Module 15 · magnetic fields and forces
Magnetism looks like a separate subject and is not. It is what electricity does when charges move — and if you push that sentence hard enough, as Einstein did, magnetism turns out to be electricity seen from a moving train. On the way: why a compass works, how every motor on Earth works, and why the magnetic force is the only force in this course that never does any work.
Put a stationary charge in a magnetic field and nothing at all happens. Set it moving and it is pushed — but not along the field, and not along its own velocity. It is pushed sideways to both. No other force in this course behaves like that, and the strangeness is worth sitting with for a moment before we tame it.
A charged particle flies into a region of uniform magnetic field, at right angles to it. What path does it follow, and how much speed does it gain or lose?
Two results fall straight out of Module 6. Set qvB = mv²/r and cancel one v to get r; then write T = 2πr/v and substitute r:
And the first result is a way of weighing atoms. Pass the ions first through a velocity selector — crossed electric and magnetic fields that cancel for exactly one speed, v = E/B — so v is pinned before the measurement begins. Then fire them into the analysis field, measure the radius of the arc they bend into, and m = qBr/v gives the mass directly. That is a mass spectrometer, and it is how isotopes — atoms of the same element that come in different masses — were discovered, how drugs are traced in athletes' samples, and how the composition of a comet's tail is read from a spacecraft.
The Earth's magnetic field traps charged particles streaming from the Sun and funnels them down toward the poles, where they spiral tightly around the field lines and slam into the upper atmosphere. The gas glows, and that glow is the aurora. Its shape — curtains and rays hanging vertically — is a picture of the field lines themselves. Bench 1 shows the circular orbit; the drift along the field lines toward the poles is driven by the field's gradient, which the bench's uniform field does not model. Without that field the solar wind would strip the atmosphere away, which is one respectable theory of what happened to Mars.
A current is charges moving. Put a current-carrying wire in a magnetic field and every one of those moving charges is pushed sideways — and since they cannot leave the wire, the whole wire is pushed:
Now bend the wire into a loop, mount it on an axle down its middle, and put it in a field. One side carries current one way and is pushed up; the other side carries it the opposite way and is pushed down. The two forces are equal and opposite, so the loop does not go anywhere — but they act at different places, which is Module 10's definition of a torque. The loop turns.
Untick the commutator and the loop swings to vertical, stops, and rocks. Why does a real motor need one?
That is the whole of a DC motor: a loop, a magnet, and a switch that flips the current twice a revolution.
One thing has to be settled before we leave it, because it looks like a contradiction and a careful reader will already have spotted it. We said the magnetic force never does any work. Here is a motor, hauling a lift full of people up a shaft, doing work all day. Both statements are true, and the way out is worth seeing. Each electron in the wire has two motions, not one: it drifts along the wire, and it is carried sideways as the whole wire sweeps round. The magnetic force is at right angles to the total velocity, so it leans slightly backwards along the wire — it pushes the electrons the wrong way round the circuit. The forward push on the sideways motion does positive work on the electron; the backward push on the drift does exactly as much negative work. They cancel: the magnetic force still does no net work, exactly as promised. What the magnetic force does is act as a perfectly honest middleman. Energy leaves the battery, because the battery has to shove current against that backward push, and it arrives at the shaft. The magnetic force carries it across without keeping a joule. Module 16 gives that backward push its name.
Faraday built the first one in 1821, a wire dipping into a bath of mercury with a magnet in the middle, and it turned continuously — the first time anyone had turned electricity into sustained motion. Everything since is refinement: many loops instead of one, iron cores to concentrate the field, and better switching.
The same forces in Bench 2 drive two other machines directly: a moving-coil meter (the needle turns until a spring balances the torque) and a loudspeaker (a coil in a magnet's gap, pushed back and forth by an audio current, dragging a paper cone with it). Run the argument truly backwards — turn the coil by hand instead of driving it with a current — and the coil generates a current instead. That is a generator, and it is Module 17.
In 1820, in the middle of a lecture, Hans Christian Ørsted noticed that a compass needle near his apparatus twitched whenever he closed a circuit. He had found the connection that two thousand years of natural philosophy had missed: a current makes a magnetic field. Within months Ampère had worked out the rules, and electricity and magnetism have been one subject ever since.
The field around a straight wire is not a set of lines pointing outward, as an electric field would be. It goes round in circles, with no beginning and no end:
Wind that wire into a coil and every turn's field adds up inside. The field down the middle is strong and uniform; the field outside is the spitting image of a bar magnet's — a large hint about what a bar magnet actually is. Slide an iron core inside and the field multiplies by hundreds, because iron's own atomic magnets line up with yours and add their strength to it. That is an electromagnet: a magnet you can switch off, which is why scrapyards can pick up a car and put it down again.
A bar magnet has two ends and they are not the same as each other: one end of a compass needle is pulled towards one of them and pushed away from the other. We call those ends the north pole and the south pole, and the field runs out of one and back into the other. Notice that the coil has them too — the field leaves one face of it and returns to the other — and that a coil quite obviously does not have a north half and a south half; the poles are just the two ends of one continuous set of loops. Keep that in mind for this one.
Cut a bar magnet in half. What do you get?
Why is a lump of iron magnetic at all, when there is no battery in it? Because every electron carries an intrinsic magnetic moment — it has spin. The classical picture of a spinning charge as a current loop gives the right magnitude, though the full account is quantum mechanical. In most materials those moments point in random directions and cancel. In iron, cobalt and nickel, quantum mechanics makes neighbouring atoms line up with each other in regions called domains, each already magnetised. In an unmagnetised nail the domains point every which way; stroke it with a magnet and they swing into line, and now the nail is a magnet. Heat it or hit it hard and they scramble again — which is why a magnet dropped often enough stops working, and why every magnet loses its magnetism above a certain temperature — its Curie temperature (770 °C for iron).
That also explains a compass. The needle is a small permanent magnet — its domains locked in one direction. The Earth's liquid iron core, carrying slow circulating currents, generates a magnetic field that runs roughly from geographic south to north. The needle swings to align with it.
| Field strength | In tesla |
|---|---|
| The Earth's field, at the surface | 0.00005 T — feeble, but it has been enough to navigate by for a thousand years |
| A fridge magnet | about 0.005 T |
| A hospital MRI scanner | 1.5 to 3 T — which is why nothing made of iron or steel goes in that room, ever |
| The strongest steady laboratory field | about 45 T |
| A neutron star's surface | up to 10¹¹ T, strong enough to distort the shape of atoms themselves |
Here is a puzzle that has no answer within this module. Take a positive charge outside a current-carrying wire, gliding along parallel to it at speed v, in the same direction as the electrons inside are drifting. In the laboratory the wire is neutral, so there is no electric force at all — but the charge is moving, so it feels a magnetic one, F = qvB. It is moving against the conventional current, so the push is away from the wire. Now run alongside it at v. In your frame the charge is at rest, and a charge at rest feels no magnetic force whatever — yet it is still being pushed away from the wire, and that is not a matter of opinion. What has changed is the wire: its positive ions are now moving and its electrons are moving more slowly than before, so Module 19's length contraction squeezes the positive row closer together and lets the negative row spread out. The wire, exactly neutral in the laboratory, carries a net positive charge per metre in your frame, and pushes the positive charge away electrically.
Both observers agree on what happens to the charge. They disagree completely about whether the force was magnetic or electric. That is the resolution: there is only one field, the electromagnetic field, and how much of it looks magnetic depends on how fast you are moving — no single frame makes the field purely electric or purely magnetic; the split is observer-dependent but neither piece vanishes entirely. Magnetism is what electricity looks like from a moving train — and this is not a poetic reading, it is a calculation you can do once you have Module 20, and the numbers come out exactly right. Relativity was hiding inside a compass needle for eighty years before Einstein noticed. His 1905 paper is titled "On the Electrodynamics of Moving Bodies", and this is the problem it opens with.
Hold a compass over a wire connected to a battery — a short piece of wire and a AA cell will do, though the wire will get warm, so do it briefly. The needle swings across the wire, not along it. Reverse the battery and it swings the other way. That is Ørsted's 1820 lecture reproduced in your kitchen, and it is the moment two thousand years of separate subjects became one.
1. A magnetic force acts on a moving charge, always at right angles to its motion. The work it does on the charge is therefore:
2. An electron and a proton enter the same magnetic field at the same speed, perpendicular to it. Compared with the electron, the proton's circular path is:
3. Two long parallel wires carry current in the same direction. They: