The inside of a conductor, where the field is exactly nothing

A conductor is a material with charges free to move in it. That is the entire input, and from it follows one of the strongest statements in electrostatics: the field inside a conductor at equilibrium is exactly zero.

The argument is one sentence. If the field were not zero somewhere inside, the free charges there would feel a force, so they would move, so the situation would not be equilibrium. Equilibrium and a non-zero interior field are incompatible by definition.

What the figure had to do #

The claim is easy; drawing it honestly is not, and it is worth describing what the generator does because the alternative — drawing an empty circle and putting lines around it — would look identical and prove nothing.

The conductor is represented by ninety-six charges spaced around its boundary. The requirement is that the total potential — from the external source plus all ninety-six — is the same at every one of them, since a conductor is an equipotential. That is a system of coupled equations, and it is solved by relaxation: sweep over the boundary, adjust each charge in proportion to how far its potential is from the mean, subtract the average so the total stays zero because the conductor is neutral and isolated, and repeat until it settles.

The lines are then traced through the sum of everything. Nothing tells them to stop at the surface or to arrive perpendicular; they do both because the field they are following does. That is the figure’s assertion, and it would fail visibly if the relaxation had not converged — the lines would push into the interior or meet the surface at an angle.

The ticks around the boundary are the solved surface charge, drawn at a length proportional to its value. They are largest where the field lines are densest, negative on the side the field arrives from and positive on the side it leaves, and they sum to zero.

Where the charge goes #

The interior field being zero has an immediate consequence that costs nothing to derive.

Take any closed surface entirely inside the conductor’s material. The field on it is zero everywhere, so the flux through it is zero, so by Gauss’s law the charge enclosed is zero. Shrink the surface to any size, put it anywhere in the bulk, and the conclusion holds.

Every excess charge on a conductor sits on its surface. Not mostly — entirely, to within the thickness of a few atoms, which is where the classical description gives out.

That is not obvious in advance. Charges repel, so it is reasonable that they spread out; it is not obvious that they go all the way to the surface and leave nothing behind. The repulsion argument gets the direction right and the extremity wrong, and the counting argument gets it exactly.

The distribution over that surface is not uniform, except for a sphere. It crowds where the surface curves most sharply, which is why a point on a conductor has an intense field around it and why every high-voltage component is built with rounded edges.

The cavity, and the direction that matters #

The most useful case is a conductor with a hole in it, and it behaves asymmetrically in a way worth being precise about.

Outside in: the cavity is shielded, completely. Whatever field is applied outside, the interior of an empty cavity within a conductor has zero field. The argument is the equipotential one: the conductor is all at one potential, so its inner surface is at that potential too, and a region bounded by a surface at constant potential with no charge inside it has constant potential throughout — hence no field. The strength and shape of the external field never enter.

Inside out: nothing is shielded at all. A charge placed inside the cavity induces an equal and opposite charge on the cavity wall and an equal charge on the outer surface, and that outer charge produces a field outside exactly as though the charge were sitting there alone. A conductor does not hide what is inside it; it only hides its inside from what is outside.

That asymmetry is the practical content. A metal box protects sensitive electronics from external fields and does nothing whatever to stop them radiating — which is why shielding an emitter requires grounding the box, so the outer surface charge has somewhere to go, while shielding a receiver does not.

The right angle at the surface #

One feature of the hero figure carries more information than it appears to, and it is the angle at which the lines arrive.

Field lines meet a conductor’s surface at exactly ninety degrees, and the reason is the equipotential argument again. A component of field along the surface would push the surface charges sideways, they would move, and the situation would not be equilibrium. So the tangential field is zero and only the perpendicular component survives.

That single condition is what makes conductors tractable. It converts an unbounded problem — find the field everywhere given a complicated arrangement of matter — into a boundary condition on a surface, and boundary conditions are what the mathematics of this subject is built to consume. Nearly every solved problem in electrostatics is solved because a conductor supplied one.

It also fixes the relationship between the surface charge and the field just outside it. Applying the counting argument to a small pillbox straddling the surface, with zero field on the inside face and a perpendicular field $E$ on the outside, gives $E = \sigma/\varepsilon_0$ — twice the field of an isolated sheet of the same density, because the conductor’s own interior contributes the other half by cancellation rather than by addition.

The cage, and what it does not do #

Every part of this is in daily use under the name of a Faraday cage, and the folk account of it contains two errors worth correcting.

A car struck by lightning protects its occupants, and it is not because of the tyres. Rubber that can withstand a strike that has already crossed a kilometre of air is not a serious proposition; the current has no difficulty completing the journey. What protects the occupants is that the shell is a conductor, the charge stays on its outer surface, and the interior field is zero. The current flows around the passengers rather than through them, and it does so through the metal because the metal is a conductor rather than because anything else is an insulator.

A mesh works as well as a solid sheet, and the criterion is a wavelength. Holes much smaller than the wavelength of whatever is arriving are, to the field, not there — the charges have plenty of room to rearrange around them. A microwave oven’s door is a perforated screen with holes of about a millimetre against radiation of 12 centimetres, a ratio of over a hundred, which is why the mesh is opaque to the microwaves and transparent to visible light, whose wavelength is ten thousand times smaller than the holes.

And there is a limit that the electrostatic argument does not reach. Shielding at low frequencies is much harder than at high ones, and magnetic fields are not shielded by this mechanism at all, since there is no magnetic surface charge to rearrange. A room that is electrically quiet to microvolts may be magnetically transparent.

The measurement that consists of finding nothing #

The zero interior field is the basis of the most precise test of the inverse-square law that exists, and the method is worth admiring.

If Coulomb’s law were $1/r^{2+\epsilon}$ rather than exactly $1/r^2$, the counting argument would not close: the flux through an interior surface would not vanish, and there would be a small field inside a charged conducting shell. So the experiment is to charge a shell, put a detector inside it, and look for nothing.

Cavendish did it in 1773 with two concentric spheres and a pith-ball electroscope, and bounded the exponent’s departure from 2 at about $10^{-2}$. Maxwell repeated it a century later at $10^{-5}$. Modern versions using lock-in detection reach around $10^{-16}$.

The reason a null test wins here is worth extracting, because it recurs. Measuring a small quantity accurately is limited by calibration, by systematic errors, by the accuracy of the standard being compared against. Confirming that a quantity is zero is limited only by the sensitivity of the detector — no standard is needed, because zero requires no calibration. Whenever a theory predicts an exact zero, the resulting test is likely to be the sharpest one available. The Michelson–Morley result has the same character, and so does the search for an isolated magnetic pole.

What the shielding costs #

Three costs, and the third is the one that decides where the model applies at all.

It takes time. “Equilibrium” is a state the conductor relaxes into, not one it is in. The relaxation time is the material’s permittivity divided by its conductivity, and for copper that is about $10^{-19}$ seconds — so fast that treating the rearrangement as instantaneous is safe for anything below optical frequencies. For a poor conductor it is not: distilled water has a relaxation time of about a tenth of a millisecond, so a body of water shields against a static field and not against a radio wave. The single number $\varepsilon/\sigma$ decides which side of that line a material falls on, and it varies over more than twenty orders of magnitude.

The interior is empty of field, not of consequence. The metal carries the current that the shielding consists of, and that current dissipates energy in any real conductor. A cage exposed to a strong alternating field warms up, and the shielding is paid for continuously.

And the boundary-value problem is expensive. The interior being trivial does not make the exterior easy. Finding the field around a conductor of arbitrary shape means finding the surface charge that makes the surface an equipotential, and that is a system with as many unknowns as the surface has patches, each coupled to every other. The relaxation in the hero figure is ninety-six unknowns solved by 260 sweeps; a realistic three-dimensional problem has hundreds of thousands, and the whole field of computational electromagnetics exists to do it efficiently. A statement about the interior that takes one sentence to prove leaves the exterior as hard as it ever was.

The vocabulary that had to be invented #

The zero interior field is also where the modern language of the subject came from, and the route is indirect enough to be worth tracing.

Faraday built a large wooden frame covered in metal foil in 1836, sat inside it with the most sensitive electrometers he had, and had the outside charged to the point where sparks jumped from it. He detected nothing at all — his own account records that he searched carefully and found no trace of an electrical effect anywhere within.

That experiment is the origin of the ice-pail work as well: a metal container, a charged body lowered into it without touching, and a measurement showing that the outside acquires exactly the charge lowered in — which is the inside-out half of the asymmetry above, established fifty years before anyone could write down Gauss’s law in the form used here.

What is worth extracting is how much of this was settled by looking for nothing and finding it. A zero field, an unchanged reading, a null. The counting argument supplied the theory afterwards, and it supplied it in a form that makes the null a prediction rather than an observation — which is what turned a demonstration into the most precise measurement in the subject.

Where the model stops #

Equilibrium, and electrostatics. Everything here is a steady state with no currents flowing. A conductor carrying a current has a field inside it — that is what drives the current — of magnitude $J/\sigma$, which for copper is tiny and not zero.

A perfect conductor. Real metals have finite conductivity, so the surface charge is not confined to a mathematical surface but to a skin whose depth depends on frequency. At mains frequency in copper that skin is about 9 millimetres; at 1 GHz it is 2 microns. The electrostatic picture is the zero-frequency limit of a family of behaviours, and the family is what shielding engineers actually work with.

A classical description. The charges are treated as a continuous fluid. At the surface of a real metal the electron density does not stop abruptly; it spills out over a fraction of a nanometre, so the effective surface sits slightly outside the ions, and for a nanoscale capacitor that offset is a measurable part of the geometry.

And “isolated” is doing work. The hero figure’s conductor is neutral and connected to nothing. Earthing it changes the problem entirely, because charge can then flow in and out and the constraint becomes fixed potential rather than fixed total charge. Which of the two is held fixed is the single most consequential detail in any conductor problem, and it is usually stated in three words that are easy to skim past.

The ladder from here #

Later rungs: the method of images, which replaces a conductor by a fictitious charge that reproduces its boundary condition and turns a hard problem into an easy one. Capacitance, and why it depends only on geometry. The energy stored in a charged conductor, and the force on its surface — which is outward, always, since like charges repel, and which is what makes a charged soap bubble expand. Conductors held at fixed potential, and the uniqueness theorem that makes the boundary-value formulation legitimate. Screening in a plasma or an electrolyte, where the same physics happens over a finite Debye length rather than a surface. The skin effect and shielding at frequency. And the surface charge on a current-carrying wire, which turns out to be what steers the field along the wire and is left out of every elementary treatment of circuits.