{"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","url":"https:\/\/pixelatedphysics.com\/nl\/products\/fields-forever-sticker-sheet","provider":"Pixelated Physics","version":"1.0","type":"link"}