Relativity & Cosmos Pixel Sticker Sheet

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

Magnetoconda

Move a magnet past a loop of wire and a current flows. Hold the magnet still and move the wire instead, at the same relative speed, and exactly the same current flows. Yet the textbook stories differ. If the magnet moves, a changing magnetic field makes an electric field that drives the charges. If the wire moves, there is no electric field in the magnet's frame, and the magnetic force on the moving charges does the job. A rod of length L moving at speed v across a field B gets the same emf either way.

ε = vBL
ε: induced emf (volt, V); v: relative speed of rod and magnet, perpendicular to the field (m/s); B: magnetic flux density (tesla, T); L: length of the rod across the field (m)

Aberra

From 1725 James Bradley tracked the star Gamma Draconis and found it shifted through the year by about 20 arcseconds, but in a pattern that did not fit parallax. In 1729 he explained it (Phil. Trans. R. Soc. 35, 637): Earth is moving, so the telescope must be tilted slightly forward to catch starlight, like slanting an umbrella when you walk in rain. The tilt is set by Earth’s orbital speed compared with the speed of light.

θ ≈ v/c ≈ 20.5″
θ: aberration angle (rad, here ≈ 20.5 arcseconds); v: Earth’s orbital speed, ≈ 30 km/s; c: speed of light (m/s)

Faintsol

As the Sun fuses hydrogen into helium, its core gets denser and hotter, so it slowly brightens. Standard solar models give about 70% of today's luminosity some 4.5 billion years ago. With today's atmosphere and albedo, that would leave the early Earth's average temperature below freezing for roughly its first two billion years. But the geological record shows liquid water on Earth more than 3.8 billion years ago. Carl Sagan and George Mullen stated this paradox in 1972.

L(t) ≈ [1 + (2/5)(1 − t/t_⊙)]⁻¹ L_⊙
L(t): solar luminosity at age t; t: age of the Sun at that time; t_⊙: present age of the Sun, about 4.57 billion years; L_⊙: present solar luminosity, 3.828 × 10²⁶ W (IAU nominal value)

Mirrorix

LIGO measures gravitational waves by comparing the lengths of two 4 km arms with laser light. At the end of each arm hangs a test mass: a 40 kg fused-silica mirror 34 cm across, hung from a four-stage pendulum to isolate it from ground vibration (Advanced LIGO, Class. Quantum Grav. 32, 074001, 2015). On 14 September 2015 a wave from two merging black holes, GW150914, stretched and squeezed the arms with a peak strain of 1.0 × 10⁻²¹ (Phys. Rev. Lett. 116, 061102, 2016).

h ≈ ΔL/L
h: gravitational-wave strain, dimensionless; ΔL: change in the difference of the arm lengths (m); L: arm length, 4 km

Skyrider

In 1912 Victor Hess carried electroscopes up in balloons to 5 350 m (Phys. Z. 13, 1084). Their discharge rate, a measure of ionising radiation, first dipped slightly, then climbed steeply with height. The radiation was coming from above, not from the ground. A flight during a near-total solar eclipse and at night showed no drop, so the Sun was not the source. He shared the 1936 Nobel Prize for cosmic rays.

ionisation rate rises with altitude
altitude: height above sea level, up to 5 350 m in 1912; ionisation rate: ions formed per cm³ per second in the electroscope

Primordix

In the first minutes after the Big Bang, the ratio of neutrons to protons froze out at a small fixed value. Nearly all those neutrons ended up in helium-4, which predicts the fraction of ordinary mass that came out as helium, matching observations.

Y_p ≃ 2(n/p)/(1 + n/p)
Y_p: primordial helium-4 fraction by mass (observed 0.245 ± 0.003); n/p: neutron-to-proton ratio; Q: rest-energy difference between neutron and proton (MeV); k_B: Boltzmann constant (eV/K); T_f: temperature when the weak reactions freeze out (thermal energy about 0.7–0.8 MeV) (K); m_n, m_p: neutron and proton masses (kg)

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)

Orbitick

GPS clocks feel weaker gravity than clocks on the ground, so they run fast, and their orbital speed makes them run slightly slow. The net effect would build up timing errors quickly, so the satellite clocks are deliberately set to tick slightly slower before launch.

Δf/f ≈ ΔΦ/c² − v²/(2c²)
Δ f/f: fractional rate of a GPS satellite clock relative to a clock on the geoid (positive: it runs fast); ΔΦ: gravitational potential at the satellite minus the effective potential on the geoid, which includes Earth's rotation term (J/kg); v: orbital speed of the satellite (m/s); c: speed of light in vacuum (m/s); f₀: factory-specified frequency of the clock's 10.23 MHz reference before launch (Hz)

Troutona

If Earth moved through an ether, a charged capacitor hanging on a fine wire should feel a torque turning its plates to face the motion. In 1903 Frederick Trouton and H. R. Noble looked for it (Phil. Trans. A 202, 165, published 1904). They found none: the torque was zero.

torque = 0
torque: turning effect on the suspended capacitor (N·m)

Mirrorspin

In 1862 Léon Foucault measured the speed of light in a laboratory with a rapidly rotating mirror (Comptes rendus 55, 501). Light reflects off the spinning mirror to a distant fixed mirror and back. In the round-trip time the rotating mirror turns a tiny angle, so the returning spot lands slightly displaced. From the displacement, the path length and the spin rate he got c = 298 000 ± 500 km/s, close to the modern value.

Δα = 2ωD/c
Δα: angle the mirror turns during the round trip (rad); the returning beam is deflected by 2Δα; ω: angular speed of the mirror (rad/s); D: distance to the fixed mirror (m); c: speed of light (m/s)

Muonaut

Muons made high in the atmosphere live only about two microseconds, too short to cross the atmosphere at face value. Moving near the speed of light, their clocks run slow by the factor γ, so many reach the ground; in the muon's frame, the atmosphere is shorter instead.

Δt = γτ, L = L₀/γ
Δ t: muon lifetime as timed by clocks on the ground (s); τ: mean lifetime in the muon's own rest frame (PDG 2024: 2.1969811 μs; the half-life is shorter, 1.52 μs) (s); γ: Lorentz factor; L₀: thickness of the atmosphere in the ground frame (m); L: the same thickness measured in the muon's frame (m); v: muon speed (m/s)