Visualizing a Student's Guide to the Maxwell Equations
Hello:
My focus has always been on the best field equations in all of physics, the Maxwell equations. My proposal for gravity is a variation on Maxwell: change the algebra for forming the action from Hamilton's quaternions to hypercomplex numbers (what I call the even representation of quaternions). Now that I have launched this site, my hope is to create visualizations of any applications of the Maxwell equations. It would be quite a site to understand visually the difference between the E and the B field!
At a physics meeting, I saw some promo material for the book "A Student's Guide to the Maxwell Equations" by Daniel Fleisch. It received 5 stars from readers at Amazon, a great complement. There is a web site that accompanies the book where they work through all the problems. I bought the book with the specific goal of developing animations for all those solved problems.
Table 1.1 featured this equation:
This equation is central to understanding EM. I have to figure out a way to animate this expression, or the project is dead. I like having clear "killer" goals - if not done, the project dies. Harsh criteria focus the mind.
There are lots of 2D graphs with charge on one axis, distance on the other. For these animations, there is no axis to label. All I can work with are events. If there was one unit of charge at position (1, 0, 0), then at position (2, 0, 0), there would be a quarter of a charge. The position for both of these are fixed - nothing can be done with space. That leaves only time to represent the difference in charge. There are at least three alternatives to representing this information visual. One is to use the charge value as the time value. Thus one would animate (1, 1, 0, 0) and (0.25, 2, 0, 0). The higher charge appears later. A negative charge would be in the past, or (-1, 1, 0, 0) and (-0.25, 2, 0, 0).
A second visualization approach would be to view the values as frequencies, so if the frequency was equal to 1, that point would be on at all times, whereas if the time was .25, it would appear on the screen a quarter of the time. The negative charge would then be 1 - value, so those points in space that used to be dim are now bright, and those that were bright would be voids.
A negative number is an additive inverse. It might be that multiplicative inverses should be used for negative numbers, so .25 goes to 4, while 1 stays at 1. I have no idea if this idea will fly, but thought I should record it for later reference.
A third way to view the magnitude of charge would be through the transparency. If the value was 1, the color would be solid. If the value is 0.25, it would be 75% see-through. Negative charge would be a 1 - value operation.
Flipping through the book, three kinds of symmetries are common for these problems: point/sphderical symmetry, line/cylinder symmetry, and plane symmetry. The software now is designed exclusively to render points. I have to do much thinking about how to easily generate spheres, cylinders, and planes.
One exciting variation on this line of work is that wherever there is an electric field, there is necessarily a gravitational field. This is because to have an electric charge, a particle must have mass. If I figure out a problem in EM, then by switching from quaternions to even quaternions which only changes multiplication, I will solve a problem with gravity. This will be a stringent test of the visual representation of this information. For example, we all know gravity only attracts, while in EM charges can attract or repel depending on their relative signs. The visualization must have this property, or they are wrong.
Wish me luck. I have no confidence I can achieve these goals, but that is the way I operate. The key thing is to make goals precise: visualizing "A Student's Guide to the Maxwell Equations".
I was wondering what would happen if I added these two together. The answer is a circle that moves along a diagonal:
The thought was that if E and B are really part of the same anti-symmetric tensor or quaternion, then we need to know what it looks like when added together as a policy.
The thing that bothers me while driving into work or walking the dogs is the the way no matter how big values of t, x, y, or z are plugged into the exponential, the values for E and B are always going to be +/-1 at most. I don't get how to distribute the values for E and B fields throughout spacetime.
Hello:
The more general form of a solution than equation 1 is this:
What is flipped is the sign on the 3-vector. In an animation, that is a mirror operation. Animated circles already contain mirror symmetry, so the two terms look exactly the same. It is kind of funny to call them 2 different solutions when one cannot tell them apart visually.
Hello:
I am not happy yet with how to represent charge since all I have to play with is time, and the time <-> charge link is certainly not obvious.
I recalled that the plane wave solution to the Maxwell equations is simple:
where on this site the capital I is a normalized 3-vector, a generalization on the imaginary i. This always has a norm equal to 1. What is amazing about that simple observation was if you were to animate eq 1, then no event in that animation could have a time greater than +/-1, and nothing could move further than +/-1. To make this problem even simplier, imagine the wave is moving along the z axis towards us, so
. E has to be orthagonal to that motion. Again, just to make things simpler, imagine the E field is in the j plane. All of its values will be of the form: (a, 0, b, 0). This will just be a circle.
The B field must be orthagonal to both the direction of the wave (k) and of the E field (j), leaving it to work exclusively in i. Multiply the E field by i to get the B field. If you want to read technical stuff on this topic, go to "Classical Electrodynamics" by Jackson, chapter 7. Here is what it looks like:
The E field is a circle in the ty complex plane, while the B field is a circle in xz space. The superposition in the upper right corner looks like a piece of art, an uber modern cross with the red circle that never makes contact with the blue horizontal line. Since a B field involves the curl, I expected it to be "blinky" - time can only equal zero - and to be curved in space.
This image is quite different from the one in Jackson, figure 7.1, which has the E field as one arrow, the B field as another arrow, and the direction as a third arrow, all at 90 degrees at an arbitrary place in spacetime. One problem with their image is it does not show the U(1) gauge symmetry which is precisely what the circle in complex plane of my representation is. Jackson's arrow could be anything, but the ty circle is U(1), no doubt about it.
There are many more things to think about. My images does not move along x yet. Yet I am not sure it should move along x. The solution looks like it is limited to values of +/-1 in spacetime. How could that be? If physics is all about the observer sitting at (0, 0, 0, 0), it may not be relevant what happens out at x = 3. If all data gets mapped back to a unit sphere surrounding the observer at zero, then all we have to do is map the data to that sphere. I am very uncertain about this issue, but it is the kind of thing I puzzle about.