Cherish (mareserinitatis) wrote,

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Maxwell's Equations, Pt. 1

I've been attempting to figure out how to write a post on some of the basics of antenna theory (since Susan asked).

However, I'm leery of recreating the same experience I had upon taking my first engineering course. I was a junior and had just finished taking electromagnetics in the physics dept. I heard about this course being offered in engineering called, "Numerical Techniques in Electromagnetics." It was a 700-level class (a truly grad-level class, not a class cross-listed for both grads and undergrads), so I had to get permission from the instructor.

The title made my heart swoon, so I felt compelled to go ask. The prof (who later became my adviser) told me I had the prerequisite, which was my emag class in physics. I was totally going to rock that course.

I got to class and nearly passed out fifteen minutes into the lecture. The first thing we discussed was Maxwell's Equations, but they were nothing like I'd ever seen before.

(If you've made it this far and are math phobic, keep going...I'll try to keep it simple enough to understand that you don't need the math.)

The first thing I noticed was that there were no integral signs. In my emag course, we tended to use the integral form rather than the derivative form. I suspect the reason for this is a difference in what engineers are trying to solve versus physicists.

So let's begin with the familiar stuff. If I whip out my undergrad emag text, Griffiths' "Introduction to Electrodynamics" (which I have, in fact, whipped out), I can turn to the inside of the back cover to see Maxwell's Equations (which I, of course, have done). I have a quibble with Griffiths. He has two cases listed as being "In general" and "In matter". I personally think "In general" should be labeled "In vacuum" and "In matter" should be labeled as "In general". However, he is a textbook author, and I am not, so we'll have to leave it as that.

Interestingly enough, while most of the book uses the integral form, he shows the equations in their differential form in the back of the book. I haven't figured this one out.

I'm going to use his "In matter" equations because, as I said, I tend take those as being the most general case. Starting with Gauss' Law, we have

The funny upside down triangle thingy (called "del" or "nabla", depending on your preference) is used to find something. What it is finding depends on the symbol after it. Since we have a dot after it, this means we are looking for something that flows or changes in a straight line (i.e. as we move in a particular direction). So , or the divergence, is what we would call a flow source finder.

D is the electric flux density. (If you don't like my terminology, please read this first.) A simpler, though less precise, way of stating it would be the electric field in a material.

The equation above says that we're trying to find a flow source for an electric field in matter. And what is the flow source? According to the equation, , which is the free charge density.

In other words, an electric field which flows straight out of something is created by free charge. This is the same thing as rubbing your hair on a balloon and putting it on a wall. You rub your hair on the balloon to create free charge. This charge generates an electric field which, when placed on a wall, attracts charges of the opposite sign and makes the balloon stick to the wall. (Next time someone does this, you should exclaim, "Gauss' Law!" to impress these people with your keen intellect. There is an exception: if they are under the age of 10, you will just confuse them.)

That's the first law. There's another law that uses the divergence:

B is the magnetic field intensity, so this equation says that there's nothing that makes a magnetic field which will flow in a straight line. In other words, there are no "magnetic charges" (also known as magnetic monopoles). You don't have a positive magnetic charge and a negative magnetic charge, at least not in nature. Magnetic "charges" show up in pairs: a magnet always has a north end and a south end.

That's the easy stuff, and the notation and terminology is similar to what engineers use. I'll do another post later discussing the other two laws and show how the the differences between how physicsts and engineers tend to approach them.
Tags: electric fields, electromagnetics, emag, engineering, magnetic fields, physics, science

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