## Tuesday, April 23, 2013

### I have seen the ∂2E(x,t)/∂x2 = µ0ε0∂2E(x,t)/∂t2!

(There's some vector notation that I wasn't able to figure out how to get into a title. And that's an exclamation point, not a factorial.)
So, I'm reading ahead in my physics textbook, and I've reached the culmination of all this electricity and magnetism stuff. That equation up there, a nice little second-order partial differential equation, is ostensibly the pinnacle of 19th century physics and Maxwell's greatest contribution. (It's my understanding, however, that Maxwell didn't actually use that notation, and that there are plenty of other possibly more useful ways to write the equation. Nevertheless, that's my textbook's presentation.)
So what does that equation say? Well, a more or less literal translation says that the way in which an electric field is distributed through space is related to the way in which an electric field is distributed through time by the constants µ0 and ε0. This relationship arises from Maxwell's equations. The ones that are most relevant here are Faraday's Law and Ampere's Law.
Faraday's Law tells us that a changing magnetic field induces an electric field. I mentioned that in this post. The most frequent application of this law is in, well, almost every method of power generation we have. Some process (burning coal, burning gas, burning uranium) causes water to boil, and that water spins a turbine connected to a magnet, and that magnet's magnetic field moves through space, which sets up an electric field (and a corresponding current) in some conveniently placed wires. The faster the magnetic field changes, the stronger the current. But as soon as the magnetic field stops changing, the current dissipates. It is only while the field is changing that an electric field is generated.
Ampere's Law, in this context, tells us something similar. It says that a changing electric field, multiplied by µ0 and ε0, produces a magnetic field.
I suspect you can see the symmetry here. A changing electric field produces a magnetic field, and a changing magnetic field produces an electric field. If the rate at which an electric field changes is increasing, then the magnetic field it produces will also be increasing. And a magnetic field that is increasing in strength will produce an electric field that increases in strength in a direction opposite to the first electric field, which will tend to diminish the strength of the first electric field. This symmetry creates a sort of back and forth seesaw effect between electric and magnetic fields.
The way this connects back to the equation of the title is that the rate at which a rate is changing is known as a second derivative. The most common example is acceleration. Velocity is the rate at which position changes. Acceleration is the rate at which velocity changes, or the rate at which the rate at which position changes. And as you saw, an "accelerating" electric field produces a changing magnetic field, and vice versa. The 2 in ∂2E(x,t)/∂t2 means we're talking about second derivatives.
Okay, what's the point of all that? The point is that, traditionally, an electric field is set up by charged particles. An electron sitting by itself creates an electric field that extends radially beyond it. And magnetic fields are usually caused by moving charges. An electron flying off by itself at a constant speed will have a magnetic field encircling it. But Maxwell's equations say that you can get an electric field just by shaking a magnetic field around, and you can get a magnetic field just by shaking an electric field around. You don't need any charges at all (beyond an initial one to set up whichever field comes first). Electric and magnetic fields sustain each other, giving rise to electromagnetism.
On a basic level, I knew this beforehand. I didn't know the details or the math, however. But what really made the concept click for me was a discussion in an earlier chapter about LC circuits--that is, circuits composed of inductors and capacitors. I already discussed what a capacitor does in my last post, but an inductor is something new. An inductor is a circuit element that takes advantage of Faraday's Law in order to modulate the current in a circuit.
So, a current is just a bunch of moving charges, which means that all currents create magnetic fields. But when the current changes, as it does in AC circuits, you set up a changing magnetic field, which in turn creates an electric field that opposes the current change. The faster the current changes, the stronger the resultant magnetic and electric fields are. In essence, energy is being taken out of the current and put into the magnetic field. An inductor is an element designed to maximize the energy pumped into the magnetic field. This sounds a lot like a capacitor, where energy is being taken out of a current and stored in an electric field.
What happens when you put these two together? Well, as we saw, when you discharge a capacitor, it expels its electric energy very quickly at first and then slows down. But an inductor opposes current change, so the sharp increase from zero current is curbed by the creation of a strong magnetic field in the inductor. The effect is to take the energy stored in the electric field and place it into the magnetic field.
Once the capacitor is fully discharged, the current should stop, but again, inductors oppose current change. So instead, the inductor dissipates the energy from its magnetic field to increase the flow of current. The capacitor, down to zero charge, now begins to charge negatively. That is, it builds up electrons on the opposite plate. The energy from the magnetic field is transferred back to the electric field. With no resistance, this oscillation continues indefinitely, trading energy between the electric and magnetic fields of the circuit. The rate at which this happens depends on the properties of the circuit, but the general shape of the interaction is going to be a sine wave.
Yes, that's right, when you move energy between electric and magnetic fields, you get a wave. One might even be tempted to call it an electromagnetic wave. In fact, the equation in the title takes the form of the general "wave equation" that can also apply to the wave-like motion of a spring or a sound wave or a whole host of other physical phenomena. The general equation looks like this: ∂2u/∂x2 = 1/v2 * ∂2u/∂t2, where u is some wave-like phenomenon, and v is the speed at which that wave propagates.
If that's the case, then µ0ε0 takes the place of 1/v2 in an electromagnetic wave. Carrying out the algebra means that the speed of an electromagnetic wave should be 1 divided by the square root of µ0ε0. µ0 = 4πx10-7 and ε0 = 8.85x10-12. Multiply these and your product is 1.11x10-17.Take the square root of that and you get 3.33x10-9. Find the reciprocal of that and you arrive at 2.99x108. This, of course, is the speed of light in vacuum. Light is a self-propagating wave of electromagnetic energy.
I have seen the light.

(There's a bit of cheating here. Those constants are now defined in relation to the speed of light, so of course the algebra works. But way back in the 19th century, εwas a proportionality constant that described the ability of the vacuum to act as a capacitor, and µwas a proportionality constant related to the magnetic force between two lengths of current-carrying wires a meter apart. And it was thought of as a rather interesting coincidence that putting those two constants together got you the speed of light. Maxwell set 'em all straight.)