Module 13 · electrostatics
Gravity took Modules 5 and 6. Now the other long-range force — the one that actually holds you together. It follows the same inverse-square shape as gravity, and it is stronger than gravity by a factor with thirty-nine zeros in it. This module is also where physics quietly stops talking about forces between objects and starts talking about fields, which turns out to be the most important change of subject in the whole course.
There are two kinds of charge. We call them positive and negative because Benjamin Franklin, in the 1750s, had to guess which of the two actually moved when things were rubbed together — and guessed wrong. The names stuck anyway, and nothing in the physics depends on which is which. Like charges repel, unlike attract, and the force between two of them obeys a law that will look extremely familiar:
Two charged spheres sit 1 cm apart and repel with a force of 90 N. You move them to 3 cm apart. The force is now:
The last cell of that ledger is the point of the bench: even for two gram-sized balls with the default microcoulombs on them, the ratio is already around 10¹⁵. Do the same sum for a proton and an electron in a hydrogen atom, at the charges and masses nature actually uses, and the electric attraction comes out about 10³⁹ times the gravitational one. Written out, that is a 1 with thirty-nine zeros after it — a thousand billion billion billion billion. Gravity is, by an enormous margin, the feeblest force in physics.
Which raises an obvious question: if electricity is that much stronger, why does gravity run the universe? Because charge comes in two signs and cancels. Any lump of ordinary matter contains almost exactly equal amounts of positive and negative charge. Step back even a few atomic diameters and the two have already cancelled, so there is almost nothing left to feel. Very close up, within a molecule or two, the cancellation is not yet complete — which is exactly why a charged balloon can grip an uncharged wall. Mass has no negative version, so it simply accumulates. Gravity wins the large scale by never cancelling out — not by being strong.
Rub a balloon on your hair and hold it against a wall. It sticks. The wall is not charged — so what is holding it? The balloon's negative charge pushes the electrons in the wall's molecules slightly away. Each molecule ends up with its positive side facing the balloon and its negative side facing away. The two sides carry equal amounts of charge, but the positive side is nearer, so by 1/r² the balloon's pull on it wins over the push on the negative side further off. Every molecule in the wall tugs a little, and the balloon stays up. That is polarisation, and it is why a charged object attracts anything, charged or not: bits of paper, a thin stream of water from a tap, your own hair.
Two facts about charge that gravity has no equivalent of. First, it is quantised — it comes in fixed lumps. Every free charge ever measured is a whole-number multiple of e = 1.6 × 10⁻¹⁹ coulombs, the charge on a proton. Millikan measured it in 1909 by watching charged oil drops hover between charged plates, and no one has ever found a fraction of it drifting about. (Quarks carry thirds of it, but they are never found alone.) Second, the total charge of an isolated system never changes. You can create a positive and a negative together, or destroy them together, but the sum stays put. Rubbing a balloon does not make charge; it moves electrons from your hair to the balloon, and your hair is left exactly as positive as the balloon is negative.
Coulomb's law answers "how hard do these two charges push each other?" But it dodges a harder question: how does one charge know the other is there? Across empty space, with nothing in between?
Faraday's answer was to stop thinking about pairs. It struck the physicists of his day as mystical, and it turned out to be the deepest idea of nineteenth-century physics. A charge, he said, changes the space around it: at every point nearby the space is now in a state it was not in before, ready to push on any charge that arrives. That state of the space is what we call a field. Any other charge responds only to the field where it actually is. Nothing reaches across a gap; the gap itself has been changed.
Field lines are drawn to show the field's direction. Why can two field lines never cross?
The rules for reading a field diagram, all of which follow from the definition:
The lines also let you say something about a whole region at once. Draw any closed surface — a bag, a balloon, a box — around some charges, and count the net number of lines coming out through it. That number depends on nothing except how much charge is enclosed: two positive charges give twice as many outgoing lines as one, and negative charge inside cancels part of the outward count. In symbols, Φ = Q/ε₀, where Φ is the total lines out (technically, the electric flux). This is Gauss's law, and much of the rest of this section rests on it — including the shell result below.
In a metal, roughly one electron per atom is free to wander the whole object. Put a metal block in an electric field, and those electrons move — instantly, and until the field they create exactly cancels the one you applied. The result is a rule with no exceptions: the electric field inside a conductor in equilibrium is zero. Not small; zero. Any leftover field would still be pushing electrons, so by definition it would not be equilibrium yet.
Two consequences follow. All excess charge on a conductor sits on its outside surface — if any of it were buried inside the metal, that buried charge would make a field of its own, and we have just established that the interior field is zero; the outer surface is the only place charge can sit without breaking that rule. And any hollow space inside a conductor is completely shielded from outside fields — a Faraday cage. That is why a car struck by lightning is safe (the charge runs round the shell, not through you), why an MRI room is lined with mesh, and why your phone loses signal in a lift. A mesh works as well as a solid sheet, provided the holes are much smaller than the wavelength of the wave you are keeping out — the trick Module 18 uses on a microwave oven door.
Module 5 found gravitational potential energy: lift a mass against gravity and you have banked mgh. The same story works here — with one thing to watch. Gravity only ever pulls, so lifting always costs energy. Charge has two signs, so pushing a positive charge toward another positive charge costs work against the repulsion (let go and you get the energy back as kinetic energy), while pushing it toward a negative charge gives energy back. Everything from Module 5 carries over except that sign.
And as with the field, it is cleaner to strip out the test charge and talk about the space itself. Potential is the energy per coulomb — how much work it takes to bring one coulomb from far away to that point. Its unit is the volt: one joule per coulomb.
Two flat plates facing each other, one charged positive and one negative, make the most useful arrangement in electrostatics. The field between them is uniform — the same everywhere, straight across — which is why it is the standard way to accelerate or steer a charged particle, from an old television tube to a particle accelerator.
The pair is called a capacitor, and its job is storing charge. How much it stores per volt is its capacitance:
A capacitor is charged from a battery and then disconnected. You now pull the plates further apart. The charge, the voltage and the stored energy:
The energy bookkeeping there is worth pausing on, because it is the first hint of something Module 18 turns into the main event: the field as a thing in its own right, with energy of its own. Where is the stored energy of a capacitor? Not "in the charges" — the neatest way to account for it is to say the energy is spread through the gap itself, so much of it in every cubic metre of space. That amount per cubic metre is called the energy density, and it turns out to be proportional to E²: double the field and you have four times as much energy sitting in the same volume. Pull the plates apart, with the charge held fixed, and the field is no stronger than before but it now fills a bigger gap; more space full of field means more stored energy, which is exactly why pulling costs work.
That is not bookkeeping for its own sake. When Maxwell's equations produce a wave of pure field, racing away from its source with no charges anywhere near it, the energy it carries has to live somewhere — and it lives in the field. The field stops being a way of describing forces and becomes a thing in its own right, with energy and momentum of its own. Every argument for that begins here, with a capacitor.
Push the electric field in air past about 3 million volts per metre and the air stops insulating: electrons are torn off its molecules and it suddenly conducts. That is called breakdown, and it is the small blue spark you get off a door handle in dry weather. Check the number on the bench: 3000 V across a 1 mm gap gives E = V/d = 3000 ÷ 0.001 = 3 million V/m, right at the limit. A thundercloud does it on a scale of kilometres, with a potential difference of perhaps 100 million volts. And a capacitor is why an unplugged television was dangerous for decades and why a camera flash can hurt you: it holds its charge, quietly, long after the power is off.
1. A hollow metal sphere carries a charge of 5 µC. Where does that charge sit, and what is the field inside the hollow?
2. A cricket ball and the Earth pull on each other gravitationally. Both are packed with charged particles, and the electric force between two charges is vastly stronger than the gravitational one. So why does nobody have to allow for an electric force between the ball and the Earth?
3. Two parallel plates are 2 mm apart with 100 V across them. A charge is released at the positive plate. The field it experiences is: