003e\u003cbr\u003ed: distance fallen from rest (m); g: gravitational acceleration (m\/s²); t: time (s); d_n: distance covered in the n-th equal time interval (m)\u003c\/p\u003e","brand":"Printify","offers":[{"title":"White \/ Kiss-Cut \/ 5.83″ × 8.27″","offer_id":50966951362663,"sku":"87767616830786361956","price":19.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/9567080901688241067_2048.jpg?v=1793676488"},{"product_id":"everyday-physics-sticker-sheet","title":"Everyday Physics Pixel Sticker Sheet – 12 Physics Stickers","description":"\u003cp\u003e12 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: Everyday Physics. Every character below has its own sticker page with the full physics panel.\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/cheeriosa-cheerios-effect\"\u003eCheeriosa\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eFloating cereal clumps together and sticks to the bowl's edge. It is not magnetic or electric: each floating ring bends the liquid surface into a meniscus. A neighbouring ring sitting on that tilted surface is pushed along it, by buoyancy and surface tension, towards the other, so rings drift together. Dominic Vella and L. Mahadevan analysed this in “The ‘Cheerios effect’” (Am. J. Phys. 73, 817, 2005). The distance over which a meniscus reaches is set by the capillary length.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eℓ\u003csub\u003ec\u003c\/sub\u003e = √(γ \/ ρg)\u003c\/strong\u003e\u003cbr\u003eℓ\u003csub\u003ec\u003c\/sub\u003e: capillary length (m); γ: surface tension (N\/m); ρ: liquid density (kg\/m³); g: gravitational acceleration (m\/s²)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/winetear-tears-of-wine\"\u003eWinetear\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eSwirl a glass of strong wine and a thin film climbs the wall above the liquid, gathers into a ridge and falls back as drops: tears, or legs. Alcohol evaporates from the film faster than water, so the film becomes more watery, and water has a higher surface tension than alcohol. The difference in surface tension pulls liquid up the glass from the bulk below, where there is more alcohol. James Thomson, Lord Kelvin's brother, described it in 1855 (“On certain curious motions observable at the surfaces of wine and other alcoholic liquors”, Phil. Mag. 10, 330). A flow driven like this is now called a Marangoni flow.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eτ = ∇γ\u003c\/strong\u003e\u003cbr\u003eτ: shear stress on the liquid surface (Pa); ∇γ: gradient of surface tension along the surface (N\/m²)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/teacupa-tea-leaf-paradox\"\u003eTeacupa\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn a stirred cup, the swirl is held in balance by a pressure that rises toward the wall (the surface is higher there). Near the bottom, friction slows the swirl, but the pressure difference set by the surface is still there. So near the floor the inward pressure push wins, and water flows toward the centre, rises up the middle and returns outward near the top. This secondary flow carries the sunken leaves to the centre, where they settle.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003e∂p\/∂r = ρ v\u003csub\u003eθ\u003c\/sub\u003e²\/r\u003c\/strong\u003e\u003cbr\u003ep: pressure (Pa); r: distance from the axis (m); ρ: density of the water (kg\/m³); v\u003csub\u003eθ\u003c\/sub\u003e: swirl (azimuthal) speed (m\/s)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/scatterina-rayleigh-scattering-pixel-sticker\"\u003eScatterina\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eSmall particles and molecules scatter short wavelengths much more strongly, in proportion to one over wavelength to the fourth power. Blue light is scattered about 5.9 times more than red, which is why the sky looks blue.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eI\u003csub\u003es\u003c\/sub\u003e ∝ 1\/λ⁴\u003c\/strong\u003e\u003cbr\u003eI_s: power scattered by a molecule much smaller than the wavelength (W); λ: wavelength of the light (m)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/halotra-22-degree-halo\"\u003eHalotra\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eThe real halo is made by millions of tiny hexagonal ice crystals in thin, high cirrus cloud, not by water droplets. Light entering one side face of a crystal and leaving by another passes through what is in effect a 60° prism. Every such prism has a minimum angle of deviation, and light piles up near it, so a bright ring appears at that angle from the Sun.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eD\u003csub\u003emin\u003c\/sub\u003e = 2 arcsin(n sin 30°) − 60°\u003c\/strong\u003e\u003cbr\u003eD\u003csub\u003emin\u003c\/sub\u003e: minimum deviation, the halo radius (degrees); n: refractive index of ice, about 1.31 for visible light; 30°: half the 60° prism angle between alternate faces\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/terminara-terminal-velocity-pixel-sticker\"\u003eTerminara\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA falling object speeds up until air drag, which grows with the square of speed, balances its weight. From then on it falls at a steady terminal velocity, approaching it along a tanh curve.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003em dv\/dt = mg − ½ρC\u003csub\u003ed\u003c\/sub\u003eAv²\u003c\/strong\u003e\u003cbr\u003em: mass of the skydiver (here 80 kg) (kg); v: falling speed (m\/s); t: time since leaving the plane, falling from rest (s); g: gravitational acceleration (m\/s²); ρ: air density (here 1.225 kg\/m^3, sea-level standard) (kg\/m³); C_d A: drag coefficient times frontal area (here 0.45 m^2, belly-down) (m²)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/magnara-magnus-effect-pixel-sticker\"\u003eMagnara\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA spinning ball moving through air is pushed sideways, toward the side where its surface moves with the airflow. That Magnus force, proportional to spin crossed with velocity, makes spinning balls curve.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003e\u003cb\u003eF\u003c\/b\u003e\u003csub\u003eM\u003c\/sub\u003e = α \u003cb\u003eω\u003c\/b\u003e×\u003cb\u003ev\u003c\/b\u003e\u003c\/strong\u003e\u003cbr\u003eF_M: Magnus force on the spinning ball (N); α: empirical coefficient (depends on size, spin regime, fluid) (kg); ω: spin angular-velocity vector (rad\/s); v: centre-of-mass velocity through the air (m\/s); m: mass of the ball (kg); b: linear drag coefficient (kg\/s)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/saltara-hydraulic-jump-pixel-sticker\"\u003eSaltara\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA fast, shallow flow can suddenly jump to a slow, deep one. Momentum conservation across the jump gives the ratio of the two depths in terms of the incoming Froude number, which compares flow speed with the speed of shallow-water waves.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eFr = v\/√(gh)\u003c\/strong\u003e\u003cbr\u003eFr: Froude number: flow speed over the speed of shallow-water waves; v: depth-averaged flow speed (m\/s); g: gravitational acceleration (m\/s²); h: depth of the flow (m); h₁, h₂: depths just before and just after a planar jump in a straight channel (m); Fr₁: Froude number of the incoming flow\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/orbitick-gps-clock-correction-pixel-sticker\"\u003eOrbitick\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eGPS clocks feel weaker gravity than clocks on the ground, so they run fast, and their orbital speed makes them run slightly slow. The net effect would build up timing errors quickly, so the satellite clocks are deliberately set to tick slightly slower before launch.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eΔf\/f ≈ ΔΦ\/c² − v²\/(2c²)\u003c\/strong\u003e\u003cbr\u003eΔ f\/f: fractional rate of a GPS satellite clock relative to a clock on the geoid (positive: it runs fast); ΔΦ: gravitational potential at the satellite minus the effective potential on the geoid, which includes Earth's rotation term (J\/kg); v: orbital speed of the satellite (m\/s); c: speed of light in vacuum (m\/s); f₀: factory-specified frequency of the clock's 10.23 MHz reference before launch (Hz)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/lenzor-lenzs-law-pixel-sticker\"\u003eLenzor\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA changing magnetic flux induces a voltage, and the induced current always flows so as to oppose the change. A magnet dropped through a copper pipe therefore drives eddy currents that brake it, so it drifts down at a steady terminal speed.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eℰ = −dΦ\u003csub\u003eB\u003c\/sub\u003e\/dt\u003c\/strong\u003e\u003cbr\u003emathcalE: EMF induced around one ring of the pipe (V); Φ_B: magnetic flux through that ring (Wb); t: time (s); m: mass of the falling magnet (kg); g: gravitational acceleration (m\/s²); F_(eddy): upward drag from the eddy currents; it grows with the magnet's speed (N)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/spinwell-angular-momentum-pixel-sticker\"\u003eSpinwell\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eAngular momentum is moment of inertia times angular speed, and without an outside torque it stays constant. Pull your arms in and your moment of inertia drops, so you spin faster.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eL = Iω\u003c\/strong\u003e\u003cbr\u003eL: angular momentum about the spin axis (kg m² \/ s); I: moment of inertia about that axis (kg m²); ω: angular speed (rad\/s); τ: external torque about the axis (N m)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/vortessa-karman-vortex-street-pixel-sticker\"\u003eVortessa\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eFlow past a cylinder sheds vortices from alternate sides, at a frequency set by the flow speed divided by the cylinder's diameter, times the Strouhal number. Von Kármán showed that the staggered vortex street is stable only for a particular spacing ratio.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ef = St·U\/D, St ≈ 0.2\u003c\/strong\u003e\u003cbr\u003ef: shedding frequency: full left-right cycles per second (Hz); St: Strouhal number, about 0.2 to 0.21 for Re from about 300 to 200,000 (lower near onset); U: speed of the oncoming flow (m\/s); D: diameter of the cylinder (m); Re: Reynolds number; ν: kinematic viscosity of the fluid (m²\/s)\u003c\/p\u003e","brand":"Printify","offers":[{"title":"White \/ Kiss-Cut \/ 5.83″ × 8.27″","offer_id":50966959587431,"sku":"23338763287246566533","price":19.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/12688476124670001992_2048.jpg?v=1793676494"},{"product_id":"classic-lab-experiments-sticker-sheet","title":"Classic Lab Experiments Pixel Sticker Sheet – 12 Physics Stickers","description":"\u003cp\u003e12 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: Classic Lab Experiments. Every character below has its own sticker page with the full physics panel.\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/wimsha-wimshurst-machine\"\u003eWimsha\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn the early 1880s James Wimshurst, a London shipwright and amateur experimenter, perfected an influence machine: two insulating discs carrying metal sectors, turned in opposite directions on one axle. Neutralising brushes and collecting combs let a tiny starting charge on one disc induce opposite charge on the other, over and over, so the charge grows with each turn. No rubbing is involved, so unlike older friction machines it works reliably in damp weather and starts itself.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eQ grows by induction each turn\u003c\/strong\u003e\u003cbr\u003eQ: charge stored on the collectors, usually Leyden jars (C); V = Q\/C: voltage across the spark gap (V)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/jumpring-thomson-jumping-ring\"\u003eJumpring\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn 1887 Elihu Thomson showed that a metal ring slipped over the iron core of a coil leaps into the air when AC is switched on. The changing flux induces a current in the ring; by Lenz's law it opposes the change, and because the ring current lags behind, the net force on average pushes the ring away. With steady AC a ring can float.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eF ∝ −I\u003csub\u003ecoil\u003c\/sub\u003e I\u003csub\u003ering\u003c\/sub\u003e\u003c\/strong\u003e\u003cbr\u003eI\u003csub\u003ecoil\u003c\/sub\u003e: current in the coil (A); I\u003csub\u003ering\u003c\/sub\u003e: induced current in the ring (A)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/meldea-melde-experiment\"\u003eMeldea\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eAround 1859 Franz Melde tied a string to the prong of a tuning fork and pulled it taut (Ann. Phys. 185, 193, 1860). At the right tensions the string broke into steady standing waves, with nodes that stood still and loops that blurred. Change the tension and the number of loops jumps.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ef = n√(T\/μ)\/(2L)\u003c\/strong\u003e\u003cbr\u003ef: frequency of the string vibration (Hz); n: number of loops; T: tension (N); μ: mass per unit length (kg\/m); L: length of the string (m)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/ruchardta-ruchardt-experiment\"\u003eRuchardta\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn 1929 Eduard Rüchardt measured the heat-capacity ratio of a gas with a simple trick (Phys. Z. 30, 58). A ball fits snugly in the neck of a large flask. Push it and it bounces up and down on the springy gas below. The squeezes are quick, so no heat flows and the gas acts adiabatically. Timing the oscillation gives γ.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eγ = 4π²mV\/(A²pT²)\u003c\/strong\u003e\u003cbr\u003eγ: ratio of heat capacities C\u003csub\u003ep\u003c\/sub\u003e\/C\u003csub\u003ev\u003c\/sub\u003e; m: mass of the ball (kg); V: volume of gas (m³); A: cross-section of the neck (m²); p: gas pressure under the ball (Pa); T: oscillation period (s)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/paddlewheel-joule\"\u003ePaddlewheel\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eJames Prescott Joule let weights fall on pulleys, turning a paddle wheel inside an insulated vessel of water. Friction of the paddles warmed the water a tiny amount. Dividing the work done by the falling weights by the heat produced, he reported in 1850 (“On the Mechanical Equivalent of Heat”, Phil. Trans. R. Soc. 140, 61) that 772 foot-pounds of work raise one pound of water by 1 °F.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eW = J Q\u003c\/strong\u003e\u003cbr\u003eW: mechanical work (ft·lbf); J: mechanical equivalent of heat (ft·lbf per BTU); Q: heat produced (BTU)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/radiometra-crookes-radiometer\"\u003eRadiometra\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA Crookes radiometer has four vanes, each black on one side and silver on the other, inside a bulb pumped down to a partial vacuum. In light, the vanes spin with the dark sides retreating. If light pressure drove it, the reflective sides would be pushed harder (a reflected photon gives twice the push of an absorbed one) and it would spin the other way. Instead, the dark side gets warmer, and the thin gas does the work. James Clerk Maxwell (Phil. Trans. R. Soc. 170, 231, 1879) and Osborne Reynolds, in his study of thermal transpiration (Phil. Trans. R. Soc. 170, 727, 1879), explained it: gas creeps around each vane's edge from the cold side to the hot side, and the reaction pushes the vane towards its cooler, silver side.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003edark side warmer ⇒ vane moves dark side trailing\u003c\/strong\u003e\u003cbr\u003edark side: absorbs light and heats up; silver side: reflects light and stays cooler; gas: thin enough for molecules to travel far between collisions, but not a hard vacuum\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/galvana-mirror-galvanometer\"\u003eGalvana\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eWilliam Thomson, later Lord Kelvin, developed the mirror galvanometer around the 1858 transatlantic telegraph cable, whose signals were too faint for ordinary needles. A tiny mirror carrying small magnets hangs inside a coil; a current twists it slightly, and a beam of light reflected from it sweeps across a scale. By the law of reflection, turning a mirror by an angle θ turns the reflected beam by twice that angle.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ebeam deflection = 2θ\u003c\/strong\u003e\u003cbr\u003eθ: rotation angle of the mirror (rad); 2θ: rotation angle of the reflected beam (rad)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/rowlanda-rowland-disc\"\u003eRowlanda\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn 1876, in Helmholtz's Berlin laboratory, Henry Rowland charged an ebonite disc and spun it fast. A magnetic needle nearby deflected (Am. J. Sci. 15, 30, 1878). A moving charge makes a magnetic field, just as a current in a wire does: current is charge in motion.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eB ∝ q v\u003c\/strong\u003e\u003cbr\u003eB: magnetic field near the disc (T); q: charge on the disc (C); v: rim speed (m\/s)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/inertix-tolman-stewart\"\u003eInertix\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eSpin a coil of wire fast and stop it suddenly: the free charges inside keep going for a moment, like passengers in a braking bus, and a pulse of current flows. Richard Tolman and Thomas Stewart measured it in 1916 (Phys. Rev. 8, 97). From the charge that flowed they got the carriers' charge-to-mass ratio, close to that of the electron, showing that current in metals is carried by electrons.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eQ = (m\/e) v\u003csub\u003e0\u003c\/sub\u003e L \/ R\u003c\/strong\u003e\u003cbr\u003eQ: charge that flows after stopping (C); m\/e: mass-to-charge ratio of the carriers (kg\/C); v\u003csub\u003e0\u003c\/sub\u003e: rim speed of the wire before stopping (m\/s); L: length of the wire (m); R: resistance of the coil (Ω)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/hemisphera-magdeburg\"\u003eHemisphera\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eOtto von Guericke, mayor of Magdeburg, invented an air pump and in 1654 demonstrated it at Regensburg before the Imperial Diet. He fitted two copper hemispheres together and pumped the air out. With the inside nearly empty, the outside air pressed them together so hard that teams of horses could not pull them apart. Let the air back in, and they fell apart by hand.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eF = πr²Δp\u003c\/strong\u003e\u003cbr\u003eF: force needed to pull the hemispheres apart (N); r: radius of the hemispheres (m); Δp: pressure difference, outside minus inside (Pa)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/quincka-quincke-rotation\"\u003eQuincka\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn 1896 Georg Quincke saw that an insulating sphere in an oil, placed in a strong steady electric field, can start spinning on its own. Charges pile up on its surface; above a threshold field the arrangement is unstable and a small tilt grows into steady rotation.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eE \u0026gt; E\u003csub\u003ec\u003c\/sub\u003e → rotation\u003c\/strong\u003e\u003cbr\u003eE: applied DC electric field (V\/m); E\u003csub\u003ec\u003c\/sub\u003e: threshold field for spontaneous rotation (V\/m)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/gravitas-big-g\"\u003eGravitas\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eG sets the strength of gravity between any two masses. It is measured with torsion balances (Cavendish, 1798) and other lab methods, where the pull between kilogram-scale masses is about 10⁻⁷ N or less.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eG = 6.674 30(15) × 10⁻¹¹ m³ kg⁻¹ s⁻²\u003c\/strong\u003e\u003cbr\u003eG: Newtonian constant of gravitation; (15): standard uncertainty in the last two digits, 2.2 × 10⁻⁵ relative\u003c\/p\u003e","brand":"Printify","offers":[{"title":"White \/ Kiss-Cut \/ 5.83″ × 8.27″","offer_id":50966967812199,"sku":"18220986879033880463","price":19.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/3211099581384067850_2048.jpg?v=1793676500"},{"product_id":"it-s-all-spin-sticker-sheet","title":"It's All Spin Pixel Sticker Sheet – 12 Physics Stickers","description":"\u003cp\u003e12 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: It's All Spin. Every character below has its own sticker page with the full physics panel.\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/echoa-spin-echo\"\u003eEchoa\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn a magnetic field, nuclear spins precess at slightly different rates because the field is not perfectly uniform, so after a pulse knocks them over they fan out and the signal fades. Erwin Hahn found in 1950 (“Spin Echoes”, Phys. Rev. 80, 580) that a second pulse makes them run back together, and a signal reappears: the spin echo. In the version by H. Y. Carr and E. M. Purcell (Phys. Rev. 94, 630, 1954), a π\/2 pulse is followed after a time τ by a π pulse that flips the fan over. The fast spins are now behind the slow ones, catch up, and all line up again at 2τ.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003et\u003csub\u003eecho\u003c\/sub\u003e = 2τ\u003c\/strong\u003e\u003cbr\u003eτ: time between the first and second pulse (s); t\u003csub\u003eecho\u003c\/sub\u003e: time of the echo after the first pulse (s)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/dehaasa-einstein-de-haas\"\u003eDehaasa\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eMagnetise an iron rod and it starts to turn. In 1915 Albert Einstein and Wander de Haas hung an iron cylinder on a fine fibre inside a coil (Verh. Dtsch. Phys. Ges. 17, 152). Reversing the magnetisation flips the tiny magnets inside the iron, and their angular momentum flips with them. Angular momentum is conserved, so the whole rod twists the other way. The ratio of magnetic moment to angular momentum sets the g-factor.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eL = 2mμ\/(ge)\u003c\/strong\u003e\u003cbr\u003eL: angular momentum of the carriers (J·s); μ: magnetic moment (A·m²); m: electron mass (kg); e: elementary charge (C); g: gyromagnetic g-factor\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/barnetta-barnett-effect\"\u003eBarnetta\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eSpin an unmagnetised iron rod and it becomes magnetised. Samuel Barnett showed this in 1915 (Phys. Rev. 6, 239): rotation acts on the electrons' angular momentum like a magnetic field, so their spins tend to line up. It is the reverse of the Einstein–de Haas effect, and gave g ≈ 2 for iron.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eM ∝ Ω\u003c\/strong\u003e\u003cbr\u003eM: magnetisation (A\/m); Ω: angular speed of rotation (rad\/s)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/magnetonix-bohr-magneton\"\u003eMagnetonix\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eAn electron with one unit of orbital angular momentum ħ has a magnetic moment of one Bohr magneton. This is the natural unit for atomic and electron magnetism.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eμ\u003csub\u003eB\u003c\/sub\u003e = eħ\/(2m\u003csub\u003ee\u003c\/sub\u003e) = 9.274 010 0657(29) × 10⁻²⁴ J\/T\u003c\/strong\u003e\u003cbr\u003eμ\u003csub\u003eB\u003c\/sub\u003e: Bohr magneton (J\/T); e: elementary charge (exact); ħ: reduced Planck constant (exact); m\u003csub\u003ee\u003c\/sub\u003e: electron mass (measured)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/helicita-goldhaber-neutrino-helicity\"\u003eHelicita\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn 1958 Maurice Goldhaber, Lee Grodzins and Andrew Sunyar measured the handedness of the neutrino (“Helicity of Neutrinos”, Phys. Rev. 109, 1015). Europium-152m captures an electron and emits a neutrino; the daughter nucleus recoils and then emits a gamma ray. Selecting gammas emitted against the recoil and measuring their circular polarisation with magnetised iron tied the neutrino's spin to its motion. Neutrinos turned out left-handed: spin opposite to motion.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eh = −1\u003c\/strong\u003e\u003cbr\u003eh: helicity, the spin component along the direction of motion, in units of the maximum\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/satyendra-nath-bose-sticker-particle-guy-major-stats-physics-sticker\"\u003eSatyendra Nath Bose\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eBose counted photons as truly identical particles and recovered Planck's law. The resulting Bose–Einstein distribution gives the average number of particles in a state of given energy, and it allows many bosons to pile into the same state.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003en̄(ε) = 1\/(e\u003csup\u003e(ε−μ)\/(k\u003csub\u003eB\u003c\/sub\u003eT)\u003c\/sup\u003e − 1)\u003c\/strong\u003e\u003cbr\u003ēn: mean occupation of a single-particle state; ε: state energy (J); μ: chemical potential (J); k_B: Boltzmann constant (J\/K); T: temperature (K)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/hanleva-hanle-effect\"\u003eHanleva\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eShine polarised light at the mercury resonance line (253.7 nm) into mercury vapour and the atoms re-emit it as polarised resonance fluorescence. Add a weak magnetic field and the polarisation drops. Wilhelm Hanle reported this in “Über magnetische Beeinflussung der Polarisation der Resonanzfluoreszenz” (Z. Phys. 30, 93, 1924). The excited atoms behave like little oscillating dipoles that precess around the field at the Larmor frequency. If they turn through a large angle before they emit, the emitted light comes out with a scrambled direction of polarisation.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eΔB ≈ ħ \/ (g μ\u003csub\u003eB\u003c\/sub\u003e τ)\u003c\/strong\u003e\u003cbr\u003eΔB: field scale over which the polarisation falls (tesla, T); ħ: reduced Planck constant; g: Landé g-factor of the excited state; μ\u003csub\u003eB\u003c\/sub\u003e: Bohr magneton; τ: lifetime of the excited state (s)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/bifurcor-stern-gerlach-experiment-pixel-sticker\"\u003eBifurcor\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA magnetic field that changes with height pushes on an atom's magnetic moment. Because an electron's spin can only point up or down, silver atoms in the Stern–Gerlach magnet split into exactly two beams instead of smearing out.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eF\u003csub\u003ez\u003c\/sub\u003e = μ\u003csub\u003ez\u003c\/sub\u003e ∂B\u003csub\u003ez\u003c\/sub\u003e\/∂z\u003c\/strong\u003e\u003cbr\u003eF_z: force on the atom along the field gradient (N); μ_z: z-component of the atom's magnetic moment (for silver, set by its one outer electron) (J\/T); B_z: z-component of the magnetic field (T); z: position along the gradient, up the page (m); g_s: electron spin g-factor, about 2.0023; μ_B: Bohr magneton (J\/T)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/revolvix-spinor-rotation-pixel-sticker\"\u003eRevolvix\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eRotating a spin-½ particle by one full turn multiplies its wavefunction by −1; only two full turns bring it back to its original state. Neutron interferometers measured this sign change directly in 1975.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eR(2π)ψ = −ψ\u003c\/strong\u003e\u003cbr\u003eR(θ): operator that rotates a spin-1\/2 state by angle θ about the axis n; ψ: two-component spinor state of a spin-1\/2 particle; n̂: unit vector along the rotation axis; σ: the three Pauli matrices (σx, σy, σz); 2π, 4π: one full turn (360°) and two full turns (720°) (rad)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/wolfgang-pauli-sticker-one-per-state-physics-sticker\"\u003eWolfgang Pauli\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eTwo identical fermions must have a wavefunction that changes sign when they are swapped. If both tried to occupy the same state, the wavefunction would be zero everywhere, so the arrangement is forbidden: one fermion per state.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eψ(1,2) = [φ\u003csub\u003ea\u003c\/sub\u003e(1)φ\u003csub\u003eb\u003c\/sub\u003e(2) − φ\u003csub\u003ea\u003c\/sub\u003e(2)φ\u003csub\u003eb\u003c\/sub\u003e(1)]\/√2\u003c\/strong\u003e\u003cbr\u003eψ: two-particle antisymmetrised wavefunction; φₐ, φ_b: single-particle orbitals (including spin); a, b: full sets of quantum numbers (n,ℓ,m_ℓ,m_s)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/barkhausa-barkhausen-noise\"\u003eBarkhausa\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eMagnetise a piece of iron slowly and smoothly and its magnetisation does not change smoothly. It grows in many tiny, sudden jumps as the walls between magnetic domains snag on defects and then snap free. Heinrich Barkhausen heard this in 1919 by winding a coil around the iron and amplifying the induced pulses into a telephone receiver: a crackle (“Zwei mit Hilfe der neuen Verstärker entdeckte Erscheinungen”, Physikalische Zeitschrift 20, 401, 1919). Each jump induces a small voltage pulse in the coil.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eV = −N dΦ\/dt\u003c\/strong\u003e\u003cbr\u003eV: voltage induced in the pick-up coil (V); N: number of turns of the coil; Φ: magnetic flux through the coil (Wb); t: time (s)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/spinwell-angular-momentum-pixel-sticker\"\u003eSpinwell\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eAngular momentum is moment of inertia times angular speed, and without an outside torque it stays constant. Pull your arms in and your moment of inertia drops, so you spin faster.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eL = Iω\u003c\/strong\u003e\u003cbr\u003eL: angular momentum about the spin axis (kg m² \/ s); I: moment of inertia about that axis (kg m²); ω: angular speed (rad\/s); τ: external torque about the axis (N m)\u003c\/p\u003e","brand":"Printify","offers":[{"title":"White \/ Kiss-Cut \/ 5.83″ × 8.27″","offer_id":50966967877735,"sku":"32478955944929435155","price":19.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/3245213422971511818_2048.jpg?v=1793676506"},{"product_id":"fields-forever-sticker-sheet","title":"Fields Forever Pixel Sticker Sheet – 12 Physics Stickers","description":"\u003cp\u003e12 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: Fields Forever. Every character below has its own sticker page with the full physics panel.\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/nikola-tesla-sticker-ac-slays-dc-pixel-art-physics-sticker\"\u003eNikola Tesla\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003ePower lost in a wire is the current squared times the resistance, so sending power at high voltage and low current wastes much less. Transformers, which only work with alternating current, step the voltage up or down in proportion to the number of turns on each coil.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eV(t) = V\u003csub\u003e0\u003c\/sub\u003e sin(ωt)\u003c\/strong\u003e\u003cbr\u003eV: voltage (V); V₀: peak voltage (V); ω: angular frequency (rad\/s); V_s, V_p: secondary, primary voltage (V); N_s, N_p: turns on secondary, primary; P_(loss): resistive power loss (W)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/james-clerk-maxwell-sticker-field-goals-achieved-physics-sticker\"\u003eJames Clerk Maxwell\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eMaxwell added one missing term, the displacement current, to the known laws of electricity and magnetism. With it the four equations are consistent and predict waves moving at the speed of light, which showed that light itself is an electromagnetic wave.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003e∇·\u003cb\u003eE\u003c\/b\u003e = ρ\/ε\u003csub\u003e0\u003c\/sub\u003e, ∇·\u003cb\u003eB\u003c\/b\u003e = 0\u003c\/strong\u003e\u003cbr\u003eE: electric field (V\/m); B: magnetic flux density (T); ρ: charge density (C\/m³); J: current density (A\/m²); ε₀, μ₀: vacuum permittivity, permeability (F\/m, N\/A²); c: speed of light (m\/s)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/michael-faraday-sticker-all-quiet-in-here-physics-sticker\"\u003eMichael Faraday\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eGauss's law relates the electric flux through a closed surface to the charge inside it. In a closed conductor at equilibrium the charges sit on the outside, so the field in an empty cavity is zero, which is why Faraday was safe inside his charged cage.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003e∮\u003csub\u003eS\u003c\/sub\u003e \u003cb\u003eE\u003c\/b\u003e·d\u003cb\u003eA\u003c\/b\u003e = Q\u003csub\u003eenc\u003c\/sub\u003e\/ε\u003csub\u003e0\u003c\/sub\u003e\u003c\/strong\u003e\u003cbr\u003eE: electric field (V\/m); dA: outward area element of a closed surface S (m²); Q_(enc): charge enclosed by S (C); ε₀: vacuum permittivity (F\/m)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/lenzor-lenzs-law-pixel-sticker\"\u003eLenzor\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA changing magnetic flux induces a voltage, and the induced current always flows so as to oppose the change. A magnet dropped through a copper pipe therefore drives eddy currents that brake it, so it drifts down at a steady terminal speed.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eℰ = −dΦ\u003csub\u003eB\u003c\/sub\u003e\/dt\u003c\/strong\u003e\u003cbr\u003emathcalE: EMF induced around one ring of the pipe (V); Φ_B: magnetic flux through that ring (Wb); t: time (s); m: mass of the falling magnet (kg); g: gravitational acceleration (m\/s²); F_(eddy): upward drag from the eddy currents; it grows with the magnet's speed (N)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/gyrolla-cyclotron-motion-pixel-sticker\"\u003eGyrolla\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA magnetic field bends a charged particle into a circle without changing its speed. Faster particles make bigger circles, but each lap takes the same time, so the cyclotron frequency depends only on charge, mass and field. That is what makes the cyclotron accelerator work.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003er = mv\/(qB)\u003c\/strong\u003e\u003cbr\u003eq: charge of the particle (C); v: speed perpendicular to the field (m\/s); B: magnetic flux density (T); m: mass of the particle (kg); r: radius of the circle (m); f_c: cyclotron frequency, laps per second (non-relativistic; at high energy it falls by 1\/γ) (Hz)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/displacera-displacement-current\"\u003eDisplacera\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eAmpère's law broke down at a charging capacitor: current flows in the wire but not across the gap. James Clerk Maxwell fixed it in 1861–65 by adding a term for the changing electric field in the gap, which makes a magnetic field just as a real current does (“A Dynamical Theory of the Electromagnetic Field”, Phil. Trans. 155, 459, 1865). With it, the equations predicted electromagnetic waves travelling at the speed of light.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eJ\u003csub\u003ed\u003c\/sub\u003e = ε\u003csub\u003e0\u003c\/sub\u003e ∂E\/∂t\u003c\/strong\u003e\u003cbr\u003eJ\u003csub\u003ed\u003c\/sub\u003e: displacement current density (A\/m²); ε\u003csub\u003e0\u003c\/sub\u003e: permittivity of free space (F\/m); ∂E\/∂t: rate of change of the electric field (V m⁻¹ s⁻¹)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/jumpring-thomson-jumping-ring\"\u003eJumpring\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn 1887 Elihu Thomson showed that a metal ring slipped over the iron core of a coil leaps into the air when AC is switched on. The changing flux induces a current in the ring; by Lenz's law it opposes the change, and because the ring current lags behind, the net force on average pushes the ring away. With steady AC a ring can float.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eF ∝ −I\u003csub\u003ecoil\u003c\/sub\u003e I\u003csub\u003ering\u003c\/sub\u003e\u003c\/strong\u003e\u003cbr\u003eI\u003csub\u003ecoil\u003c\/sub\u003e: current in the coil (A); I\u003csub\u003ering\u003c\/sub\u003e: induced current in the ring (A)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/wimsha-wimshurst-machine\"\u003eWimsha\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn the early 1880s James Wimshurst, a London shipwright and amateur experimenter, perfected an influence machine: two insulating discs carrying metal sectors, turned in opposite directions on one axle. Neutralising brushes and collecting combs let a tiny starting charge on one disc induce opposite charge on the other, over and over, so the charge grows with each turn. No rubbing is involved, so unlike older friction machines it works reliably in damp weather and starts itself.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eQ grows by induction each turn\u003c\/strong\u003e\u003cbr\u003eQ: charge stored on the collectors, usually Leyden jars (C); V = Q\/C: voltage across the spark gap (V)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/bremsella-bremsstrahlung-pixel-sticker\"\u003eBremsella\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eAny accelerating charge radiates. Larmor's formula says the power grows with the square of the charge and the square of its acceleration, which is why electrons that slow down sharply in matter give off braking radiation.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eP = q²a²\/(6πε\u003csub\u003e0\u003c\/sub\u003ec³)\u003c\/strong\u003e\u003cbr\u003eP: power radiated by the accelerating charge (non-relativistic Larmor formula) (W); q: charge of the particle (C); a: magnitude of its acceleration (m\/s²); ε₀: vacuum permittivity (F\/m); c: speed of light (m\/s)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/azurwake-cherenkov-radiation-pixel-sticker\"\u003eAzurwake\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA charged particle moving through a medium faster than light travels in that medium emits light in a cone, like a sonic boom. The cone angle depends on the particle's speed and the refractive index, and Frank and Tamm's formula gives how many photons come out per unit length.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ecos θ = 1\/(nβ), β \u0026gt; 1\/n\u003c\/strong\u003e\u003cbr\u003eθ: angle between the emitted light and the particle's path (the wavefront cone has half-angle 90° − θ) (rad); n: refractive index of the medium, about 1.33 for water; β: particle speed as a fraction of c; N: number of photons emitted; x: distance travelled by the particle (m); λ: wavelength of the emitted light (m)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/penninga-penning-trap\"\u003ePenninga\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eEarnshaw's theorem says no arrangement of static electric charges alone can hold a charge in stable equilibrium. A Penning trap gets around it by adding a strong, uniform magnetic field to an electric quadrupole field. A trapped particle then has three motions: a fast modified cyclotron circle (ω\u003csub\u003e+\u003c\/sub\u003e), an axial bounce along the field (ω\u003csub\u003ez\u003c\/sub\u003e), and a slow magnetron drift (ω\u003csub\u003e−\u003c\/sub\u003e). L. S. Brown and G. Gabrielse showed that, even in an imperfect trap, the three combine to give the true cyclotron frequency (“Geonium theory”, Rev. Mod. Phys. 58, 233, 1986).\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eω\u003csub\u003ec\u003c\/sub\u003e² = ω\u003csub\u003e+\u003c\/sub\u003e² + ω\u003csub\u003ez\u003c\/sub\u003e² + ω\u003csub\u003e−\u003c\/sub\u003e²\u003c\/strong\u003e\u003cbr\u003eω\u003csub\u003ec\u003c\/sub\u003e: free-space cyclotron frequency, qB\/m (rad\/s); ω\u003csub\u003e+\u003c\/sub\u003e: modified cyclotron frequency (rad\/s); ω\u003csub\u003ez\u003c\/sub\u003e: axial frequency (rad\/s); ω\u003csub\u003e−\u003c\/sub\u003e: magnetron frequency (rad\/s)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/capacitwin-two-capacitor-paradox\"\u003eCapacitwin\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eCharge one capacitor C to voltage V and join it to an identical empty one. Charge is conserved, so each ends at V\/2 with half the charge. But the stored energy is now Q²\/4C instead of Q²\/2C: half the energy is gone. This holds however small the wire resistance is.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eΔE = ½ · C\u003csub\u003e1\u003c\/sub\u003eC\u003csub\u003e2\u003c\/sub\u003e\/(C\u003csub\u003e1\u003c\/sub\u003e + C\u003csub\u003e2\u003c\/sub\u003e) · (V\u003csub\u003e1\u003c\/sub\u003e − V\u003csub\u003e2\u003c\/sub\u003e)²\u003c\/strong\u003e\u003cbr\u003eΔE: energy lost when the capacitors are joined (J); C\u003csub\u003e1\u003c\/sub\u003e, C\u003csub\u003e2\u003c\/sub\u003e: capacitances (F); V\u003csub\u003e1\u003c\/sub\u003e, V\u003csub\u003e2\u003c\/sub\u003e: starting voltages (V)\u003c\/p\u003e","brand":"Printify","offers":[{"title":"White \/ Kiss-Cut \/ 5.83″ × 8.27″","offer_id":50966976102503,"sku":"13477997120803188746","price":19.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/15921913249398551895_2048.jpg?v=1793676512"},{"product_id":"starter-pack-sticker-sheet","title":"Physics Starter Pack Pixel Sticker Sheet – 12 Physics Stickers","description":"\u003cp\u003e12 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: Physics Starter Pack. Every character below has its own sticker page with the full physics panel.\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/isaac-newton-sticker-gravity-hits-different-physics-sticker\"\u003eIsaac Newton\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eNewton's second law says force equals the rate of change of momentum. His law of gravitation says every two masses attract with a force proportional to their product and inversely proportional to the square of their distance, and together they explain falling apples and planetary orbits.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003e\u003cb\u003eF\u003c\/b\u003e = d\u003cb\u003ep\u003c\/b\u003e\/dt\u003c\/strong\u003e\u003cbr\u003eF: net force (N); p: momentum (kg·m\/s); G: gravitational constant (m³·kg⁻¹·s⁻²); m₁, m₂: masses (kg); r: separation (m)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/richard-feynman-sticker-quantum-it-s-complicated-physics-sticker\"\u003eRichard Feynman\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eFeynman's rule: to get from a to b, a quantum particle takes every possible path, and each path contributes a phase set by its action. You add all those amplitudes and square the total to get the probability.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eK(b,a) = Σ\u003csub\u003epaths\u003c\/sub\u003e e\u003csup\u003eiS\/ħ\u003c\/sup\u003e\u003c\/strong\u003e\u003cbr\u003eK(b,a): amplitude to go from a to b (propagator), in d spatial dimensions (m^(-d)); S: classical action along a path (J·s); ħ: reduced Planck constant (J·s); φ_k: amplitude for alternative k; P: probability\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/marie-curie-sticker-she-literally-glowed-up-physics-sticker\"\u003eMarie Curie\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eEach radioactive nucleus has a fixed chance per second of decaying, so the number left falls exponentially. The half-life is ln 2 divided by that decay constant, and the activity is the decay constant times the number of nuclei.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eN(t) = N\u003csub\u003e0\u003c\/sub\u003e e\u003csup\u003e−λt\u003c\/sup\u003e\u003c\/strong\u003e\u003cbr\u003eN: number of undecayed nuclei; λ: decay constant (s⁻¹); t_(1\/2): half-life (s); A: activity (1 Bq = 1 decay per second) (Bq)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/stephen-hawking-sticker-black-holes-been-there-bent-that-physics-sticker\"\u003eStephen Hawking\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eHawking found that black holes radiate like warm bodies, with a temperature inversely proportional to their mass. Bekenstein's entropy is proportional to the area of the horizon, not its volume, which links gravity, thermodynamics and quantum theory.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eT\u003csub\u003eH\u003c\/sub\u003e = ħc³\/(8πGMk\u003csub\u003eB\u003c\/sub\u003e)\u003c\/strong\u003e\u003cbr\u003eT_H: Hawking temperature (K); M: black-hole mass (kg); S_(BH): black-hole entropy (J\/K); A: horizon area (m²); ħ, c, G, k_B: fundamental constants (J·s, m\/s, m³·kg⁻¹·s⁻², J\/K)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/nikola-tesla-sticker-ac-slays-dc-pixel-art-physics-sticker\"\u003eNikola Tesla\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003ePower lost in a wire is the current squared times the resistance, so sending power at high voltage and low current wastes much less. Transformers, which only work with alternating current, step the voltage up or down in proportion to the number of turns on each coil.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eV(t) = V\u003csub\u003e0\u003c\/sub\u003e sin(ωt)\u003c\/strong\u003e\u003cbr\u003eV: voltage (V); V₀: peak voltage (V); ω: angular frequency (rad\/s); V_s, V_p: secondary, primary voltage (V); N_s, N_p: turns on secondary, primary; P_(loss): resistive power loss (W)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/max-planck-sticker-he-quantized-the-vibe-physics-sticker\"\u003eMax Planck\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003ePlanck found that the glow of a hot body could only be explained if energy is exchanged in packets of hν. His formula gives the full blackbody spectrum at any temperature and started quantum theory.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eE = hν\u003c\/strong\u003e\u003cbr\u003eE: quantum of energy (J); h: Planck constant (exact since 2019) (J·s); ν: frequency (Hz); B_ν: spectral radiance (W·m⁻²·sr⁻¹·Hz⁻¹); T: temperature (K); k_B: Boltzmann constant (J\/K)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/niels-bohr-sticker-said-nothing-clearly-explained-everything-physics-sticker\"\u003eNiels Bohr\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn Bohr's model the energy levels of hydrogen are minus the Rydberg energy divided by n squared. When an electron drops from a higher level to a lower one it emits a photon carrying exactly the energy difference, which explains hydrogen's sharp spectral lines.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eE\u003csub\u003en\u003c\/sub\u003e = −R\u003csub\u003e∞\u003c\/sub\u003ehc\/n², with R\u003csub\u003e∞\u003c\/sub\u003ehc ≈ 13.6 eV\u003c\/strong\u003e\u003cbr\u003eE_n: energy of level n (infinite-nuclear-mass approximation; hydrogen uses reduced mass; quoted in eV) (J); R_∞ hc: Rydberg energy (CODATA 2022) (J); n: principal quantum number; h: Planck constant (J·s); ν: photon frequency (Hz); Eᵢ, E_f: initial and final level energies (J)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/scatterina-rayleigh-scattering-pixel-sticker\"\u003eScatterina\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eSmall particles and molecules scatter short wavelengths much more strongly, in proportion to one over wavelength to the fourth power. Blue light is scattered about 5.9 times more than red, which is why the sky looks blue.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eI\u003csub\u003es\u003c\/sub\u003e ∝ 1\/λ⁴\u003c\/strong\u003e\u003cbr\u003eI_s: power scattered by a molecule much smaller than the wavelength (W); λ: wavelength of the light (m)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/sortex-maxwells-demon-pixel-sticker\"\u003eSortex\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eLandauer argued that erasing one bit of information must release at least k_B T ln 2 of heat. Because a Maxwell's demon has to erase its records eventually, this cost is what saves the second law of thermodynamics.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eQ\u003csub\u003eerase\u003c\/sub\u003e ≥ k\u003csub\u003eB\u003c\/sub\u003eT ln 2\u003c\/strong\u003e\u003cbr\u003eQ_(erase): heat released to the surroundings when one bit of memory is erased (a lower bound, not a fixed cost) (J); k_B: Boltzmann constant (J\/K); T: temperature of the surroundings that absorb the heat (K); ln 2: natural logarithm of 2: one bit is a choice between two states\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/relicta-cosmic-microwave-background-pixel-sticker\"\u003eRelicta\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eThe cosmic microwave background is light left over from when the universe became transparent. It has a nearly perfect blackbody spectrum, and its temperature has fallen in proportion to the stretching of space, to about 2.7 K today.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eT(z) = T\u003csub\u003e0\u003c\/sub\u003e(1 + z)\u003c\/strong\u003e\u003cbr\u003eT(z): temperature of the background radiation at redshift z (K); z: redshift (last scattering: about 1090); T₀: temperature today (Fixsen 2009: 2.72548 ± 0.00057 K) (K); B_ν: blackbody spectral radiance per unit frequency (W·m⁻²·Hz⁻¹·sr⁻¹); ν: frequency (Hz); h: Planck constant (J·s)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/condensa-bose-einstein-condensate-pixel-sticker\"\u003eCondensa\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eBelow a critical temperature, a gas of bosons starts collecting in its lowest quantum state. The fraction in that state grows as the temperature drops, following 1 − (T\/Tc)^(3\/2), and the critical temperature depends on the gas's density and particle mass.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eN\u003csub\u003e0\u003c\/sub\u003e\/N = 1 − (T\/T\u003csub\u003ec\u003c\/sub\u003e)\u003csup\u003e3\/2\u003c\/sup\u003e (T ≤ T\u003csub\u003ec\u003c\/sub\u003e)\u003c\/strong\u003e\u003cbr\u003eN₀: number of bosons in the ground state; N: total number of bosons; T: temperature (K); T_c: critical temperature for condensation (K); n: number density (m⁻³); m: boson mass (kg)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/libratrix-lagrange-points-pixel-sticker\"\u003eLibratrix\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eTwo bodies orbiting each other have five points where a small third body can keep pace with them. For the Sun and Earth, L2 lies a little beyond Earth, and the triangular points L4 and L5 are stable only when one mass greatly outweighs the other.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003er\u003csub\u003eL2\u003c\/sub\u003e ≈ R(M\u003csub\u003e2\u003c\/sub\u003e\/(3M\u003csub\u003e1\u003c\/sub\u003e))\u003csup\u003e1\/3\u003c\/sup\u003e\u003c\/strong\u003e\u003cbr\u003er_(L2): distance from the smaller body to L2 (second line: Sun–Earth L2, beyond Earth) (m); R: distance between the two bodies (m); M₁, M₂: masses of the larger and the smaller body (kg); M₁\/M₂: mass ratio; above about 24.96, L4 and L5 are stable (Routh)\u003c\/p\u003e","brand":"Printify","offers":[{"title":"White \/ Kiss-Cut \/ 5.83″ × 8.27″","offer_id":50966984327271,"sku":"33930378019787642819","price":19.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/3235699826954520802_2048.jpg?v=1793676519"},{"product_id":"atom-smashers-sticker-sheet","title":"Atom Smashers Pixel Sticker Sheet – 12 Physics Stickers","description":"\u003cp\u003e12 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: Atom Smashers. Every character below has its own sticker page with the full physics panel.\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/lise-meitner-sticker-split-happens-physics-sticker\"\u003eLise Meitner\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eWhen a uranium nucleus splits, the pieces weigh less than the original, and the missing mass appears as energy. Meitner and Frisch estimated it from the electric repulsion between the two fragments at the moment they part: about 200 MeV per fission.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eQ = (M\u003csub\u003einitial\u003c\/sub\u003e − M\u003csub\u003efinal\u003c\/sub\u003e)c²\u003c\/strong\u003e\u003cbr\u003eQ: energy released (quoted in MeV) (J); M: total rest mass before\/after (kg); c: speed of light (m\/s); E_C: Coulomb repulsion energy of two touching fragments (J); Z₁, Z₂: fragment proton numbers; e: elementary charge (C)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/hideki-yukawa-sticker-heavy-ball-short-game-physics-sticker\"\u003eHideki Yukawa\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eYukawa proposed that nuclear forces come from exchanging a massive particle, so they fade out exponentially beyond a range of about ħ\/mc. Turning the observed short range around predicted the mass of the carrier, later identified as the pion.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eV(r) = −(g²\/4π) e\u003csup\u003e−r\/R\u003c\/sup\u003e\/r, R = ħ\/(mc)\u003c\/strong\u003e\u003cbr\u003eV(r): potential energy between two nucleons (J); g: coupling strength (g²\/4π has units of energy times length here) (J^(1\/2)·m^(1\/2)); r: separation (m); R: range of the force (m); λ: Yukawa's range parameter, the inverse of R (he took 5×10¹² per centimetre) (m⁻¹); ħ: reduced Planck constant (J·s)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/maria-goeppert-mayer-sticker-a-full-floor-is-magic-physics-sticker\"\u003eMaria Goeppert Mayer\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eProtons and neutrons fill energy shells in the nucleus, much like electrons in an atom. Adding strong spin-orbit coupling moves the shell gaps to the observed magic numbers 2, 8, 20, 28, 50, 82 and 126, where nuclei are unusually stable.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eE\u003csub\u003eN\u003c\/sub\u003e = ħω(N + 3\/2)\u003c\/strong\u003e\u003cbr\u003eE_N: energy of oscillator shell N (J); N: oscillator shell number; ħω: oscillator level spacing (J); g_N: states in shell N per nucleon species (spin included); V_(ℓ s): spin–orbit potential (J); C(r): radial strength of the spin–orbit term (with orbital and spin angular momentum measured in units of ħ) (J)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/moderon-neutron-moderation-pixel-sticker\"\u003eModeron\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eA neutron loses the largest share of its energy when it hits a nucleus of about its own mass. The average logarithmic energy loss per collision tells you how many collisions a moderator needs to slow fission neutrons down to thermal energies.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eα = ((A − 1)\/(A + 1))²\u003c\/strong\u003e\u003cbr\u003eA: mass of the target nucleus in neutron masses (hydrogen: 1); α: smallest possible ratio of neutron energy after to before one collision (a head-on hit); ξ: average loss of ln E per collision (hydrogen: 1); n_(coll): average number of collisions to slow a neutron to thermal; E₀: starting neutron energy, about 2 MeV from fission (MeV); E_(th): thermal neutron energy, about 0.025 eV (eV)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/vevette-higgs-field-pixel-sticker\"\u003eVevette\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eThe Higgs field's potential has its lowest point away from zero, so the field takes a nonzero value v of about 246 GeV everywhere. Particles that interact with it gain masses proportional to that value and their coupling strength.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eV(φ) = −μ²|φ|² + λ|φ|⁴\u003c\/strong\u003e\u003cbr\u003eV: Higgs potential (energy density; natural units) (GeV⁴); φ: Higgs field; on the art the horizontal axis is the field's size (natural units) (GeV); μ²: parameter of the potential, positive (GeV²); λ: self-coupling of the field, positive; ⟨φ⟩: vacuum expectation value: where the field settles, in the valley (GeV); v: electroweak scale, about 246.22 GeV (GeV)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/cptera-cpt-theorem\"\u003eCptera\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eFlip all charges (C), mirror space (P) and run time backwards (T), all at once, and any local, Lorentz-invariant quantum field theory looks exactly the same. This CPT theorem was worked out by Julian Schwinger (1951), Gerhart Lüders (1954) and Wolfgang Pauli (1955). It forces a particle and its antiparticle to have exactly equal mass and lifetime.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eCPT: C × P × T is an exact symmetry\u003c\/strong\u003e\u003cbr\u003eC: charge conjugation, particle ↔ antiparticle; P: parity, mirror reflection of space; T: time reversal\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/weinberga-weak-mixing-angle\"\u003eWeinberga\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn the electroweak theory (Glashow, Weinberg, Salam), the neutral gauge fields B and W³ mix. Rotating them by the weak mixing angle θ\u003csub\u003eW\u003c\/sub\u003e gives the massless photon and the massive Z boson. The angle also links the couplings: e = g sin θ\u003csub\u003eW\u003c\/sub\u003e.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003esin²θ\u003csub\u003eW\u003c\/sub\u003e ≈ 0.231 (MS-bar at M\u003csub\u003eZ\u003c\/sub\u003e)\u003c\/strong\u003e\u003cbr\u003eθ\u003csub\u003eW\u003c\/sub\u003e: weak mixing (Weinberg) angle, about 28.7°; MS-bar: modified minimal-subtraction renormalisation scheme; M\u003csub\u003eZ\u003c\/sub\u003e: Z boson mass scale, about 91.19 GeV\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/massratio-proton-electron\"\u003eMassratio\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eThe proton is about 1836 times heavier than the electron. This dimensionless ratio sets why atoms have a tiny heavy nucleus with a light electron cloud, why nuclei barely move in chemistry (the Born–Oppenheimer approximation), and the ratio of vibrational to electronic energies in molecules. It is measured with Penning-trap mass comparisons and precision spectroscopy of molecular hydrogen ions such as HD⁺.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eμ = m\u003csub\u003ep\u003c\/sub\u003e\/m\u003csub\u003ee\u003c\/sub\u003e = 1836.152 673 426(32)\u003c\/strong\u003e\u003cbr\u003eμ: proton-to-electron mass ratio (dimensionless); m\u003csub\u003ep\u003c\/sub\u003e: proton rest mass, measured; m\u003csub\u003ee\u003c\/sub\u003e: electron rest mass, measured; (32): standard uncertainty in the last two digits, about 1.7 × 10⁻¹¹ relative\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/splitterra-cockcroft-walton\"\u003eSplitterra\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn April 1932 at the Cavendish Laboratory, John Cockcroft and Ernest Walton fired protons from a high-voltage accelerator at lithium. Scintillation screens flashed with pairs of alpha particles flying apart (Proc. R. Soc. A 137, 229, 1932). It was the first nucleus split by artificially accelerated particles, and the energy released matched E = mc².\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ep + ⁷Li → 2α + 17.3 MeV\u003c\/strong\u003e\u003cbr\u003ep: proton; ⁷Li: lithium-7 nucleus; α: alpha particle, a helium-4 nucleus\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/fogplate-becquerel\"\u003eFogplate\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eHenri Becquerel suspected that uranium salts, made to glow by sunlight, might give off X-rays. Paris stayed cloudy in late February 1896, so he put plates wrapped in black paper, with uranium salt on top, in a drawer. On 1 March he developed them anyway and found strong images: the uranium emitted penetrating rays with no sunlight at all (Comptes rendus 122, 501, 1896). It was radioactivity.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003euranium → rays, no light needed\u003c\/strong\u003e\u003cbr\u003euranium salt: potassium uranyl sulfate in his first tests; plate: photographic plate wrapped in black paper\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/helicita-goldhaber-neutrino-helicity\"\u003eHelicita\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn 1958 Maurice Goldhaber, Lee Grodzins and Andrew Sunyar measured the handedness of the neutrino (“Helicity of Neutrinos”, Phys. Rev. 109, 1015). Europium-152m captures an electron and emits a neutrino; the daughter nucleus recoils and then emits a gamma ray. Selecting gammas emitted against the recoil and measuring their circular polarisation with magnetised iron tied the neutrino's spin to its motion. Neutrinos turned out left-handed: spin opposite to motion.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eh = −1\u003c\/strong\u003e\u003cbr\u003eh: helicity, the spin component along the direction of motion, in units of the maximum\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/penninga-penning-trap\"\u003ePenninga\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eEarnshaw's theorem says no arrangement of static electric charges alone can hold a charge in stable equilibrium. A Penning trap gets around it by adding a strong, uniform magnetic field to an electric quadrupole field. A trapped particle then has three motions: a fast modified cyclotron circle (ω\u003csub\u003e+\u003c\/sub\u003e), an axial bounce along the field (ω\u003csub\u003ez\u003c\/sub\u003e), and a slow magnetron drift (ω\u003csub\u003e−\u003c\/sub\u003e). L. S. Brown and G. Gabrielse showed that, even in an imperfect trap, the three combine to give the true cyclotron frequency (“Geonium theory”, Rev. Mod. Phys. 58, 233, 1986).\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eω\u003csub\u003ec\u003c\/sub\u003e² = ω\u003csub\u003e+\u003c\/sub\u003e² + ω\u003csub\u003ez\u003c\/sub\u003e² + ω\u003csub\u003e−\u003c\/sub\u003e²\u003c\/strong\u003e\u003cbr\u003eω\u003csub\u003ec\u003c\/sub\u003e: free-space cyclotron frequency, qB\/m (rad\/s); ω\u003csub\u003e+\u003c\/sub\u003e: modified cyclotron frequency (rad\/s); ω\u003csub\u003ez\u003c\/sub\u003e: axial frequency (rad\/s); ω\u003csub\u003e−\u003c\/sub\u003e: magnetron frequency (rad\/s)\u003c\/p\u003e","brand":"Printify","offers":[{"title":"White \/ Kiss-Cut \/ 5.83″ × 8.27″","offer_id":50966992552039,"sku":"23827837638882759772","price":19.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/9510169551877603359_2048.jpg?v=1793676525"},{"product_id":"light-time-relativity-sticker-sheet","title":"Light, Time \u0026 Relativity Pixel Sticker Sheet – 9 Physics Stickers","description":"\u003cp\u003e9 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: Light, Time \u0026amp; Relativity. Every character below has its own sticker page with the full physics panel.\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/muonaut-muon-time-dilation-pixel-sticker\"\u003eMuonaut\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eMuons made high in the atmosphere live only about two microseconds, too short to cross the atmosphere at face value. Moving near the speed of light, their clocks run slow by the factor γ, so many reach the ground; in the muon's frame, the atmosphere is shorter instead.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eΔt = γτ, L = L\u003csub\u003e0\u003c\/sub\u003e\/γ\u003c\/strong\u003e\u003cbr\u003eΔ t: muon lifetime as timed by clocks on the ground (s); τ: mean lifetime in the muon's own rest frame (PDG 2024: 2.1969811 μs; the half-life is shorter, 1.52 μs) (s); γ: Lorentz factor; L₀: thickness of the atmosphere in the ground frame (m); L: the same thickness measured in the muon's frame (m); v: muon speed (m\/s)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/orbitick-gps-clock-correction-pixel-sticker\"\u003eOrbitick\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eGPS clocks feel weaker gravity than clocks on the ground, so they run fast, and their orbital speed makes them run slightly slow. The net effect would build up timing errors quickly, so the satellite clocks are deliberately set to tick slightly slower before launch.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eΔf\/f ≈ ΔΦ\/c² − v²\/(2c²)\u003c\/strong\u003e\u003cbr\u003eΔ f\/f: fractional rate of a GPS satellite clock relative to a clock on the geoid (positive: it runs fast); ΔΦ: gravitational potential at the satellite minus the effective potential on the geoid, which includes Earth's rotation term (J\/kg); v: orbital speed of the satellite (m\/s); c: speed of light in vacuum (m\/s); f₀: factory-specified frequency of the clock's 10.23 MHz reference before launch (Hz)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/troutona-trouton-noble\"\u003eTroutona\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIf Earth moved through an ether, a charged capacitor hanging on a fine wire should feel a torque turning its plates to face the motion. In 1903 Frederick Trouton and H. R. Noble looked for it (Phil. Trans. A 202, 165, published 1904). They found none: the torque was zero.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003etorque = 0\u003c\/strong\u003e\u003cbr\u003etorque: turning effect on the suspended capacitor (N·m)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/magnetoconda-moving-magnet\"\u003eMagnetoconda\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eMove a magnet past a loop of wire and a current flows. Hold the magnet still and move the wire instead, at the same relative speed, and exactly the same current flows. Yet the textbook stories differ. If the magnet moves, a changing magnetic field makes an electric field that drives the charges. If the wire moves, there is no electric field in the magnet's frame, and the magnetic force on the moving charges does the job. A rod of length L moving at speed v across a field B gets the same emf either way.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eε = vBL\u003c\/strong\u003e\u003cbr\u003eε: induced emf (volt, V); v: relative speed of rod and magnet, perpendicular to the field (m\/s); B: magnetic flux density (tesla, T); L: length of the rod across the field (m)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/mirrorspin-foucault-speed-of-light\"\u003eMirrorspin\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIn 1862 Léon Foucault measured the speed of light in a laboratory with a rapidly rotating mirror (Comptes rendus 55, 501). Light reflects off the spinning mirror to a distant fixed mirror and back. In the round-trip time the rotating mirror turns a tiny angle, so the returning spot lands slightly displaced. From the displacement, the path length and the spin rate he got c = 298 000 ± 500 km\/s, close to the modern value.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eΔα = 2ωD\/c\u003c\/strong\u003e\u003cbr\u003eΔα: angle the mirror turns during the round trip (rad); the returning beam is deflected by 2Δα; ω: angular speed of the mirror (rad\/s); D: distance to the fixed mirror (m); c: speed of light (m\/s)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/ole-romer-sticker-io-isnt-late-light-is-physics-sticker\"\u003eOle Rømer\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eIo's eclipses behind Jupiter ran later when Earth was farther from Jupiter, because the light had farther to travel. Rømer used that delay to show that light has a finite speed, crossing the width of Earth's orbit in a matter of minutes.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eΔt = Δd\/c\u003c\/strong\u003e\u003cbr\u003eΔ t: extra light travel time (s); Δ d: extra distance light must travel (m); c: speed of light in vacuum (m\/s); AU: astronomical unit, about the mean Earth–Sun distance (exactly 149 597 870 700 m since 2012) (m)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/aberra-bradley-aberration\"\u003eAberra\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eFrom 1725 James Bradley tracked the star Gamma Draconis and found it shifted through the year by about 20 arcseconds, but in a pattern that did not fit parallax. In 1729 he explained it (Phil. Trans. R. Soc. 35, 637): Earth is moving, so the telescope must be tilted slightly forward to catch starlight, like slanting an umbrella when you walk in rain. The tilt is set by Earth’s orbital speed compared with the speed of light.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eθ ≈ v\/c ≈ 20.5″\u003c\/strong\u003e\u003cbr\u003eθ: aberration angle (rad, here ≈ 20.5 arcseconds); v: Earth’s orbital speed, ≈ 30 km\/s; c: speed of light (m\/s)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/minkowskara-photon-momentum\"\u003eMinkowskara\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eHow much momentum does light carry inside a medium of refractive index n? Hermann Minkowski (1908) gave a momentum that grows with n; Max Abraham (1909–1910) gave one that shrinks as 1\/n. Both follow from consistent electrodynamics, and experiments have seemed to support each in different setups.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003ep\u003csub\u003eMinkowski\u003c\/sub\u003e = nħω\/c,   p\u003csub\u003eAbraham\u003c\/sub\u003e = ħω\/(nc)\u003c\/strong\u003e\u003cbr\u003ep: momentum of one photon in the medium; n: refractive index of the medium; ħ: reduced Planck constant (exact in the 2019 SI); ω: angular frequency of the light; c: speed of light in vacuum (exact)\u003c\/p\u003e\u003ch3\u003e\u003ca href=\"https:\/\/pixelatedphysics.com\/products\/mirrorix-ligo-test-mass\"\u003eMirrorix\u003c\/a\u003e\u003c\/h3\u003e\u003cp\u003eLIGO measures gravitational waves by comparing the lengths of two 4 km arms with laser light. At the end of each arm hangs a test mass: a 40 kg fused-silica mirror 34 cm across, hung from a four-stage pendulum to isolate it from ground vibration (Advanced LIGO, Class. Quantum Grav. 32, 074001, 2015). On 14 September 2015 a wave from two merging black holes, GW150914, stretched and squeezed the arms with a peak strain of 1.0 × 10⁻²¹ (Phys. Rev. Lett. 116, 061102, 2016).\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eh ≈ ΔL\/L\u003c\/strong\u003e\u003cbr\u003eh: gravitational-wave strain, dimensionless; ΔL: change in the difference of the arm lengths (m); L: arm length, 4 km\u003c\/p\u003e","brand":"Printify","offers":[{"title":"White \/ Kiss-Cut \/ 5.83″ × 8.27″","offer_id":50967001464935,"sku":"26281982849352452674","price":19.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/11092993536271825152_2048.jpg?v=1793676531"}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/collections\/sticker-sheets.png?v=1793679132","url":"https:\/\/pixelatedphysics.com\/nl\/collections\/sticker-sheets.oembed","provider":"Pixelated Physics","version":"1.0","type":"link"}