I’m covering we’re working on Ampere’s law in my calc-based general physics course. In preparation I was looking around at different ways to present it, and I realized that I was getting crabby about most of the presentations I found in the various texts that I looked at. Authors were saying “trust me, this is true so let’s use it”, “See how it works for infinite lines? It always works” or “Here’s a whole different truth about magnetic fields so let’s use it!” Ok, I’m still crabby as you can tell.
Since the last time I taught this course I’ve taught our advanced electromagnetism course and I work hard to get the students to understand the connections between current and magnetic fields. In that class we do all the ugly multivariable calculus to show that Ampere’s law follows directly from the Biot-Savart law. In other words, it’s not any new physics, it’s just a restatement that helps out in very symmetric situations.
That last point about symmetry gives a great connection with electric fields. Students learn how to calculate any field using Coulomb’s law and then are later shown how the structure of electric fields is such that Gauss’s law can come in handy in symmetric situations. I want to do the same thing tomorrow, but there’s a problem with extending the analogy. In electrostatics, the notion that Gauss’s law is the same as Coulomb’s law is easy for a single charge, and then I work with the students to see how any charge distribution is really just a combination of little charges. Therefore Gauss’s law always works. But in magnetostatics (I hate that word), while it’s easy to show that Ampere’s law works for an infinite straight wire, I don’t see an easy way to convince students that any current setup can be thought of as a collection of infinite (parallel – since that’s all our book does) straight lines of current. I’m so nervous that a student will say “but what if the wires bend just above the surface we’re considering?”
So, what to do? I went over and talked with my colleague who teaches multivariable calculus this semester and asked what he thought about me doing a little of what I do in our advanced electromagnetism course. He said they haven’t quite gotten to curls and they’re a long way away from Stoke’s theorem, but he figured it’d be good if I gave a good hand-wavy try at it along with saying something like “and you’ll see how this works in math later this semester.” So I thought I’d get some thoughts down here about how I might try to do this tomorrow.
One thing I could do is show them a simulation of a couple of nearby infinite, parallel currents. I’ll make one of them have an adjustable current and I want to get the students to talk about what happens to the overall field as you change the current. I guess I’m going for comments about circularity, but I’m willing to see where that goes.
Another idea, playing off something we did yesterday, is to continue to look at the analogy between electrostatics and magnetostatics (I still hate that word). If Coulomb’s law leads to Gauss’s law, what could the Biot-Savart law lead to? Magnetic field lines never begin or end, so a straight Gauss’s law isn’t very useful (unless there were magnetic monopoles – oh well). Electric field lines poke out, so let’s see how many poke out (Gauss’s law). Maybe something like: Magnetic fields go around in circles, let’s see what happens if you follow them around (weak, I know).
Big idea: local (non-infinite) currents cause magnetic field circularity. So a measure of the circularity should tell you about the local currents. If we add up all the curl in an area, we’d have all the current poking through that area. But the fundamental theorem of calculus tells us that we ought to be able to get interior information from the border. Maybe just adding up the circulating field at the edge will tell us about the current poking through:
Another approach (less likely, though). What is the curl of the Biot-Savart law? This is tough because you need at least one vector identity and the use of at least one delta function. I guess I’m not sure what it means to handwave this.
And, of course, I want them to see that this’ll only be useful in symmetric situations, where the integral above is a near write-me-down.
Thoughts? Some starters for you
- Why do you hate the word magnetostatics? I love it and here’s why . . .
- I’m in this class and I’m super excited for tomorrow. Mostly I can’t wait to . . .
- I’m in this class but I’m skipping tomorrow because this seems dumb.
- If the students are going to eventually take advanced electromagnetism, why do you care?
- Just give them the formula and show them the three cases that it works for. Then they’ll be all set to do tons of
- I love Ampere’s law. My favorite thing to do when teaching it is . . .
- I hate Ampere’s law. I avoid teaching it by . . .
- Whatever you do, make sure you give them 17 more right-hand-rules to memorize.
- Why didn’t you put the accent in everywhere you typed Ampère?