Forces & Fields Vol. 1 Pixel Sticker Sheet

7 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: Forces & Fields Vol. 1. Every character below has its own sticker page with the full physics panel.

Azurwake

A 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.

cos θ = 1/(nβ), β > 1/n
θ: 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)

Bremsella

Any 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.

P = q²a²/(6πε₀c³)
P: 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)

Capacitwin

Charge 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.

ΔE = ½ · C₁C₂/(C₁ + C₂) · (V₁ − V₂)²
ΔE: energy lost when the capacitors are joined (J); C₁, C₂: capacitances (F); V₁, V₂: starting voltages (V)

Diagonix

A 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.

I = I₀ cos²θ
I: 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)

Gyrolla

A 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.

r = mv/(qB)
q: 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)

Kibbla

A 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.

mgv = UI
m: test mass (kg); g: local gravitational acceleration (m/s²); v: coil velocity (m/s); U: induced voltage (V); I: coil current (A)

Lenzor

A 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.

ℰ = −dΦ_B/dt
mathcalE: 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)