With the semester winding down, I think it’s a good time to discuss a slightly annoying idiosyncrasy I’ve observed this spring. E&M has, as a co-requisite, vector calculus. My school expects these two classes to be taken together, and E&M involves a fair amount of vector calc. But despite this connection, or perhaps because of it, my math and physics have been, you might say, out of phase.
What do I mean by out of phase? I think a thousand words can sum it up best…
As you can see here, Lt. Cmdr. Geordi La Forge’s hand is out of phase with the engineering console. They’re in the same place at the same time, and yet somehow they’re not actually interacting. The solution, of course, is a concentrated burst of anyons. Yep.
Or maybe I mean something a little more like this:
We just got through a unit on AC circuits, which are significantly more complex (heh) than DC circuits. The main reason is that the three basic elements of a circuit all respond differently to alternating current. Resistors don’t much care. They reduce the voltage across them, but (ideally) their response isn’t dependent on the incoming voltage. Thus, the voltage across a resistor looks like whatever current is coming from the AC source.
A capacitor, on the other hand, has its maximum voltage when it’s fully charged. And when it’s fully charged, the current across it is zero. Thus, the voltage across a capacitor is at its peak when the blue line crosses zero.
And inductors resist changes in current, which means an inductor will always have a voltage that is positive when the source is decreasing, and negative when the source is increasing.
Why did I go through all of that? Well, for two reasons. First, the differences between those circuits give rise to the different voltage graphs above, and those differences are known as phase shifts. Resistors are “in phase” with their voltage source, whereas inductors and capacitors are “out of phase.” That is, they don’t line up.
When two waves that are in phase with each other combine, they produce one wave that is bigger than either of the two waves. Conversely, out of phase waves that hit cancel each other out. This is known as interference, and maximizing the acoustics of a room is all about making sure sound waves are in phase when they get to the audience.
Secondly, a complete description of the wave-like behavior of AC circuits involves complex numbers—the square roots of negative numbers—a topic which has only been given the briefest of overviews in any math class I’ve ever taken. Consequently, the entirety of the complex aspect of AC circuits was covered in a single line from my professor: “And if we increase this factor over here, you see that we get imaginary numbers.” So, you see, my math and physics classes are out of phase.
But it doesn’t stop there. Near the beginning of the semester, we learned that the electric field can be thought of as the change in voltage across space. The text and our professor briefly mentioned that this is referred to as a gradient and involves partial derivatives. But at that moment in math, we were learning how to plot three-dimensional vectors.
Later on, we learned about Gauss’ law for electricity, which lets you figure out the amount of charge enclosed by a surface if you know what the electric fields coming out of that surface look like (and vice versa). When you first learn how to use Gauss’ law, you’re only asked to apply it to surfaces that are highly symmetric, which makes the math a lot easier. But it turns out that Gauss’ law is a very general principle in vector calculus involving surface integrals. And what, pray tell, were we learning about in math at that moment? Partial derivatives, of course.
Now, as the semester draws to a close, our professor has shown us Maxwell’s equations in all their glory. All their integral glory, anyhow. We’ve seen them in one form or another already, and we’ve used three of them. The one we haven’t used is Gauss’ law for magnetism, which is like the electricity law except it says there’s no such thing as magnetic charge and there’s never any net flow of magnetic field lines through a surface. Why is there a law saying the answer to your question is no? Because, our professor explained, the law is more useful in its differential form, where it uses a vector calculus principle known as divergence. We’ll be getting to divergence next week in math. Right now we’re studying surface integrals.
Maybe I'm misinterpreting all this. Maybe learning about the idea in physics first primes us for the material in math later. But I'm not sure I buy that argument. I feel as if we'd be better served by having to take vector calculus first, or by having E&M and calculus classes that were more, well, in phase with each other.
Anywho, so when are we going to learn the math behind complex numbers and phases? Apparently it involves phasor arithmetic and phasor diagrams. I don't know about you, but I encountered phasor diagrams waaaay before I started taking math classes...