Physics Starter Pack Pixel Sticker Sheet

12 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.

Isaac Newton

Newton'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.

F = dp/dt

F: net force (N); p: momentum (kg·m/s); G: gravitational constant (m³·kg⁻¹·s⁻²); m₁, m₂: masses (kg); r: separation (m)

Richard Feynman

Feynman'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.

K(b,a) = Σ_paths e^(iS/ħ)

K(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

Marie Curie

Each 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.

N(t) = N₀ e^(−λt)

N: number of undecayed nuclei; λ: decay constant (s⁻¹); t_(1/2): half-life (s); A: activity (1 Bq = 1 decay per second) (Bq)

Stephen Hawking

Hawking 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.

T_H = ħc³/(8πGMk_B)

T_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)

Nikola Tesla

Power 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.

V(t) = V₀ sin(ωt)

V: 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)

Max Planck

Planck 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.

E = hν

E: 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)

Niels Bohr

In 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.

E_n = −R_∞hc/n², with R_∞hc ≈ 13.6 eV

E_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)

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)

Sortex

Landauer 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.

Q_erase ≥ k_BT ln 2

Q_(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

Relicta

The 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.

T(z) = T₀(1 + z)

T(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)

Condensa

Below 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.

N₀/N = 1 − (T/T_c)³ᐟ² (T ≤ T_c)

N₀: 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)

Libratrix

Two 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.

r_L2 ≈ R(M₂/(3M₁))¹ᐟ³

r_(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)