Built on Facts

Yesterday in my recitation section I went through the chapter on electromagnetic induction, covering Faraday’s law and the displacement current term in Ampere’s law before assigning a quiz. Though this quiz really doesn’t need those concepts, it was a good opportunity to break out my all-time favorite Intro E&M quiz question.

Consider two parallel wires of infinite length, separated by a distance r, each with a uniform positive charge density λ. Both wires are moving in the same direction with velocity v, also parallel to the wires. If the electrical repulsion is exactly balanced by the magnetic attraction caused by the current, what is v?

I’ll assume you know how to get the electric field of a long wire. It’s

And the force felt by a length l of the other wire is thus

Which is a little awkward because l is infinite, but the force per length is a fine thing to use:

Ok, that’s that for the electrical repulsion. The magnetic field of a current-carrying wire is

Doing the same sort of jazz as above to find the force per length, we get

Now we need to figure out what the current is. Current is (charge/time) which is the same thing as (charge/length)*(length/time). And that is just the charge density times the velocity. We get as our final expression for the magnetic force per length:

Since the magnetic force must exactly cancel with the electric force, we have

Solve for v and you finally get the answer

Gasp! It’s the speed of light. The goal of the quiz is of course to show in dramatic fashion that the speed of light is naturally a consequence of the very same equations we’ve been using to characterize mundane charged particles and wires. When I first saw this problem as an undergrad it felt like an epiphany. Yes of course you can show that Maxwell’s equations satisfy the wave equation, but it’s quite another thing to see light metaphorically blaze forth from a problem with nothing to do with light at all.

Lurking in the background is some even deeper physics. After all, in velocity regimes less than c the electrical repulsion will be partially canceled by the magnetic attraction and thus the acceleration of the wires apart from each other will not be as fast. But in a frame of reference moving along with the wires, there’s no velocity at all and thus there’s no magnetic field at all. The wires should accelerate apart at full speed in that frame. Einstein ended up dramatically dispatching this discrepancy when he wrote the relativistic versions of Maxwell’s equations. Turns out (simplifying, and making a long story short) that in fact relativistic effects resolve the difficulty. The premise of the question - that the electric and magnetic effects completely cancel - is not actually realizable.

The question has some real depth, and it’s not too often you have the opportunity to do that in a question that a Physics 208 student can be legitimately expected to do. So I’m always looking forward to being able to assign this particular problem!