Imagine a world where there was no magnetism at all — just electrostatic forces. It turns out that by adding Einstein’s laws of special relativity to that world, and looking at the effects that special relativity has on moving charges, the laws of magnetism appear — just by mashing electrostatics and relativity together.
In other words, the moving electrons in an electric motor’s coils see the world from a different frame of reference to the copper atoms making up the windings of the motor — and this imbalance is what makes the motor turn!
There’s a bunch of stuff around on the internet on this, but I find it hard to process because it deals with general relationships between magnetic and electric fields, not specific examples. I’m sure this is actually very general and useful, but it’s also beyond my immediate understanding as well. So, I tried approach the problem in my own way using just special relativity and high-school level physics, and it works!
0. Ampère’s force law: the result we are trying to arrive at
We’re aiming for the most basic formulation of Ampère’s force law, which gives the magnetic force between two parallel wires:
Where is the magnetic constant, and are the currents through the two wires, is the length of the wires, and is the distance between the two wires.
It’s worth noting that the formula above only holds true when the first wire is of length , and the second wire is of infinite length. This is convenient, because infinite integrals have a tendency to work out more nicely anyway.
(This same equation is often used to calculate the force between two wires both of length , but this is only accurate when is much greater than .)
1. Our first axiom: Coulomb’s law
The force on a point with charge due to another point with charge in a vacuum is given by Coulomb’s law:
where is the electric constant, and is the distance between the two points. This force is an attraction if positive, a repulsion if negative. If we move over to Cartesian coordinates by substituting , the vertical component of the force is given by
This term represents the vertical force between an electron or proton in the top wire and an electron or proton in the bottom wire. We could figure out the horizontal component as well, but it would get cancelled out in the end.
2. Electrostatic attractions with charges distributed along lines
Next, we need to take charge and replace it with an infinite, evenly distributed horizontal line of charge. Each unit length of the line has charge . We’ll call this force . This represents the force between an electron or proton in the top wire, and all of the electrons or all of the protons in the bottom wire. The term inside the integral is the same as above, but with the bottom point charge replaced by the charge on a infinitesimal length of the line of charge: .
This is where the calculus gets fun: this integral looks pretty nasty as it stands. But the hypotenuse-looking term makes it look susceptible to a trigonometric substitution:
The following identity also comes in handy:
We end up with:
Now we can take the top charge and spread it out along a line as well — although this time, the line is finite in length. This time we substitute for . Since the integrand is not a function of , this integral is trivial:
So, we’re halfway through and we haven’t touched relativity, magnetism, or even motion of any kind yet. But we do have an expression for the electrostatic force between two lines of evenly distributed charge, which we’ll need later.
3. Special relativity: length contraction
Einstein’s theory of special relativity states that when an object is moving, a stationary observer sees some strange effects — a clock on the moving object appears to run slowly (time dilation), and the object appears to be compressed along the direction that it’s travelling (length contraction). Length contraction is described as follows:
Where is the length of an object at rest and is the observed length of the object when moving at velocity . The Lorentz factor is given by:
So why does length contraction have any relevance to charges moving down a wire? Let’s consider the top wire, which has Coulombs of charge per unit length. A section of length will contain a charge . When current starts flowing, that section of charge will start moving — and will consequently appear to contract according to the formula above. This means the charges will appear to be bunched more closely together, since the same amount of charge is now packed into the contracted length. The new amount of charge per unit length will be:
This is the central point of this post — if you start off with a wire that is perfectly neutral (i.e., charges are balanced,) simply allowing current to flow through the wire will cause the electrons to appear to bunch up, while the protons will stay still. This imbalance leads to a net negative charge appearing on the wire! This is all assuming that you’re observing from a stationary frame of reference (with respect to the protons in the wire.) If you move along at the same speed as the electrons, then it’s the protons that will appear to be moving, and bunching up, and you’ll observe a net positive charge on the wire.
We’ll find later that the velocity of charges within a wire is extremely small, so a first-order approximation of the Lorentz factor is acceptable. Since the gradient of at is , we can use the following approximation:
4. Drift velocity of electrons in conductors
In order to calculate the magnitude of the charge contraction, we need to know the velocity of the electrons in the wire. If one unit of wire contains free electrons totalling a charge of , and the velocity of the electrons is , then the total charge moving per unit of time is:
5. Combining everything together
Let’s now tally up the forces acting on the top wire. First, the protons in the top wire. The protons are at rest, so they see the world from a stationary frame of reference. The protons in the bottom wire are also at rest, so the repulsion force acting on the bottom wire’s protons is not affected by relativity:
The electrons in the bottom wire are moving though, at a velocity of , so the attraction between the top protons and bottom electrons is given a boost:
Next, what do the electrons in the top wire see? We’re now travelling along with these electrons, so the protons in the bottom wire appear to be moving backwards at :
And the electrons in the bottom wire appear to be moving at a velocity of :
The total force acting on the top wire is the sum of these four terms:
Applying the approximation of the Lorentz factor from section 3:
Incorporating the expression for current from section 4:
And applying the definition of the electric constant :
And there you have it — combine electrostatics and special relativity, and you end up with the magnetic formula from section 0.
Although special relativity has been experimentally verified over and over again, it’s often incredibly sensitive instruments (usually atomic clocks) travelling at high speed doing the measuring. Other times, it’s astronomical observations. Even the wonderful point that the GPS system would rapidly fall apart (losing 10 kilometers of accuracy per day) without taking relativity into account seems out of this world (literally). The idea that the relatively familiar force of magnetism is a side effect of relativity seems a lot closer to home.
What’s more amazing is that this happens, despite the incredibly low velocities involved. A length contraction of just 0.1% requires a speed of 13,400,000 m/s. Although electrical signals flow down a wire at very high speed, the charge carriers themselves drift along very slowly — the distinction is much like that between the speed of sound in air vs the speed that the wind blows. A 1 mm diameter copper wire carrying 3 A, has a drift velocity of around 1 metre per hour. At that speed, the Lorentz factor, and the apparent charge imbalance is 1 + 4.3 × 1025. That’s an excess of 0.000000000000000000000043%. And yet, that’s all you need to give to the electrostatic force for it to drive an electric car around — it just shows how incredibly strong the electrostatic force is.
Take two normal 1mm diameter copper wires 1 meter long, and 1 meter apart. If you were to remove the atoms of copper, leaving just the free electrons behind, the electrostatic repulsion between the two remaining clouds of electrons would be about 1017 kilograms worth — enough to lift a small moon. Somehow, these ridiculously huge and tiny numbers multiply together to give the moderate magnetic force that we use every day.
On the other hand, for a long time, I’ve found both magnetism and special relativity a bit mysterious and strange. It’s somehow comforting to know that one is just a side-effect of the other — one less thing to keep me up at night.