# Built on Facts

## The Light Fantastic

#### July 15th, 2008 · 7 Comments

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!

### 7 responses so far ↓

• 1 Ville Lindholm // Jul 15, 2008 at 9:39 am

That’s cool! :D

Does it have to be wires though, since this looks quite general?

Matt replies: No it doesn’t! In fact the grad school version of this problem is just two charged particles moving parallel to each other. The wires just make things conceptually simpler and more relatable to the real world.

• 2 Alex M // Jul 15, 2008 at 1:26 pm

This is pretty cool! I wish we had found out about “c” this way, instead of through an obtuse and somewhat dull wave equation lecture at the end of E&M.

• 3 Nick // Jul 15, 2008 at 1:54 pm

This is just downright interesting, your recitation students are lucky.

Of course, I’d like to request for a post about the relativistic effects that make things all better at speeds near c.

• 4 CCPhysicist // Jul 15, 2008 at 2:26 pm

So you just proved that the force depends on the coordinate system used. (For any value of v < c.) Any problem with that?

Oh, and the simplest analysis of why having a wire makes a difference (when the charges are moving instead of the wire) is in Berkeley 2.

• 5 Anonymous // Jul 22, 2008 at 8:48 am

best site ever came to .

• 6 An Interesting Worked Physics Problem « Twisted One 151’s Weblog // Jul 27, 2008 at 1:23 pm

[...] An Interesting Worked Physics Problem There’s an interesting worked E&M problem over at Built on Facts: The Light Fantastic. [...]

• 7 Jimmy // May 29, 2009 at 5:55 pm

After doing this problem as it was given in your blog, I am extremely disappointed with myself. I had been given an almost exactly identical problem in my physics class a few weeks ago, and I had experienced the same amazing feeling of epiphany upon solving it. The problem I had solved involved a charged capacitor with plates of equal surface charge density, and I had to figure out the speed v at which the magnetic force of repulsion balanced out the electric force of attraction. So, the problem I solved was essentially equivalent to your problem. However, when I was doing your problem, I DIDN’T SEE THE ANSWER COMING! Which just tells me that I need to work on my physical intuition. I feel like I should have recognized that your problem was my problem in disguise…