Illustrated Physics
physics, drawn
How this site is made

The figure library

Every picture here is generated from code at build time. This page lists each family of figures in its plainest form, with every essay that draws on it.

No figure here is a drawing that was made once and saved. Each one is a function: it takes parameters and returns SVG, so the same generator produces the 27° incline and the 5° incline without either being redrawn.

That is the reason the collection can keep growing without the illustrations drifting apart. A generator is written once, checked once, and every essay that calls it inherits the same line weights, the same colour roles, and the same behaviour in dark mode. There are 103 of them so far.

Each is listed with the essays that call it, most-reused first. That ordering is not decoration: a generator used once is a design failure rather than a neutral cost, and a generator used twenty times has a blast radius worth knowing before it is edited. The count below is what npm run phase gates — it refuses to advance a phase in which figures per generator has fallen.

relativistic-energy

29 essays

Total energy against speed, in units of the rest energy. The total energy of a moving body divided by its rest energy, against speed as a fraction of the speed of light. The Newtonian answer, one plus half v squared over c squared, is drawn beside it: the two agree to 0.004 per cent at a tenth of light speed and disagree by 39 per cent at nine-tenths. The relativistic curve has a vertical asymptote at c, which is why nothing with mass reaches it.

Every branch relativistic-energy draws, and what checks it

microstates

25 essays

Ways to arrange 10 coins. The number of distinct arrangements giving each number of heads, for 10 coins. Every individual arrangement is equally likely; the middle wins because there are more ways to reach it.

Every branch microstates draws, and what checks it

energy-levels

22 essays

The energy ladder of hydrogen. The first 6 energy levels of hydrogen, drawn to scale in eV, at -13.61, -3.40, -1.51, -0.85, -0.54, -0.38. The levels crowd toward zero rather than spreading out, so the levels have a top and an atom has an ionisation energy. The arrow marks a transition: 3 to 2 releases 1.890 eV, a photon at 656.1 nm.

Every branch energy-levels draws, and what checks it

viscosity-temperature

22 essays

One word, two mechanisms, opposite signs. Viscosity on a logarithmic axis against temperature over the range 280 to 360 kelvin, where gases and liquids can both be measured. The gases rise and the liquids fall, and the two families are separated by three decades of magnitude as well as by sign. The logarithmic slopes at the middle of the range are 0.76 for air, 0.69 for helium, -6.13 for water, -5.41 for ethanol, -23.00 for glycerol, so the steepest liquid responds 30 times more strongly than the gas and in the other direction. Nothing about the word viscosity requires this: what is being measured in both cases is the ratio of a shear stress to a shear rate, and that definition says nothing about what carries the momentum. In a gas it is molecules in free flight, so heating speeds up the carriers; in a liquid the molecules are permanently in contact and what has to happen is one of them getting past its neighbours, so heating removes an obstacle rather than adding a carrier. The obvious question this raises is what a dense gas near its critical point does, where neither picture holds, and the honest answer is that neither formula on this chart applies there at all.

Every branch viscosity-temperature draws, and what checks it

phase-boundary

19 essays

The phase boundary of water, from one equation. Pressure against temperature for water on a logarithmic pressure axis spanning 6.6 decades. The vaporisation curve is integrated from Clausius–Clapeyron between the triple point at 273.16 kelvin and 611.7 Pa and the critical point at 647.096 kelvin, with a single latent heat of 43.32 kilojoules per mole — the value the two published points on the curve imply. The measured latent heats are 45.05 at the triple point and 40.65 at the reference point, and the fitted value sits between them, because a constant latent heat is an average over the interval. The sublimation curve below the triple point is not measured but predicted, from the two latent heats adding where all three boundaries meet: 51.1 kilojoules per mole, which reaches 103.2 Pa at 253.1 kelvin against a measured 253.15. The melting curve is drawn at the slope Clapeyron gives it, -13.5 megapascals per kelvin, which is a volume ratio and nothing else: water's solid is 917 against 1000 kilograms per cubic metre for its liquid, so melting shrinks it and the line leans backwards. Across the whole 6.6 decades of this axis that line moves 5.2 kelvin, and one atmosphere shifts the melting point by 0.0075 kelvin. At 1 atmosphere the boundary is crossed at 373.1 kelvin, where water boils. The one place the curve fails is its top end: a constant latent heat reaches 37.4 MPa at the critical temperature where the measured critical pressure is 22.1 MPa, 70 per cent high, because the latent heat falls to zero at the critical point and this curve does not know that.

Every branch phase-boundary draws, and what checks it

band-formation

17 essays

One well, two wells, and the band they become. The energy levels of a chain of identical wells, for 1, 2, 3, 6, 12, 40 of them, with an on-site energy of -4 eV and a coupling of -0.9 eV between neighbours. One well has one level. Two split it into two, 1.80 eV apart. By 40 the levels have filled a band 3.59 eV wide, which is closing on the limit of four times the coupling, 3.60 eV — and no further widening happens however many more wells are added. The count of levels grows with the number of wells; the width of the band does not.

Every branch band-formation draws, and what checks it

em-wave

17 essays

The same loop, spanned two ways. A 100 cm² capacitor with a 2 mm gap, charged at 1.0 million volts per second. A loop drawn round the wire can be spanned by a flat surface, which the current of 44.271 µA passes through, or by a bag-shaped surface that passes between the plates, which no charge crosses at all. Ampère's law as it stood gave two different answers for one circulation. The rate of change of electric flux between the plates, multiplied by ε₀, is 44.271 µA — the same number to every figure, because C = ε₀A/d is the same ε₀A/d either way. That is the term, and it is not an approximation or a correction: it is what makes the law consistent at all.

Every branch em-wave draws, and what checks it

spacetime

17 essays

A spacetime diagram at β = 0.5. Position across, time up, in units where light travels at 45°. The shaded wedges are the future and past reachable by light; the tilted axes belong to an observer moving at 0.5 of the speed of light.

Every branch spacetime draws, and what checks it

flux-freezing

16 essays

What a collapse does to a field. A body of radius 700,000 km carrying a field of 0.01 T, collapsing to 10 km. The flux through every comoving loop is fixed, so B goes as 1/R² and the field reaches 4.9·10⁷ T — a compression of 7·10⁴ in radius bought a factor of 4.9·10⁹ in field. The line has slope exactly −2 and that is the only claim being made: what a real object ends up with also depends on how well the flux was held, and on what generated it.

Every branch flux-freezing draws, and what checks it

interferometer

16 essays

Two ports, and one of them dark. The two outputs of a balanced two-path interferometer against the phase added to one arm, in turns. With the arms equal, every photon leaves by the same port and the other receives nothing at all — not a little, nothing, to 6.7e-16 across the whole sweep. The photon has not chosen a path and then been redirected; the two amplitudes for reaching the second port cancel, and cancellation is only available because both paths were taken. Half a turn moves every photon to the other port. This is the apparatus the interaction-free measurement is built on, and the dark port is the whole of the mechanism: a detector at a place where nothing ever arrives is an instrument of enormous sensitivity, because anything at all that arrives there is news.

Every branch interferometer draws, and what checks it

thin-lens

16 essays

A converging lens making a real image. An object 2.44 focal lengths from a thin converging lens. The image sits where the construction rays cross, at 1.69 focal lengths, magnified -0.69×.

Every branch thin-lens draws, and what checks it

chain-fall

15 essays

What a scale reads while a chain falls onto it. The reading of a scale, in units of the whole chain's weight, against the length of chain that has already landed, for two ways of putting the same chain down. Lowered gently, the scale reads the weight of what is resting on it and nothing else, so the reading climbs along the diagonal to one and stops. Dropped from rest with its lower end just touching, the scale reads three times that at every instant of the fall: one part is the pile's weight and two parts is the force needed to stop the links that are arriving, which is λv² with v² = 2gx and is therefore exactly twice λgx however far the fall has got. The peak, read off the drawn curve, is 3.00 chain weights. It is reached at the instant the last link lands, and the reading then falls discontinuously to one, because the momentum flux stops all at once. The discontinuity is the part a real experiment does not show — a real chain has links of a finite size and a scale has a response time — and it is the reason a chain dropped into a bucket on a kitchen scale reads high and then settles.

Every branch chain-fall draws, and what checks it

fourier-packet

15 essays

A packet, and the wavenumbers it is made of. Above: a wave packet built by adding a continuum of plane waves centred on wavenumber 12 with a spread of 1.6. Below: the weight given to each wavenumber. The packet's width, measured as the standard deviation of its probability, is 0.442; the spread of wavenumbers is 1.131; their product is 0.500, which is a half and cannot be less. Narrowing one bracket widens the other by exactly as much. Nothing quantum has been used to draw either panel.

Every branch fourier-packet draws, and what checks it

gradient-reflection

15 essays

Reflection against the thickness of the boundary. How much of the wave comes back, against the width of the transition between media of wavenumber 1 and 2.2, on a logarithmic scale. At zero width the curve reaches 1.406e-1, which is the abrupt-interface value computed from the two end values and nothing else — so the smooth calculation contains the sharp one as a limit rather than contradicting it. Past a width of about a wavelength the fall is a straight line on this scale, of measured slope -12.551 per unit width against the -12.566 the asymptotic form gives, which is 4π times the smaller of the two wavenumbers. Exponential, not merely small: three wavelengths of transition costs eight decades of reflection. The circles are the same quantity computed by slicing the profile into uniform layers, agreeing with the curve to a factor of 1.0115.

Every branch gradient-reflection draws, and what checks it

planck-spectrum

15 essays

The blackbody spectrum, against what classical physics predicted. Spectral exitance against wavelength for a blackbody at 3000, 4000, 5000 kelvin, in kilowatts per square metre per nanometre. Each curve peaks at the wavelength Wien's displacement law gives — 966 nm at 3000 K, 724 nm at 4000 K, 580 nm at 5000 K — and falls to nothing at short wavelengths.

Every branch planck-spectrum draws, and what checks it

solenoid-field

15 essays

The end of the winding is where the field is half. Field on the axis of a solenoid, in units of the infinite-solenoid value μ₀nI, with distance measured in half-lengths so that ±1 is the mouth of the winding whatever its length. At 1:1 the centre reaches 0.7075 of μ₀nI and the end plane 0.4472, a ratio of 0.6322; At 2.5:1 the centre reaches 0.9285 of μ₀nI and the end plane 0.4903, a ratio of 0.5280; At 5:1 the centre reaches 0.9806 of μ₀nI and the end plane 0.4975, a ratio of 0.5074; At 10:1 the centre reaches 0.9950 of μ₀nI and the end plane 0.4994, a ratio of 0.5019. Every point is a sum of Biot–Savart contributions from each of the turns, and the centre value agrees with the closed form for a continuous winding to 0.049 per cent. The half is exact only in the limit: a winding as long as it is wide gives 0.63, and the familiar rule is a statement about a solenoid nobody has.

Every branch solenoid-field draws, and what checks it