Illustrated Physics
physics, drawn
Concept

Momentum — where it appears

Mass times velocity, conserved in every collision and read directly off the curvature of a charged track in a known field. Its conservation follows from space being the same everywhere, and it survives into relativity in the form γmv while mass times velocity does not.

Named by 20 essays across 7 fields — each of them below, with the objects they name alongside it.

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 Same field, same charge, three momenta. Electrons entering a 10 mT field at right angles to it, at 1, 4, 9 keV, each drawn for a quarter of its turn. The radius is mv/qB — 10.7 mm, 21.3 mm, 32.0 mm — so measuring the curvature of a track measures the momentum of whatever made it, which is how every particle detector since the cloud chamber has worked. The time to go once round is 2πm/qB = 3.57 ns for all three: the faster particle travels a proportionally longer way round and arrives at the same moment.

The force that does no work

A magnetic field can turn a moving charge through any angle at all and cannot add a joule to it. Everything a magnet is good for follows from that one prohibition — including the fact that a bent track is a reading of momentum, and that a machine built on it stops working at five kilovolts for an electron.

electromagnetism · Magnetism A collision with restitution 0.4. Two bodies before and after a head-on collision. Momentum is the same on both rows by construction; kinetic energy is only preserved when the collision is elastic.

The point that keeps moving as if nothing had happened

Newton's third law makes every internal force cancel against its own partner, which leaves the external sum governing a single mass-weighted average of positions. In the collision below the total momentum stays at 4.00 kg·m/s while 63 per cent of the kinetic energy leaves, and the average travels at 1.00 m/s throughout, before and after.

mechanics · Momentum The viscosity of a gas, over 8 decades of pressure. The viscosity of 3 gases at 300 K against pressure, on a logarithmic pressure axis and a linear viscosity one. The lines are flat, and that is the whole figure. Viscosity is the rate at which momentum is carried across a shear, which is the density of carriers times the distance each one carries it: ⅓ρv̄λ. Doubling the pressure doubles the density and halves the free path, and the two cancel exactly, so the same gas at a hundredth of an atmosphere is exactly as viscous as at one — which is not what anybody expects of a thinner gas and is what is measured. nitrogen comes out at 17.9 μPa·s against a measured 17.9, helium comes out at 19.3 μPa·s against a measured 19.9, argon comes out at 21.7 μPa·s against a measured 22.7. The flatness ends when the free path reaches the apparatus rather than the next layer of gas: at a vessel 10 mm across that is around 0.68 Pa for nitrogen, 1.94 Pa for helium, 0.69 Pa for argon, below which there is no gas-to-gas hand-off left to make.

The viscosity that does not care how much gas there is

Pump most of the air out of a vessel and the air that is left is exactly as viscous as it was. Maxwell derived that in 1860, did not believe it, and spent six years building an apparatus to measure it — which is a better description of how a prediction becomes knowledge than any amount of agreement would have been.

thermodynamics · Kinetic theory Six observers, six energies, one loss. The kinetic energy of the same collision before and after it, as measured by observers moving at 6 different speeds. No two of them agree about how much energy there was: the totals here range from 5.50 to 25.50 in the same units. Every one of them agrees about how much was lost — the gap between the two curves is 3.413 for all of them, varying by 6.2e-15. Energy is a quantity an observer owns; a change in it is not, and that is why heat, deformation and sound can be counted at all.

The energy that depends on the observer

A moving train has kinetic energy. Watched from a second train alongside it, it has none. Both statements are correct, neither can be corrected, and the whole of mechanics still works — because what conservation laws constrain is not how much energy there is but how much of it changes.

mechanics · Energy Pushing one way and going another. The angle between an applied force and the acceleration it produces, against the angle between the force and the body's velocity, at four speeds. At 0° and 90° the two are parallel, because those are the two directions the γ³ and γ divisors do not mix. Everywhere between, they are not: at 0.99c the worst case is 73.9°, reached with the force at 8.0°. A body under a steady sideways-ish push does not travel along it.

The push that does not point where the body goes

Newton's second law survives relativity in the form F = dp/dt and in no other. Written as F = ma it fails outright, and not merely by a factor — at high speed a body pushed at forty-five degrees accelerates at eighty, because the same force is divided by γ³ along the motion and by γ across it.

relativity · Relativistic dynamics What a collision has to spend, against what it is given. The energy available in a proton–proton collision, against the energy of one beam, on logarithmic axes. Against a stationary target the available energy is √(2mE) and the line has slope one half; head-on it is 2E and the slope is one. Bevatron at 6.2 GeV per beam reaches 3.7 GeV; SPS fixed target at 450 GeV per beam reaches 29.1 GeV; LEP at 104.5 GeV per beam reaches 209.0 GeV; Tevatron at 980 GeV per beam reaches 1960.0 GeV; LHC at 6500 GeV per beam reaches 13000.0 GeV. The gap is the whole architecture of the subject: the LHC's beams give 13000 GeV head-on and would give 110 GeV against a stationary proton, a factor of 118. Reaching the same 13000 GeV in fixed-target mode would need a beam of 90.1 million GeV. What the missing energy has gone into is not lost: it is the kinetic energy of the centre of mass, which every product has to carry away and which no experiment can use.

The collision that wastes most of the energy

The LHC's two beams carry 6,500 GeV each and 13,000 GeV are available. Fire one of those beams at a stationary block of copper instead and 110 GeV are available — the other 12,890 have gone into the motion of the wreckage and cannot be used for anything. The difference is a square root, and every accelerator built since 1970 is a consequence of it.

relativity · Relativistic dynamics The mass of two things that have none. The invariant mass of a pair of photons of equal energy, in units of E/c², against the angle between them. It is computed from the total energy and the vector sum of the two momenta, and agrees with 2E·sin(θ/2) to 1.0e-14. at 0° the pair weighs 0.000 E/c²; at 30° the pair weighs 0.518 E/c²; at 60° the pair weighs 1.000 E/c²; at 90° the pair weighs 1.414 E/c²; at 120° the pair weighs 1.732 E/c²; at 180° the pair weighs 2.000 E/c². Two photons flying in the same direction have no mass between them at all, because their momenta add to exactly the energy over c; anything else and they do. Nothing has been added: the constituents are massless at every angle, and the mass of the system is a property of the arrangement. At 180° the pair weighs 2E/c², which is every joule it contains — the case of a sealed box of light, where the two beams cancel in momentum and the whole energy shows up on the scales.

The box of light that weighs something

Two photons flying apart have a mass between them, though neither has one. Seal them in a mirrored box and the box is heavier than it was empty, by exactly the energy inside divided by c². Mass is not a property of stuff and it does not add up — it is a property of a system, and 99 per cent of the mass of everything anybody has ever weighed is of this kind.

relativity · Mass-energy Thrust against the air it is supposed to be pushing on. The thrust of one F-1 engine against ambient pressure, in atmospheres. It is 6.78 meganewtons at sea level and 7.77 in vacuum — 14.6 per cent more with the air taken away. The line falls at exactly 9.787 newtons per pascal, which is the nozzle's exit area, because the term is (p_e − p_a)·A_e: the ambient pressure pushes on the exit plane from outside and there is nothing to push back on it. The account in which the exhaust shoves against the atmosphere makes the opposite prediction — thrust falling with the pressure and vanishing in vacuum — and it is not a small disagreement about a coefficient. It has the sign wrong. A rocket works better in vacuum than in air, and every engine ever fired has said so.

The push that needs nothing to push against

A rocket engine produces more thrust in vacuum than at sea level — fifteen per cent more, for the engine drawn here, and the extra is exactly the ambient pressure times the nozzle's exit area. The account in which the exhaust shoves against the atmosphere does not merely overstate a coefficient. It has the sign wrong, and every engine ever fired has said so.

mechanics · Momentum A band 3.60 eV wide, and the curvature is the whole story. Energy against wavenumber for one tight-binding band, α + 2β·cos(ka), with α = -4 eV, β = -0.9 eV and a repeat of 300 pm. The band is 3.60 eV from floor to ceiling and periodic in k, so the zone boundary at ka = π is not an edge of anything: it is where the curve turns over. Near the bottom it is a parabola, which is the free-electron dispersion with a different coefficient — a mass of 0.470 m_e rather than one — and near the top it is a parabola the other way up, a mass of -0.470. The inflection between them is at ka = π/2, and it is the point at which a constant force stops producing any acceleration.

The mass a curve decides

An electron in a solid answers a force with a mass that is nothing to do with the mass of an electron. It is set by how sharply the band bends, it is smaller than the free value in a wide band and larger in a narrow one, and near the top of any band it is negative — which is why aluminium's Hall voltage has the sign of a positive carrier and no adjustment to an electron count can repair it.

quantum · Bands 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.

The pile that lands heavier than it weighs

Drop a chain onto a scale and the reading is three times the weight of the part that has landed — not approximately, exactly, all the way through the fall. The extra two parts are the force needed to stop links that are still arriving, and the same arithmetic run backwards says that picking a chain up wastes exactly half the energy it takes to get it moving.

mechanics · Momentum Three quantities that do not move while everything else does. On the left, an orbit in an inverse-square attraction, integrated from its equation of motion over 2.4 revolutions at an eccentricity of 0.55. On the right, three quantities computed from that same trajectory at every step and plotted against time: the energy, the angular momentum, and the length of the eccentricity vector that points at periapsis. Every one is flat to better than 1.2e-11 in units where the circular speed at r = 1 is 1, and none of them was constrained to be — the integrator was given the force and nothing else. Each is a symmetry seen sideways. The energy is constant because the force law does not mention the time; the angular momentum is constant because it does not mention the direction; and the eccentricity vector is constant because of a symmetry that is not a motion of space at all, which is why the inverse square closes its orbits and its neighbours do not. What the picture cannot show is the direction of the argument: it demonstrates that these three are constant here, and the theorem says something much stronger, that a constant exists for every continuous symmetry whatever the system.

The conservation law a symmetry hands over

Energy, momentum and angular momentum are usually presented as three separate empirical facts that happen to hold. They are one fact three times: every continuous symmetry of a system's action supplies a quantity that does not change, the correspondence is exact, and it runs both ways.

mechanics · Least action The fuel a starship needs, which is most of the universe. The mass of fuel a rocket must start with, divided by the mass it ends with, against the final speed as a fraction of light's — the vertical axis being the number of decades in that ratio, because the numbers do not fit on any other scale. Each solid curve is one exhaust speed, and each dashed one beside it is what Tsiolkovsky's Newtonian formula would have said. Below about a tenth of light speed the two are indistinguishable; above it they part, and the relativistic curve turns upward without limit as the final speed approaches light's, because what adds linearly is the rapidity rather than the velocity. The photon rocket — exhaust at exactly the speed of light, which is the best any engine can do — needs a mass ratio of 1.7321 to reach half light speed, which is √3 exactly, and 4.4 to reach nine tenths. Those are modest numbers and they are the whole of the good news. A chemical exhaust at 4 km/s needs 10^2968 even to reach a tenth of light speed, which is not a difficult engineering problem but an arithmetic impossibility — there are about 10⁵⁰ atoms in the Earth. What the chart cannot show is the other half of the trip: stopping at the far end squares the ratio, and coming home squares it again.

The fuel a starship needs

Tsiolkovsky's logarithm survives relativity with one substitution: what adds is the rapidity rather than the velocity. The result is that a photon rocket reaches half light speed on a mass ratio of the square root of three, and a chemical one reaches a tenth of it on a mass ratio with three thousand digits.

relativity · Mass-energy A loop at rest carrying 2.2e-12 kg m/s. A square loop of area 100 cm² carrying 20 amps, sitting still in a uniform electric field of 1.00 megavolts a metre. The carriers going up the field on one side cross 100 kilovolts on the way, so the ones in the top wire are less energetic than those in the bottom by that much per unit charge. The current is the same all the way round, so the same number of carriers pass per second in each wire — but they carry different energy, and momentum is energy times velocity over c². The two wires therefore contribute unequally, and the difference is 2.23e-12 kilogram metres a second, pointing across both the field and the dipole. Nothing in the picture is moving as a whole. The electromagnetic field round the loop carries exactly that momentum the other way.

The momentum of something that is not moving

A current loop sitting still in an electric field has momentum in the space around it. Nothing is moving, so something must be carrying an equal and opposite amount — and it is the loop, whose carriers on the high-potential side are more energetic than those on the low.

electromagnetism · Field energy The point that does not notice the collision. Two bodies of rest mass 1 and 2, approaching at 0.8c and -0.3c, colliding elastically and leaving at -0.6168c and 0.5969c — speeds obtained by reversing the motion in the zero-momentum frame, with the total energy and momentum checked to a part in a million million. The third line is the energy-weighted centre. It runs straight through the collision at 0.1872c, which is the total momentum divided by the total energy, and it has no kink — verified at two hundred instants. Nothing here is the centre of mass: the rest masses are unchanged by the collision but the energies are redistributed, and it is the energies that do the weighting.

The centre that is not a place

The centre of mass is replaced in relativity by the centre of energy, which moves uniformly and does everything the old point did — except be the same point for everybody. Boost a spinning body and its centre moves, so a spinning object has no centre at all.

relativity · Relativistic dynamics The motion that arrives late and does not go far. A flat plate sliding back and forth in its own plane, with the fluid above it drawn at 0°, 60°, 120°, 180° of the cycle. The speed at depth y is exp(−y/δ) times a cosine whose phase lags by y/δ, and both halves are checked: the largest speed reached at any depth follows exp(−y/δ) to a part in a thousand million, and the fluid one skin depth out is fastest a full radian after the plate is. Two skin depths out the motion is an eighth of the plate's and a quarter of a cycle behind; four skin depths out there is essentially nothing. Alternating shear does not diffuse away without limit — it fills a fixed depth and stops.

The shear that only reaches so far

Viscosity carries momentum sideways without limit when the driving is steady. Reverse the driving and it stops: the motion fills a depth set by the viscosity and the frequency, arrives there late, and beyond that depth the fluid does not know the wall exists.

fluids · Viscosity What the plane between them carries. The stress transmitted across the plane halfway between two charges of 10 nC held 1 cm apart, against distance from the axis in units of the half-separation. For two like charges the field on that plane lies entirely in it — checked here rather than assumed — so the plane sees only the pressure across the lines, the stress is negative everywhere, and the two halves are pushed apart. For opposite charges the field on the plane is entirely perpendicular to it, the plane sees only the tension along the lines, the stress is positive, and the halves are pulled together. Faraday's two words for a field line, tension along it and pressure across it, are exactly these two curves; the whole content of the tensor is that they are the same quantity, ε₀E²/2, wearing two signs. The force is what is left after integrating either curve over the plane, and both integrals come to the same magnitude — the Coulomb force — which is the next figure.

The force read off a surface that touches nothing

Draw any closed surface through empty space, measure the field on it, and the sum of one expression over that surface is the total force on everything inside — whatever the contents are, and without knowing anything about them. The expression is Maxwell's stress tensor, and it turns Faraday's guess about tension along a field line into an exact statement.

electromagnetism · Field energy The floor does no work and the jumper leaves the ground. A 70-kilogram person pushing off the floor: the floor's force in units of body weight against time, with the centre of mass's height and speed drawn on the same axis, each scaled. The force reaches 2.6 body weights, the contact lasts 260 milliseconds, and the take-off speed that comes out of integrating it is 1.67 metres a second — a jump of 14 centimetres. Integrating the floor's force over the centre of mass's rise gives 247 joules. The work the floor does is zero, because the patch of floor under the foot never moves and work is a force times the displacement of its own point of application. Both numbers are correct and they are answers to different questions: the first is what Newton's second law integrated over the centre of mass gives, and the second is what crosses the boundary between the floor and the person, which is nothing.

The floor that does no work

A jumper leaves the ground with three hundred joules of kinetic energy, supplied by a floor that does exactly zero work — because work is a force times the displacement of its own point of application, and the patch of floor under the foot never moves. Newton's second law integrated over the centre of mass gives the right kinetic energy and is not the work-energy theorem, and telling the two apart is what the first law of thermodynamics is for.

mechanics · Energy 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 The action at one instant, drawn as a map. Trajectories leaving one point at the same moment, at launch speeds 0.6, 1 and sixteen directions each, in a uniform field pulling downward, drawn up to time 1. Behind them, dashed, are the level curves of the action at that instant, regarded as a function of where a trajectory ends. They are circles, and their common centre is neither the launch point nor anywhere the particles have reached: it is 0.500 above the launch point, while the whole swarm has fallen by the same 0.500. Every arriving velocity points straight out from that centre, so every trajectory crosses the level curves at right angles, and the arriving momentum equals the gradient of the action to one part in ten thousand. The action integrated along each path agrees with the map's value at its end.

The action that knows where every path ends

The action is usually a number attached to one path. Treat it instead as a function of where the true path ends, and a single function of position and time holds every trajectory at once — its slope is the momentum, its rate of change is the energy, and its level curves are wavefronts, drawn about a point that sits above the source while everything falls.

mechanics · Least action

Named alongside it

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

Centre of massConservationDissipationEnergyReference framesConservation lawsFour-momentumRelativistic dynamicsActionAngular momentumBlackbodyCollision

All concepts