Illustrated Physics
physics, drawn
Concept

Blackbody — where it appears

A body that absorbs everything falling on it, whose emitted spectrum depends on its temperature and on nothing else about it. Its total output goes as the fourth power of temperature and its peak wavelength as the reciprocal of it, so a hotter body is both brighter and bluer.

Named by 16 essays across 6 fields — each of them below, with the objects they name alongside it.

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.

The curve that would not come down

Classical physics predicted that a warm object radiates infinite power at short wavelengths. Every step of the derivation was correct, the prediction was absurd, and closing the gap required assuming that energy comes in lumps.

quantum · Blackbody Photocurrent against applied voltage, at two intensities. Photocurrent against retarding voltage for caesium lit at 12×10¹⁴ hertz, at relative intensities of 1 and 2. Both curves reach zero at the same stopping voltage of 2.82 volts, because the brighter light delivers more photons and not more energetic ones. The saturation currents are in the ratio of the intensities. The slope between the stopping voltage and zero assumes the emitted electrons' energies are spread uniformly below the maximum, which they are not; the crossing point does not depend on that assumption.

Light arrives in lumps, and brightness only changes how many

Shine dim blue light on a metal and electrons come out. Shine intense red light and none do, however long the wait. The frequency decides whether anything happens; the intensity decides only how much.

quantum · Photon Which grains the light wins. The radiation force on a spherical grain divided by the gravitational force on it, against the grain's radius, on logarithmic axes. Both forces fall as the inverse square of the distance, so the ratio does not depend on how far away the grain is — only on how big it is. Light acts on the cross-section and gravity on the volume, so the ratio goes as 1/a, and the two are equal at 287 nm for material of density 2000 kg/m³. Anything smaller than that is expelled; anything larger stays.

Light has a pressure

Sunlight pushes on a square metre with about the weight of a grain of sand, which sounds like a curiosity until the object being pushed is small enough. The demonstration in every school cupboard turns the wrong way, and the reason it does is more interesting than the effect it is supposed to show.

astrophysics · Radiation pressure The most any concentrator is allowed. The greatest concentration a receiver can be given, against the half-angle it accepts, both logarithmically, for a receiver in a medium of index 1. The upper curve is a three-dimensional concentrator, n²/sin²θ; the lower is a trough, which concentrates in one direction only, n/sin θ. The Sun's angular radius is 0.267°, so sunlight can be concentrated by at most 46165 times in a dish and 215 times in a trough — marked. Nothing about glass, mirrors, wavelength or aperture appears in either expression. The bound is thermodynamic: a receiver that accepts light out to θ also radiates out to θ, and the concentration at which what it emits balances what it absorbs is exactly where these curves are.

The brightness no lens can increase

A lens can make an image smaller and therefore hotter, and there is a temperature at which it stops — the temperature of the source. Every arrangement of glass and mirrors ever built obeys a bound that contains no wavelength, no aperture and no material — only the angle the receiver is allowed to accept — and the bound comes from thermodynamics rather than from optics.

optics · Etendue What a thermal camera reads off a shiny surface. The temperature a camera calibrated for a black body reports, against the emissivity of the surface it is pointed at, for surfaces truly at 323 K, 373 K, 473 K in a room at 293 K. Every curve begins at the true temperature when the emissivity is one and ends at the room's temperature when it is zero, because a surface that emits nothing reflects everything and the camera is then looking at the room. polished aluminium at 0.05 reads 312 K; stainless steel, oxidised at 0.8 reads 451 K; matt black paint at 0.95 reads 468 K; human skin at 0.98 reads 471 K for a surface truly at 473 K. That is not a fault in the instrument. It follows from Kirchhoff's law — a poor emitter is a poor absorber and therefore a good reflector — so the shortfall in a shiny surface's own glow is made up almost exactly by whatever is reflected in it, and no measurement of the light leaving a surface can separate the two without knowing the emissivity in advance.

The glow that says nothing about the surface

A thermal camera pointed at a saucepan of boiling water reads a hundred degrees. Pointed at a polished aluminium block at the same temperature it reads about twenty-five, and the instrument is working perfectly. What it is measuring is emissivity as much as temperature — and a surface's emissivity is forced to equal its absorptivity, at every wavelength and every angle, by an argument with no physics of matter in it at all.

thermodynamics · Blackbody How many modes a square has below a given wavenumber. The number of vibration modes of a square with wavenumber below k, counted exactly — every eigenvalue of this region is a closed form, so the staircase is the true count and not an estimate. There are 265 of them below k = 60. The smooth curves are what Weyl's law predicts. The upper one is the leading term alone, the area times k² over 4π, and it is too high by 21.5 modes at the right-hand edge; the lower one subtracts the perimeter term, the perimeter times k over 4π, and is out by 2.4. The content of the law is that the count depends on the region through its area and its perimeter and — to this order — through nothing else at all: not through its shape, not through where its corners are, not through whether it is convex. The staircase's steps are the individual modes, and they cluster where two different pairs of indices give the same wavenumber. That the count is smooth in the large while being a staircase in the small is what makes a mode count usable in thermodynamics, where it appears as a density of states and never as a list.

How many ways there are to vibrate

A drum has infinitely many modes, and below any given frequency it has a finite number of them. That number turns out to depend on the drum's area and the length of its rim and — to the accuracy anybody uses — on nothing else about its shape. Almost every result in thermal physics that involves waves is an application of that count.

waves · Standing waves Energy, pressure and entropy of a gas nobody counted. The energy density, pressure and entropy density of blackbody radiation against temperature, on logarithmic axes, together with the pressure a monatomic gas of the same energy density would have. Every curve is a power of the temperature — the fourth for energy and pressure, the third for entropy — because the only length in the problem is the thermal wavelength and the only energy is kT. The pressure is exactly a third of the energy density, where an ordinary gas's is two thirds, a factor of 2: a photon carries momentum E/c and a slow molecule carries √(2mE), and that difference is the whole of it. Some values: at room temperature the radiation pressure is 1.86e-6 pascals, which is a ten thousand millionth of an atmosphere; at 1e+7 kelvin it is 2.52e+12, which is where radiation rather than matter holds a star up.

The gas that nobody counted

A box of gas holds however many molecules were put in it. A box of radiation holds however many photons the temperature says, because the walls make and destroy them until the free energy is least — and one dropped assumption changes every result. The pressure becomes a third of the energy density instead of two thirds, the entropy goes as the cube of the temperature, and the adiabatic index comes out at exactly four thirds.

thermodynamics · Blackbody Colder the bigger it is. The Hawking temperature against mass, on logarithmic axes, with the microwave background drawn across it. The slope is minus one exactly, so a heavier hole is colder — a negative heat capacity, which is the fact everything else here follows from. The two lines cross at 4.50e+22 kg, about a hundredth of the Moon's mass. Anything heavier than that is colder than the sky it sits in and absorbs more than it emits, so it grows rather than evaporates. A stellar-mass hole is at 6.2e-8 kelvin and will not begin to lose mass until the background has cooled below that, which takes something like 10¹² years. Evaporation is not something happening now to any hole anybody has observed.

The hole that outlives everything and then does not

A black hole radiates at a temperature that rises as it shrinks, so losing energy makes it lose faster. The whole history follows from that one sign: a life proportional to the cube of the mass, nearly nothing happening for almost all of it, and an end that arrives in a second.

astrophysics · Horizons The temperature a planet ought to be. Each body's measured surface temperature against the temperature at which it would radiate away exactly the sunlight it absorbs — computed from two numbers, the sunlight reaching it and the fraction it reflects, with no other property of the body used. Points on the diagonal are bodies the one-line argument gets right. the Moon: balance 270 K, surface 250 K, −20 K with no atmosphere; Mercury: balance 433 K, surface 340 K, −93 K with no atmosphere; Mars: balance 210 K, surface 210 K, +0 K; Earth: balance 255 K, surface 288 K, +33 K; Venus: balance 227 K, surface 737 K, +510 K. The airless bodies fall below the line rather than on it, and the reason is the fourth power: a surface running from noon heat to night cold radiates like its hottest parts and averages like its coldest, so a mean thermometer reading is lower than the temperature that matches the emitted flux. Mars, whose atmosphere is thin and whose surface is nearly isothermal by comparison, sits on the line. Everything with a substantial atmosphere sits above it, by tens of kelvin on Earth and hundreds on Venus, and always in the same direction. Nothing here explains why. What the figure fixes is the size and the sign of what has to be explained.

The height a planet is seen from

A body in sunlight settles where it radiates away what it absorbs, and that takes two numbers and one line of arithmetic. It gets the Moon right and Earth wrong by thirty-three kelvin. The correction is not that the atmosphere traps heat but that it moves the level space sees the planet from, and the rest is done by a lapse rate that is not a radiative quantity at all.

thermodynamics · Blackbody Two temperatures for the same sunlight. Sunlight described two ways, against how much it has been concentrated. The flat line is the temperature its spectrum belongs to — 5,762 K, the Sun's surface, which concentration does not change because a mirror does not alter a photon's energy. The rising curve is the temperature a blackbody would need in order to radiate the flux actually arriving: 394 K unconcentrated, 2213 K under a parabolic dish, and 5771 K at the geometric limit, where the two meet — computed here and checked against the Sun's own temperature, because a perfect concentrator reproduces the source's radiance and cannot exceed it. The gap between the two curves is the dilution, and dilution is entropy: the same energy spread over a hundred thousand times more directions occupies a hundred thousand times more modes. That is what a converter has to carry, and it is why the ceiling on solar conversion is not the Carnot efficiency between 5,762 K and 300 K.

The work a diluted beam will not do

Sunlight at the top of the atmosphere has the spectrum of a body at 5,762 kelvin and the energy flux of one at 394. The mismatch is not an accident of units: the light has been spread over a hundred thousand times more modes than it left in, and that dilution is entropy. Run it through a heat engine unconcentrated and five per cent of it is available as work.

optics · Etendue The modes of a wire, counted. Nyquist's argument of 1928, with its count performed. Two resistors joined by a lossless line are in equilibrium, and the line is a one-dimensional cavity whose standing waves are spaced c/2L apart in frequency. Each mode has an electric and a magnetic energy, both quadratic, so equipartition gives it kT — the same half a kT per quadratic term that a heat capacity counts. The upper points are how many modes fall in a band of a hundred and thirty-seven megahertz, for five line lengths, from 17 on a thirteen-metre line to 8,558 on one of six kilometres. The lower points are the power each end therefore receives, as a fraction of kTΔf, and they converge on one: a longer line has proportionally more modes and takes proportionally longer to deliver them, so the length cancels. The longest line lands within 0.005 per cent and the shortest is 4.5 per cent low, because thirteen metres holds only seventeen whole modes in the band and the remainder is a real granularity rather than an error. What is left in the limit is kTΔf — a noise power with no resistance in it at all, and none of the line's properties either.

Half a kT in a piece of wire

Count the quadratic terms in a molecule's energy and equipartition gives a heat capacity. Count them on a transmission line instead and the same theorem gives a resistor's noise voltage — 4kTRΔf, with nothing in it about what the resistor is made of. A fifty-ohm input at room temperature says 0.91 nanovolts in every root hertz, and no design removes it.

thermodynamics · Equipartition The one prediction the Planck scale makes. The energy density the vacuum should have, from summing the zero-point energy of a field's modes up to a cutoff, against where that cutoff is put — thirty decades of cutoff energy and a hundred and thirty of density, both logarithmic. The horizontal line is what is measured: 5.34e-10 joules per cubic metre, the dark energy that accounts for sixty-nine per cent of the universe. Cutting the sum off at the Planck energy — which is where dimensional analysis says every description available runs out — overshoots it by 10^121. That is the largest disagreement between an estimate and a measurement anywhere in physics, and the slope of the line is why it cannot be argued away: the density goes as the fourth power of the cutoff, so cutting off at the electroweak scale still overshoots by 10^54 and cutting off at one electronvolt — below which no physics is in doubt at all — still overshoots by 10^8. The cutoff that would give the right answer is 8.0e-3 electronvolts, which is a wavelength of about a tenth of a millimetre and corresponds to no known physics whatever.

The estimate that misses by a hundred and twenty

Every argument about the Planck scale is an argument about consistency rather than about data, with one exception. The zero-point energy of the quantum fields gravitates, dimensional analysis at the Planck cutoff says how much, and what is measured is 10¹²¹ times smaller. It is the largest disagreement between an estimate and a measurement anywhere in physics, and lowering the cutoff does not rescue it.

astrophysics · Planck scale A blackbody in every direction, at a different temperature in each. The spectrum of a blackbody at 100 kelvin in its own frame, seen by an observer it is moving past at 0.5 of the speed of light, in 5 directions. Each curve is a Planck spectrum exactly — the Planck form survives a Doppler shift, with the temperature multiplied by the shift — and the temperatures run from 57.74 kelvin looking one way to 173.21 looking the other. So the body is a perfect blackbody in each direction and has no single temperature. A thermometer placed in the radiation reads something between, and what it reads depends on where it is put and on how much of the sky it sees — which is the reason a transformation law for temperature was argued about for sixty years without being found.

The body that has no temperature when it moves

Energy, momentum, length, duration and field strength all change when the observer moves. Temperature was argued about for sixty years, with three transformation laws proposed and each defended by people making no mistake. The resolution is that a moving blackbody is a perfect blackbody in every direction at a different temperature in each — so a thermometer's reading depends on where it is put, and the quantity the law was for is not there.

relativity · Relativistic thermodynamics The bath does push back, by an unmeasurable amount. The retarding force on a perfectly absorbing body moving through isotropic radiation at 2.725 kelvin, against its speed, for three areas. Moving through a bath of radiation is not free: the radiation arriving from ahead is blue-shifted and more intense and the radiation from behind is red-shifted and weaker, so the body absorbs more momentum from the front than from the back and decelerates. A square metre at half the speed of light feels 3.7e-14 newtons. That is the reason a preferred frame exists without relativity being violated: the laws are the same in every frame and the radiation is not — it is a physical system with a state, and its state picks out the frame in which it is isotropic, exactly as a body of water does.

The bath that pushes back

Moving through a bath of radiation is not free. The light arriving from ahead is blue-shifted and more intense and the light from behind is weaker, so a body absorbs more momentum from the front than from the back and slows down. That drag picks out the frame in which the radiation is isotropic — without violating relativity, because the laws are the same in every frame and the radiation is not.

relativity · Relativistic thermodynamics Equilibrium is where the entropy peaks, and there the temperatures differ. Two cavities of radiation, one high in a gravitational field and one low, free to exchange energy, with the clock at the bottom running at 0.8 of the rate of the one at the top. What is conserved is the energy either would deliver to a distant observer, so energy held at the bottom counts for 0.8 of its local value. Across: the share of that conserved energy held at the top. Above: the total entropy of the two gases. Below: the temperature a thermometer in the top cavity reads, as a fraction of one in the bottom cavity. The entropy peaks at a share of 0.339, found by search, and there the top cavity is at 0.800 of the bottom's temperature — the clock-rate ratio exactly. Where the two local temperatures are equal, at a share of 0.556, the entropy is 1.82 per cent below its peak and energy still flows downward, into the deeper cavity.

The column that is hotter at the bottom

Two bodies in equilibrium have the same temperature — that is what equilibrium was supposed to mean. In a gravitational field it is false. A column left alone until nothing in it changes is warmer at the bottom by exactly the factor by which clocks there run slow, a part in ten million billion per metre on the Earth and more than a per cent across the outer kilometre of a neutron star, and near a black hole's horizon the equilibrium temperature grows without limit.

relativity · Relativistic thermodynamics The light of a diode at room temperature is as bright as a surface thousands of kelvin hot. How many photons occupy each mode of the light, on a logarithmic scale, against photon energy. The lowest curve is the thermal glow of a 1.42 eV semiconductor at 300 K with no voltage across it. The solid curve above it is the same device with 1.3 V across it, emitting only above its gap. The dashed curve is a blackbody at 2571 K, the temperature whose light has the same occupation as the diode's at 1.472 eV, just above the gap. They cross there and nowhere else: the diode's occupation falls a factor e every 25.9 meV, as its lattice's temperature requires, and the blackbody's every 222 meV. No single temperature describes the diode's light. At each photon energy it has a brightness temperature, and that temperature is 300 K multiplied by ε/(ε − qV).

The glow that carries a voltage

Thermal radiation has no chemical potential, because walls make and destroy photons freely. A light-emitting diode is a body that glows at room temperature with a voltage written into its light — Planck's law with the voltage as the photons' chemical potential — which is why its light can be as bright as a surface thousands of kelvin hot, why at low voltage it can put out more light than the power it draws and cool itself doing so, and why a reverse voltage makes a surface look colder than it is.

thermodynamics · Chemical potential

Named alongside it

The objects these essays reach for when they reach for this one.

TemperatureCosmic microwave backgroundThe second lawDetailed balanceEntropyEquilibriumRadiation pressureThermal equilibriumAbsorptionChemical potentialConcentrationEmissivity

All concepts