Classical Physics Vol. 4 Pixel Sticker Sheet

10 pixel-art physics stickers on one A5 (5.83″ × 8.27″) white kiss-cut sheet: Classical Physics Vol. 4. Every character below has its own sticker page with the full physics panel.

Saltara

A fast, shallow flow can suddenly jump to a slow, deep one. Momentum conservation across the jump gives the ratio of the two depths in terms of the incoming Froude number, which compares flow speed with the speed of shallow-water waves.

Fr = v/√(gh)
Fr: Froude number: flow speed over the speed of shallow-water waves; v: depth-averaged flow speed (m/s); g: gravitational acceleration (m/s²); h: depth of the flow (m); h₁, h₂: depths just before and just after a planar jump in a straight channel (m); Fr₁: Froude number of the incoming flow

Vortessa

Flow past a cylinder sheds vortices from alternate sides, at a frequency set by the flow speed divided by the cylinder's diameter, times the Strouhal number. Von Kármán showed that the staggered vortex street is stable only for a particular spacing ratio.

f = St·U/D, St ≈ 0.2
f: shedding frequency: full left-right cycles per second (Hz); St: Strouhal number, about 0.2 to 0.21 for Re from about 300 to 200,000 (lower near onset); U: speed of the oncoming flow (m/s); D: diameter of the cylinder (m); Re: Reynolds number; ν: kinematic viscosity of the fluid (m²/s)

Weissenberga

Stir water with a spinning rod and the surface dips into a vortex, because inertia flings the liquid outward. Spin a rod in a polymer solution, such as dough or a dilute polyacrylamide solution, and the liquid climbs up the rod instead. Stretched along the circular streamlines, the long polymer chains act like stretched rubber bands. The resulting tension around the streamlines squeezes the fluid inward, towards the rod, and the only way it can go is up. Karl Weissenberg described this behaviour in “A continuum theory of rheological phenomena” (Nature 159, 310, 1947).

N₁ = τ₁₁ − τ₂₂ > 0
N₁: first normal stress difference (Pa); τ₁₁: normal stress along the flow direction (Pa); τ₂₂: normal stress across the flow, in the direction of the velocity gradient (Pa)

Winetear

Swirl a glass of strong wine and a thin film climbs the wall above the liquid, gathers into a ridge and falls back as drops: tears, or legs. Alcohol evaporates from the film faster than water, so the film becomes more watery, and water has a higher surface tension than alcohol. The difference in surface tension pulls liquid up the glass from the bulk below, where there is more alcohol. James Thomson, Lord Kelvin's brother, described it in 1855 (“On certain curious motions observable at the surfaces of wine and other alcoholic liquors”, Phil. Mag. 10, 330). A flow driven like this is now called a Marangoni flow.

τ = ∇γ
τ: shear stress on the liquid surface (Pa); ∇γ: gradient of surface tension along the surface (N/m²)

Langmuira

Stick a small electrode into a plasma and sweep its voltage. Irving Langmuir and Harold Mott-Smith worked out what the current tells you (“The Theory of Collectors in Gaseous Discharges”, Phys. Rev. 28, 727, 1926, after their 1924 papers). Below the plasma potential the probe repels most electrons, and only those with enough energy to climb the barrier arrive. For a Maxwellian electron gas that electron current rises exponentially with probe voltage, so the slope of ln I gives the electron temperature. Above the plasma potential the current saturates.

I_e ∝ exp(e(V − V_p)/(kT_e))
I_e: electron current to the probe (A); V: probe voltage (V); V_p: plasma potential (V); k: Boltzmann constant (J/K); T_e: electron temperature (K)

Candelor

The candela, the SI unit of luminous intensity, links watts of light to how bright that light looks to a standard human eye. It is defined by fixing the luminous efficacy of monochromatic radiation at 540 × 10¹² Hz (a green of about 555 nm) at exactly 683 lm/W. For other wavelengths the watts are weighted by the CIE photopic sensitivity curve V(λ).

Φ_v = K_m ∫ V(λ) Φ_e,λ dλ, K_cd = 683 lm/W
Φ_v: luminous flux (lumen, lm); K_cd: luminous efficacy of 540 THz radiation, 683 lm/W exactly (defining constant of the 2019 SI); K_m: maximum photopic efficacy, ≈ 683 lm/W; V(λ): CIE photopic luminous efficiency function, peak 1 near 555 nm; Φ_e,λ: spectral radiant flux (W/nm); λ: wavelength (nm)

Halotra

The real halo is made by millions of tiny hexagonal ice crystals in thin, high cirrus cloud, not by water droplets. Light entering one side face of a crystal and leaving by another passes through what is in effect a 60° prism. Every such prism has a minimum angle of deviation, and light piles up near it, so a bright ring appears at that angle from the Sun.

D_min = 2 arcsin(n sin 30°) − 60°
D_min: minimum deviation, the halo radius (degrees); n: refractive index of ice, about 1.31 for visible light; 30°: half the 60° prism angle between alternate faces

Nicolette

In 1828 William Nicol of Edinburgh cut a calcite rhomb along a diagonal and cemented the halves back together with Canada balsam (Edinb. New Phil. J. 6, 83). Calcite splits light into an ordinary ray (n_o ≈ 1.658) and an extraordinary ray (n_e ≈ 1.486, along the slowest direction). Balsam sits between them at about 1.55. The ordinary ray, meeting the lower-index balsam at a steep angle, is totally internally reflected out of the side; the extraordinary ray passes. What leaves the end is linearly polarised.

n_o > n_balsam > n_e
n_o: ordinary refractive index of calcite, ≈ 1.658; n_balsam: Canada balsam, ≈ 1.55; n_e: extraordinary index, ≈ 1.486 (sodium light)

Perota

Two parallel, partly reflecting mirrors trap light that bounces back and forth. Only wavelengths that fit a whole number of half-waves into the gap build up and pass through; the rest cancel out. The transmission is a comb of sharp peaks, spaced by the free spectral range.

FSR = c/(2nL)
FSR: free spectral range, the frequency spacing between peaks (Hz); c: speed of light (m/s); n: refractive index between the mirrors; L: mirror spacing (m)

Tweezra

Focus a laser beam tightly through a microscope lens and a small transparent bead is pulled towards the brightest point and held there. Light bends as it passes through the bead, and the change in its momentum pushes back on the bead. For a bead that bends light more than its surroundings, the net gradient force points up the intensity gradient, towards the focus. Arthur Ashkin accelerated and trapped particles with radiation pressure in 1970 (Phys. Rev. Lett. 24, 156), and with colleagues demonstrated the single-beam gradient trap, the optical tweezers, in 1986 (Opt. Lett. 11, 288).

F_grad ∝ ∇I
F_grad: gradient force on the bead (N); ∇I: gradient of the light intensity (W/m³)