## Faraday’s law is overused

First let me say that I thought for sure I’d written this post last year, but it appears not. That could also mean that maybe I didn’t write it because someone else had a good breakdown of my argument, but a quick check of google didn’t really find anything. So, here goes.

Faraday’s law is sometimes called the law of induction. It’s really one of the four Maxwell equations that (supposedly) explain all of electromagnetism.  In differential form it’s:

$\vec{\nabla}\times\vec{E}=-\frac{\partial \vec{B}}{\partial t}$

But most physics students first see it in integral form:

$\text{EMF}\left(=\oint \vec{E}\cdot\vec{dl}\right)=-\frac{d}{dt}\int \vec{B}\cdot\vec{dA}\equiv -\frac{d}{dt}\Phi_B$

Basically it says that you’d get current flowing around a circuit if the magnetic field flowing through the circuit changes.

However, I claim it’s way over used.

What I mean is that at least half the homework problems you’ll find in typical physics texts about Faraday’s law could be done (and often easily) using the concept of motional EMF, a concept that grows right out of the “magnetostatics” the students have been studying up to this point. Motional EMF simply has you calculate the actual magnetic forces on the charges in the wires of the circuit and has you figure out which way they’ll move. There’s no need for some strange “induced electric field” and the direction of current flow is attainable from a straightforward right-hand-rule that they’ve already learned ($\vec{F}_B=q\vec{v}\times\vec{B}$).

Any time the magnetic field is fixed in space and the wire moves (generators, rail guns, etc) you don’t need Faraday. The motional EMF gives you a perfectly fine answer, and it doesn’t require a deeper understanding of the connections between the magnetic and electric fields.

Don’t get me wrong, I think we should teach the crap out of Faraday. It’s incredible! Apparently if you have a time changing magnetic field, you get a new type of electric field produced (very much unlike the field that students have seen before!). But too often the homework doesn’t actually involve a time-varying magnetic field. It’s just a uniform, constant field that shines on a cool circuit that moves. If that’s the case, you don’t need Faraday.

Certainly I agree it’s cool that the motional EMF calculation can be rejiggered to show that it’s equal to the time rate of change of the magnetic flux. But so what? If it’s not what’s actually happening in the physical situation, why confuse our students with it?

Here’s an example. Consider a rectangular circuit where one side can be grabbed and moved further down the rails of the neighboring sides (case 1 in the image below). If there’s an unchanging magnetic field directed down through the plane of the circuit, if you move the bar, you’ll get current flowing around the loop. Cool! But it’s “just” because the free/mobile charges in the bar are responding to the magnetic force and moving along. No weird new electric field. Now consider the usual next case in a textbook (case 2 in the image below). The bar remains the same but a “magnetic field flashlight” sweeps in from the other direction. Whoa, it seems pretty similar to the first case, it’s just that the relative motion has been switched.

2 cases for a typical Faraday’s law discussion

But now the wire (and all the charges inside it) aren’t moving, so it can’t be the magnetic force getting them to go around. Instead we have something new! The field inside the circuit is not constant (nor uniform). So now it’s creating a circulating electric field which pushes the charges around. Awesome! Do you get the same answer? Amazingly yes. Though students don’t seem too surprised since they’re pretty convinced by this time of Galilean relativity.

Interestingly there’s a great footnote in chapter 7 of Griffiths’ famous electrodynamics book for undergraduates. He says that perhaps we should call Faraday’s new field the G field and point out that it pushes charges around just like an E field. However, it has a curl and no divergence!

So what do I propose? Just keep the homework problems a little more separate. Or at least ask your students if the current is due to (boring) magnetic field forces or (interesting) weirdo Faraday’s electric field forces.

Your thoughts? Here are some starters for you:

• Why do you say supposedly when talking about Maxwell’s equations?
• Why do you put vector symbols over your vector entities instead of bolding them?
• What do you mean when you say that Faraday’s electric field is different than Coulomb’s electric field?
• I’m in your class right now and I like this distinction. It helps me . . .
• I’m in your class right now and I don’t really care about this distinction. As long as I can . . .
• Why should we introduce a new field if it acts just like an E field? (it doesn’t)
• I love having students figure out whether the amount of B field is growing or falling in generator problems. Leave me alone!
• It’s all relative, students should use whatever calculation approach helps them get the right answer.
• I love this post because I like to have my students focus on what’s really happening microscopically.
• I hate this post because I don’t want to have to explain what a curl is to my introductory students.
• In motional EMF you have to hand waive the idea that the field is constant in a wire. How do you approach that?
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## About Andy "SuperFly" Rundquist

Professor of physics at Hamline University in St. Paul, MN
This entry was posted in electromagnetism, general physics, physics. Bookmark the permalink.

### 6 Responses to Faraday’s law is overused

1. Brian says:

You should check out the university of Washington tutorials in introductory physics related to this. There is a good sequence. It has students work through cases using forces on charges as you suggest, before introducing Faradays Law. At same time it also introduces them to the flux idea and gets them looking at patterns between change in flux and and current . It then introduces a scenario where flux changes (which if the pattern continues suggests we should see a current), but an analysis of the magnetic forces suggests no current should arise. It the just tells students what happens, but you could easily demo or do as a lab.

• Andy "SuperFly" Rundquist says:

cool! I figured there’d be an approach out there I’d like. I think I was motivated to post this because the text I’m using (Young and Freedman) does Faraday’s flux approach before motional emf. Weird. I didn’t do it that way in class today.

2. David Brookes says:

How is case 2 NOT a case of motional emf? As you say, there’s relative motion between the charges and the field. Please elaborate. ☺

• Andy "SuperFly" Rundquist says:

That’s the case with the sweeping b-field “flashlight”. No charges are moving (except up at the flashlight, of course).

3. Rob Ryan says:

Your first starter was my question. To wit, Why do you say supposedly when talking about Maxwell’s equations?”

• Andy "SuperFly" Rundquist says:

Ha, that’s really just my pet peeve that you also need Coulomb’s law and the Biot Savart law to understand the origin of the fields in the first place.