Forces & Fields Vol. 2 Pixel Sticker Sheet

7 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: Forces & Fields Vol. 2. Every character below has its own sticker page with the full physics panel.

Minkowskara

How 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.

p_Minkowski = nħω/c, p_Abraham = ħω/(nc)

p: 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)

Penninga

Earnshaw'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_+), an axial bounce along the field (ω_z), and a slow magnetron drift (ω_−). 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).

ω_c² = ω_+² + ω_z² + ω_−²

ω_c: free-space cyclotron frequency, qB/m (rad/s); ω_+: modified cyclotron frequency (rad/s); ω_z: axial frequency (rad/s); ω_−: magnetron frequency (rad/s)

Refracta

Light 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.

sin θ₁/v₁ = sin θ₂/v₂

θ₁: 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)

Scatterina

Small 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.

I_s ∝ 1/λ⁴

I_s: power scattered by a molecule much smaller than the wavelength (W); λ: wavelength of the light (m)

Thomsonia

When 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.

σ_T = (8π/3) r_e² = 6.652 458 7051(62) × 10⁻²⁹ m²

σ_T: Thomson cross section; r_e: classical electron radius, e²/(4πε₀m_ec²) = 2.817 940 3205(13) × 10⁻¹⁵ m; e: elementary charge (exact); ε₀: vacuum permittivity (measured); m_e: electron mass (measured); c: speed of light (exact)

Gravitas

G 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.

G = 6.674 30(15) × 10⁻¹¹ m³ kg⁻¹ s⁻²

G: Newtonian constant of gravitation; (15): standard uncertainty in the last two digits, 2.2 × 10⁻⁵ relative

Moderon

A 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.

α = ((A − 1)/(A + 1))²

A: 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)