Illustrated Physics
physics, drawn
Concept

Gauss's law — where it appears

The statement that the flux of a field out of a closed surface counts what is inside it, whatever the shape of the surface or the arrangement within. It is exact for any inverse-square field, which is why it applies to gravity as readily as to electrostatics and why symmetry makes it a calculating tool.

Named by 11 essays across one field — each of them below, with the objects they name alongside it.

A closed surface with the charge inside. Every field line from the enclosed charge crosses the surface exactly once on its way out, so the net flux counts the charge.

Counting what comes out, and never looking inside

Draw any closed surface. The field crossing it depends only on the charge enclosed — not on where that charge sits, not on its shape, not on anything outside.

electromagnetism · Gauss's law The same law, three shapes of source. Field strength against distance on logarithmic axes, for a point, a long line and a wide plane carrying charge. The exponent is the slope, and it is set by how the area of the enclosing surface grows rather than by anything about the force law.

The shape decides the falloff, and the force law never changes

A point charge gives an inverse square, a line gives an inverse, a plane gives a constant. All three come from the same law, and the exponent belongs to the geometry of the source rather than to the physics.

electromagnetism · Gauss's law Four paths, one answer. The field of a straight wire carrying 10 A, summed step by step around four closed paths in 4000 pieces each. Three of them enclose the wire and each returns 12.566 µT·m, which is μ₀I; the fourth does not enclose it and returns zero, because the outward stretch of the path and the return stretch cross the same field lines in opposite senses. Nothing about the shape survives into the answer — not the radius, not the centring, not the corners — which is what makes the law usable and also what makes it useless without a symmetry to hand.

The field that wraps a current

Ampère's law says that going once round a closed path and adding up the field counts the current threaded through it, and nothing else about the path survives into the answer. Four different loops round one wire return the same number to five decimal places — which is exactly why the law is both easy and treacherous.

electromagnetism · Ampere law Equipotentials, with the field lines that cross them. Contours of constant potential, traced by marching squares, with field lines traced along the gradient of the same potential. The two families meet at right angles everywhere, which is a consequence of the field being the gradient rather than a property of the drawing.

The attraction that needs no charge

Gauss's law says nothing comes out of a neutral molecule, and yet water's field one nanometre away reaches 1.1 × 10⁸ V/m. What survives when the monopole vanishes is a separation, and every step down the tower of falloffs below it is paid for with one more order of cancellation.

electromagnetism · The field concept The plane deleted, and one charge put in its place. A charge of 1 nC held 20 mm above an earthed conducting plane. The lines are traced through the field of the real charge plus an equal and opposite one at the mirror position, and then cut at the plane, because below it there is metal and no field whatever. Nothing in the tracing knows about the surface: each line follows the local field direction and stops where it arrives. That every one of them arrives perpendicular — the worst departure among the 9 drawn is 2.2° away from square — is the boundary condition showing itself rather than a rule imposed on the drawing. The image charge is drawn faint because it is not there: it is a way of writing a function that happens to satisfy the equation and the boundary values, which by the uniqueness theorem makes it the field and not a model of the field.

The charge that has to be somewhere else

Hold a charge above an earthed metal sheet and the field above it is exactly the field of two charges — the real one and an imaginary partner buried at the mirror position. The partner is not an analogy or an approximation. It is a legal guess, and a legal guess is a proof.

electromagnetism · Conductors Two quantities that are never computed and never change. A two-dimensional field integrated for 110 steps using only the two equations that contain a time derivative — Faraday's and Ampère's. The two that do not, Gauss's law for the electric field and the statement that there are no magnetic charges, are never imposed and never checked during the run. Their residuals are plotted: the divergence of B stays below 2.4e-16 of the field's own size and the divergence of E below 2.4e-16, over the whole run, while the field itself moves and changes by a factor of 8.59. That is not a numerical coincidence. Taking the divergence of Faraday's law gives the divergence of a curl, which vanishes identically, so ∂(∇·B)/∂t is zero whatever the fields are doing; the same manoeuvre on Ampère's law gives ∂(∇·D)/∂t = −∇·J. So the two constraints are initial conditions, propagated for ever by the two that are laws of motion, and Maxwell's four equations are two dynamical ones and two statements about how the field was set up.

The two equations that are not laws of motion

Maxwell's equations are usually presented as four laws of equal standing. Two of them contain no time derivative at all, which means they cannot be evolution equations: they are conditions on the field at one instant. What makes them consistent with the other two is that the other two preserve them exactly — and one of the two preservations holds only because charge is conserved.

electromagnetism · Maxwell equations The gravity that grows on the way down. The acceleration due to gravity inside the Earth against distance from the centre, from Gauss's law applied to the Preliminary Reference Earth Model's density: g(r) = G M(r)/r², counting only the mass inside each radius. The model reproduces the Earth's mass to 0.02 per cent, a surface gravity of 9.82 m/s² and a moment-of-inertia factor of 0.3308, none of which it was fitted to. For a uniform Earth, drawn dashed, gravity would fall in a straight line to zero at the centre. The real one does the opposite for the first 2891 km: it rises through the whole mantle to 10.69 m/s² at the core–mantle boundary, 8.8 per cent above its surface value, and only then falls to zero through the core. Going down through the mantle removes very little mass and brings the dense core much closer, and the second effect wins.

The pull that grows on the way down

Inside a uniform ball, gravity falls in a straight line from the surface to nothing at the centre, and that is the answer usually given for the Earth. It is wrong for almost three thousand kilometres. Going down through the mantle, gravity rises, reaching nearly nine per cent above its surface value where the core begins — because Gauss's law counts only the mass inside, and the local form of the law says gravity grows inward wherever the rock is lighter than two thirds of the average beneath it.

electromagnetism · Gauss's law Three bodies of one mass, one field outside, three fields inside. The gravitational field against distance from the centre, both in units of the body's surface values, for 3 spherical bodies of the same mass and radius: a uniform ball; a dense core under a light mantle; a hollow shell. Outside the surface the three curves are one curve — checked at 1.7 radii by adding up the pull of every mass element in each body, which agrees with the pull of a point of the same mass to better than a part in five hundred. Inside they part: the uniform ball's field falls in a straight line, the layered body's rises to 1.24 times the surface value at the top of its core, and the hollow shell's is zero throughout its cavity. Their moment-of-inertia factors are 0.400 (uniform ball), 0.330 (dense core under a light mantle), 0.551 (hollow shell) — a number that no measurement of the field outside can supply.

The field outside that cannot find the core

Gauss's law gives the field outside a body from what it encloses, and read backwards it is a limit. A uniform ball, a planet with an iron core and a hollow shell of the same mass have identical gravity everywhere outside. The field fixes a list of numbers — the mass, the flattening, higher moments — and leaves free everything else, including the moment of inertia; the Earth's core was weighed by its wobble, not by its pull.

electromagnetism · Gauss's law The same nudge in four kinds of space. A body on a circular orbit given 5 per cent more speed, integrated for up to 6 turns in the field a point mass would have in spaces of 2, 3, 4, 5 dimensions. In 2 dimensions it stays between 1.00 and 1.10 of the original radius; in 3 dimensions it stays between 1.00 and 1.23 of the original radius; in 4 dimensions it leaves after 1.93 turns; in 5 dimensions it leaves after 1.19 turns. In three dimensions the orbit is an ellipse that closes; in two it is a rosette that does not close but never wanders far; from four dimensions up there are no stable orbits at all, and a planet nudged by anything would spiral away or plunge.

The number of dimensions a planet can orbit in

Gauss's law says that a point source's field is its strength spread over the surface around it, and the size of that surface depends on how many dimensions space has. In three it grows as the square of the distance, and the field falls as the inverse square. Count differently and the field falls differently — and above three dimensions it falls so steeply that no planet could hold a stable orbit and no atom could hold its electron. The inverse-square law is not a fact about gravity or electricity. It is a fact about space, and it is one of the conditions for there being anything in space to orbit.

electromagnetism · Gauss's law Two ways to be wrong, and the apparatus that sees each. The fractional sag of the potential at the centre of a charged spherical shell below the shell's own, on a logarithmic axis, against the shell's radius in metres, also logarithmic, for two different departures from Coulomb's law. If the force fell as 1/r^(2+ε) with ε = 10⁻¹⁶, the sag is 3.07·10⁻¹⁷ whatever the shell's size (dashed): the law has no length in it, so neither does its failure. If the photon had a mass of 10⁻¹⁴ eV, the sag is (μR)²/6 (solid): 4.3·10⁻²⁰ for a centimetre shell and 4.3·10⁻¹⁶ for a metre. A mass introduces a length, and the signal grows as the square of the apparatus compared with it — so the strongest bounds on a photon mass come from the largest fields, not the most sensitive laboratories.

The mass that would make Gauss's law leak

Gauss's law says the flux out of any closed surface counts the charge inside, whatever the surface's size. If the photon had even a tiny mass, that would stop being true: the field of a charge would die off exponentially beyond a length ħ/mc, the flux through a large sphere would fall short of the charge, and the inside of a charged shell would no longer be field-free. There are two different ways Coulomb's law could fail, and they are caught by opposite strategies. A wrong exponent has no length in it and shows equally in any apparatus; a mass has a length, and shows as the square of the apparatus's size. That is why the tightest limit on the photon's mass comes not from a laboratory but from the magnetic field the solar wind carries past Pluto.

electromagnetism · Gauss's law How much a tiny imbalance of charge would take off a body's weight. The electric repulsion between the parts of any body, as a fraction of their mutual gravity, if every nucleon carried an unbalanced charge δ (in units of the electron's), on logarithmic axes. It is (δ/δc)², with δc = 8.93 × 10⁻¹⁹, for a grain of sand, a planet or a galaxy alike. At the imbalance Lyttleton and Bondi proposed, 2 × 10⁻¹⁸, repulsion would beat gravity 5.0 times over, everywhere — their mechanism for an expanding universe. Experiments with gas let out of tanks had bounded the imbalance near 10⁻²⁰ within a year, where the ratio is 1.3·10⁻⁴, and modern ones near 10⁻²¹, 1.3·10⁻⁶.

The imbalance of charge that would outweigh gravity

Gravity and electricity both fall as the inverse square, so Gauss's law treats them identically: the field of any round body counts what is inside it. If every nucleon in ordinary matter carried a tiny unbalanced charge, the body's charge would follow its mass everywhere, and its electric repulsion would be a fixed fraction of its gravity at every size from a grain of sand to the universe. Below about one part in a billion billion of an electron's charge, gravity always wins; above it, repulsion always does. In 1959 two astronomers proposed that the universe expands because the true imbalance is just above that line.

electromagnetism · Gauss's law

Named alongside it

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

The inverse-square lawElectric fieldFalloff exponentSymmetryDivergenceElectric potentialField linesFluxGravitational fieldHydrostatic equilibriumMoment of inertiaAmperes law

All concepts