{"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","url":"https:\/\/pixelatedphysics.com\/tr\/products\/starter-pack-sticker-sheet","provider":"Pixelated Physics","version":"1.0","type":"link"}