Induction, Induction, Induction…

Electromagnetic induction. We’ve all heard of it in all sorts of things. Power generators? You got it. Transformators? Yup, it’s there. Induction stove? It’s all in the name. And it’s always taught in physics classes. Despite these, there is still this one thing that keeps bothering me. An example that is often included in explanations of electromagnetic induction but I think not very accurate.

I’ll start with the induction itself. Electromagnetic induction is not all that complicated actually. It starts with magnetic flux, the amount of field lines through a surface. Mathematically,

where the integration is taken over the surface . Simple enough. Now, let’s say there is a conducting loop . Faraday found that a changing magnetic flux produces emf around the loop,

Yes, there are an infinite number of surface with as boundary. Which surface should I use? Any surface would do. Any surface will produce the same flux and also the same emf. And the emf itself is a measure of how much the charge carriers are being forced to move. Precisely,

where is force per unit charge acting on the charge carriers and the integral is carried out along a line.

Clear enough. Where’s the problem?

Imagine there’s a conducting piece of straight wire, at right angles with a homogenous magnetic field, moving at right angles with both the wire and the field with velocity . Free electrons inside the wire are moving in a magnetic field and feels Lorentz force as a result. They are then pulled in the direction of the wire. Voila! Emf is produced.

Magnetic field does produce emf in the moving piece of wire of length . There’s no problem with that. The problem is, people confuse this with the electromagnetic induction as in the above. The argument goes like this: the emf is electromagnetic induction with the surface from which flux is calculated taken as the area the wire swept over. Let’s get quantitative. The rate of change of the area swept is, simply,

Going further, the emf is,

Done.

What if we followed the previous argument with the Lorentz force? OK, let’s do it. The force of a charge moving at is,

Hence, the emf,

There goes the problem. Somehow, both results are identical even though the initial arguments are radically different. Personally, I prefer the second way of thinking because the following reason.

In the loop and changing magnetic flux illustration, the change in magnetic flux creates electric field. It is the electric field that acts on charge carries to produce emf. Electromagnetic induction means changing magnetic field produces electric field and vice versa. In the piece of moving wire case, there is no electric field at all. So it’s actually just magnetic field making emf.

For those of you who love physics but love the mathematics just as much, don’t worry, me too. So, let’s talk about this in a more mathematical way. The second (that one with curl of ) of Maxwell’s equations states

This is Faraday’s induction law rewritten by Maxwell. Surface integrating both sides and applying Stokes’ theorem gives Faraday’s law (with ).

Assuming the surface is constant over time (read A/N),

On the other hand, the emf can be defined in the following way,

This separates the two cases clearly. In the first case, the emf arises from the first term (induction) in the integral. While in the second one, the emf arises from the second term(Lorentz force).

That’s all in this post. I might (strong emphasis on might) add some figures to help with the illustrations but I’m just too lazy right now. Hope my post clears a few things for people out there.

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Author’s Note:

Constant over time…interesting constraint. A surface that is changing over time is probably going to be pretty bizarre but interesting anyway. I’ve almost never seen calculations with a surface that is changing over time other than Alfven’s theorem. You know what, I think I have my next blog post 🙂