What's the deal with magnetic fields?

I apologize, but there won't be any Insane Clown Posse jokes in this article.

In a recent article, I brought up an important gotcha about electricity:

“In electronic circuits, the flow of electrons is confined to conductors, but the transfer of energy doesn’t involve these particles bouncing off each other; instead, the process is mediated through electromagnetic fields. These fields originate from charge carriers, but extend freely into the surrounding space.”

This property stumps many novices, especially if they cling on to the didactical toll known as the hydraulic analogy. Without considering interactions at a distance, it’s hard to grasp the behavior of transistors, capacitors, inductors, and other electronic components.

But how do these fields behave? The electric field is the easy one: most simply, it’s the static force of attraction and repulsion between charge-bearing particles. It’s what binds electrons to the nucleus of an atom, and what causes packing peanuts to stick to cats.

Electric fields at work. By Sean McGrath via Wikipedia, CC-BY.

Magnetic fields are more mystifying. Most textbooks assert that they’re a manifestation of the same underlying phenomenon, but then proceed to treat them as as wholly separate animal governed by its own, seemingly ad-hoc rules. The authors reach into a hat and pull out a “B field” or an “H field” that acts on moving particles… but only when the textbook says it should.

A detour through space

To develop better intuition, we ought to start with the speed of light. The significance of the concept is sometimes misunderstood, but in the most basic sense, it appears to be a fundamental constraint on causality: you can’t exert influence at a distance any faster than it would take for a photon — a massless elementary particle — to travel from here to there.

This creates an interesting problem for Newtonian physics. Let’s imagine that a guy named Finn is sitting in a spaceship that’s flying away from you at 90% the speed of light. For safety, the hull of his spaceship is equipped with a pair of flashing light beacons: one in front and one in the back, roughly 100 meters apart.

Goodbye, Finn.

The rear beacon flashes on a timer. As for the other one, Finn didn’t want to add 100 meters of wiring throughout the ship, so he just rigged a photodetector to pick up the flashing from the other beacon and trigger on that.

Conventionally, your frame of reference is as good as Finn’s; in fact, he might argue that he’s stationary and you’re the one getting away. There should be nothing unusual about physics onboard his ship. If Finn is watching the beacons, the rear one should be turning on at some given time, and then at t + 330 nanoseconds, the photons traveling at the speed of light should make it across the hull and toggle the front one.

But in your frame of reference, this ain’t right! By the time the photons from the rear beacon on Finn’s ship travel 100 meters, the front of the ship has gotten away and is now 90 meters ahead. Either Finn’s photons are a lot faster than yours, or more time needs to pass before the front beacon turns on.

An early attempt to address this issue was the concept of luminiferous aether: a cosmic medium through which photons supposedly propagate at a constant speed. This theory implied the existence of a special, “aether-anchored” frame of reference. If that happened to be your frame, then Finn, tearing through the aether at breakneck speeds, would see the physics on his spaceship getting out of whack.

Alas, no proof for the existence of luminiferous aether could ever be produced; instead, the answer turned out to be special relativity. The theory posits that all inertial frames of reference are equivalent. Instead, the trick is that when the frames are in relative motion, they experience space and time in different ways. In particular, in our frame of reference, instantaneous measurements of Finn’s ship would indicate it’s shorter than expected, and the time onboard would be seemingly passing more slowly than on Earth.

But let’s stick to magnets

To offer a basic explanation of magnetism, we really only need to lean on relativistic length contraction: the apparent reduction of the size and distance between moving objects along the direction of their travel. The effect is negligible in most contexts unless the objects are moving at very high (“relativistic”) speeds, but electric fields are a notable exception — even though the average drift speed of electrons in a wire usually doesn’t exceed several centimeters per minute.

Let’s consider a section of copper wire at rest. In the metal, there’s a certain number of mobile electrons in the conduction band, skating across a lattice with a matching density of immobile, positively-charged copper ions. Internal electric fields require the electrons to stay in the conductor, but they’re not kept on a particularly tight leash:

A very simple model of a copper conductor.

In this setup, because the positive and negative charges in the conductor are in balance, there is no net electric field acting on the stray charges nearby — neither in the reference frame of the stationary charge, nor of the moving one.

For the next step, it’s helpful to imagine a toy circuit: a one-meter loop of wire with one hundred mobile electrons inside. It’s a simple necessity that in the stationary (“lab”) frame of reference, the average distance between these electrons stays constant — 1 cm — no matter if the’re staying put or circulating around the loop. The only way to change their spacing would be to add electrons or take away some; otherwise, it’s always 100 elementary particles spread throughout a meter of wire.

But electrons in motion are supposed to exhibit length contraction! That is to say, once they start moving in relation to the observer, their spacing in the direction of travel becomes tighter than the “true” distance seen in their own frame of reference. The only way to explain this discrepancy is that when electrons are set in motion in our circuit, their proper, non-contracted spacing must increase.

With this in mind, let’s revisit the earlier conductor model, this time with some current flowing through. As we established, in the lab frame of reference, the length-contracted density of electrons and copper ions must stay the same as before — so there is no net electric field acting on a nearby stationary charge:

Copper conductor with some current flowing.

But what about a random charge outside the conductor that’s traveling in the same direction as the electrons inside? Well, from its perspective, the conductor consists of a bunch of non-moving electrons spaced pretty far apart, and then a markedly higher density of positive ions moving the other way round!

In other words, the charge, in its frame of reference, would experience a net electric field that’s pulling it toward the conductor — but again, only if it was moving along it in the first place:

The perspective of a nearby moving charge.

And this is, in essence, the origin of magnetic fields. It’s not always useful to analyze them this way — but at the very least, it’s good to have a slightly more intuitive explanation of where they come from.