Introduction to Electrodynamics/Chapter 7/4
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Contents
7.4 Potential Formulations of Electrodynamics
In section 7.4, we look at the potential fields for Maxwell's Equations, the equations that completely describe the electric and magnetic fields even for dynamic systems.
7.4.1 Scalar and Vector Potentials
Back in electrostatics, we had the curl of E = 0. This let us write the potential using a scalar field, where the gradient of that field was the electric field. Now that we have the curl of E equal to the change in B, we can't use that anymore.
But because we haven't yet found any magnetic monopoles, we still have the divergence of B equal to 0. So we can still use our A field for the magnetic vector field, with the simple rule that the B field is the curl of A.
When we stick this into Faraday's Law, we get that the curl of E is equal to minus the time derivative of the curl of A. We can re-write this as the curl of the E field plus the time derivative of A is equal to 0.
Now we have something with a curl of 0, and now we can pull out a scalar potential, which we'll call V, the electric potential, because when A isn't changing very fast, it works just like it did in electrostatics.
Unlike B, which is simply the curl of A, E is now negative the sum of the divergence of V and the time derivative of A. We can rewrite it with V on one side, to suggest that a changing A is really like an E, which is what we found out earlier.
We're done with the equations 2 and 3 of Maxwell's Equations. Let's plug these values into Gauss's Law and Ampere's Law with Maxwell's Term and see what we get.
When we plugged in the electrostatic E = - grad V into Gauss's Law, we got back Poisson's Equation. Now, we have this extra d by dt A term, which, when we apply the divergence, is d by dt div A. This is all equal to minus 1 over eps0 rho, the volume charge density.
When we stick these potential fields into Ampere's Law with Maxwell's Term, we get back this fairly complicated formula. On the left side we have the double-curl of A, and on the right we have the volume current density J, and the time derivative of E, which is V and the time derivative of A. Plugging it in, we get three terms on the right, J, gradient of the change in V over time, and the second derivative, the acceleration, of A over time.
When we use the identity of the double curl is equal to the gradient of the divergence minus the vector Laplacian of the field, we can rewrite the terms this way. There's a bit of a pattern here, which we'll look into with more detail later.
So let's review the important equations we've discovered. First, the curl of A is still B, since there is no magnetic monopoles. Next, we have V expressed in terms of E and the time derivative of A. When we plugged these into Gauss's Law, we found that the charge density is related to the Laplacian of V and the change over time of the divergence of A. We also found that we could rewrite Ampere's Law with Maxwell's Correction in this complicated form.
Example 14
In Example 14, we're given the two potentials V=0 and A = this mess. Note that c is the 1 over the square root of epsilon0 mu0. Yes, this is the speed of light, and yes, as we'll see more in chapter 8, it must be that way. alpha is just a constant.
So when you've moved beyond ct from the origin along x , you're going to have an A of 0, but within that range, it's this. The term on the left increases over time linearly---kind of like a train moving at the speed of light. The term on the right is how far away you are from the origin along the x axis.
With V 0, the electric potential equation gives us that the electric field opposes the rate of change in A. That's rather easy to calculate...
The B field is simply the curl of A
And the E field depends on A.
We have our equations. Calculating the curls and divergences and time differentials is rather easy.
You can see now that we have everything we could ever want to know about any of the four Maxwell's equations. (Do it)
You'll note that you can even calculate the current. Note that there is a discontinuity at x=0. That implies you have a surface current. Remember that the curl of the normal and the current give you the discontinuity in B, so we calculate the surface current to be XYZ moving in this direction.
That's simply a current that gradually rises over time, creating this weird magnetic and electric field.
7.4.2 Gauge Transformations
Remember back in electrostatics we discovered that you can add any constant to the potential V and it has no effect whatsoever on the electric field. We also discovered with the magnetic potential vector field that you can add specific values to it and it has no effect on the magnetic field. Now, to make our lives easier, we would choose the electric potential to be 0 at infinity, or 0 on a plane, or the surface of a cylinder or a sphere, and that made solving certain problems easier. For the magnetic potential, we chose to set A so that its divergence was 0, not that we had to, but because it was easier for us.
Could it be that we can choose different V's and A's to make our lives even easier? The answer is yes, but we now have a problem: A and V are entangled in E. If we monkey with V, we may have to monkey with A.
Let's start by assuming we can add some vector potential alpha to A to create a new vector potential that gives us the same B and E. Let's also add beta to the electric potential to give us a new electric potential that will give us the same E as well. (B doesn't depend on V of course. It's just the curl of A.) Maybe alpha and beta are related. Maybe not.
So A prime and V prime need to give us the same B and E.
The curl of A prime is simply the curl of A plus the curl of alpha. Since the curl of A gives us B, and we need the curl of A prime to give us the same B, this implies that the curl of alpha must be zero. Just like the curl of E in electrostatics gave us a scalar potential, we can write alpha as the gradient of some scalar field lambda. Any lambda will work, it doesn't matter, as long as it's a scalar field.
The E vector due to our A prime and V prime cannot change as well. E is minus the gradient of V and V prime, minus the time derivative of A and A prime. A prime is just A plus the gradient of lambda. V prime is V plus beta. So distribute these out, and cancel like terms, and we get that the gradient of beta has to cancel the time derivative of the gradient of lambda. Extract the gradient, and we have that beta must cancel time derivative of lambda.
Now, the gradient of this must be zero. As for the scalar electric potential V in electrostatics, we can add any value at any time to the combination of beta and d by dt lambda, and it won't change the gradient. It will stay zero. So beta is minus the time derivative of lambda plus some function of time, and everyone's happy.
This k(t) can be pulled into lambda, simplifying everything further. If we add the integral of k dt from 0 to t to lambda, then the time derivative of lambda will include k(t).
We've just shown how we can monkey with the A and V fields. We can add the divergence of some scalar field lambda to A as long as we subtract the time derivative of it from V. As long as we do this, we can have whatever A or V field makes the problem simpler.
Remember for V in electrostatics, we chose a V of 0 at infinity, or a V at some plane to be 0, or V to be 0 on the surface of a cylinder or sphere. That made problem solving easier. Remember for magnetostatics, we chose A so that it had no divergence, although we could've chosen A to be any field that had a divergence and it wouldn't have changed B. Depending on what we choose to do with our A and V fields in electrodynamics, certain classes of problems will be easier or harder to solve. There are a number of choices, called gauge transformations. We're going to talk about the two most popular and useful ones next. You can probably find some on your own.
7.4.3 Coulomb Gauge and Lorentz Gauge
Two sections ago we talked about how A and V interact with E and B. The B field is still the curl of A, but the E field now depends on the gradient of V and the time derivative of A.
Then last section, we saw how we could modify and A and V combination and still get the same E and B fields. We found that we could simply add the gradient of this function lambda to A, and subtract the time derivative of the same function from V, and obtain A's and V's that didn't change our E's and B's. What we choose for lambda, or rather, what kinds of A's and V's we look for, are the gauge transformations. We're going to talk about the two most important ones here.
Coulomb Gauge
For the Coulomb Gauge, we're going to stick with the convention from magnetostatics and choose A to have a divergence of 0. That changes Gauss's Law in temrs of the potential to simplify to Poisson's Equation.
Well, we've already solved Poisson's Equation back in chapter 3. We're back in our electrostatic universe, it seems. Except now the E field is going to have this dependence on the time derivative of A, which it didn't before.
This should bother you. That's because the V potential is determined everywhere all at the same time, depending on how the charge is configured right now. We know that it takes time for electric and magnetic fields to propagate, or at least we will in chapter 8 and 9. To understand why this is absurd, pretend you setup your electrostatic measuring device that can detect the electric potential, which is based solely on how charges are configured anywhere in the universe at that time. If someone moves a charge on Mars or halfway across the galaxy, you should be able to see it right away. You've effectively skirted the laws of relativity, namely, that things don't go faster than the speed of light, including things like news on how charges are positioned.
The trick is you cannot measure V itself, you can only measure E and B. And E is dependent on both V and A. What is A doing? Well, it relates to the current, and a moving charge constitutes a current.
We can write an equation for solving A. We start with what we got when we pluffed in A and V to Ampere's Law with Maxwell's Equation. Remember this mess? Well, we set the divergence of A to be zero, so the divergence of A term falls out. Looks a bit easier to solve, but I would rather not.
So, the Coulomb Gauge gives us a V that is easier to solve, but an A that is not. As long as we're willing to remind ourselves that news of changes to V take time to propagate, we can use the Coulomb Gauge for some approximations.
Lorentz Gauge
The Lorentz Gauge is when we choose the divergence of A to be minus the time derivative of V.
This makes Ampere's Law with Maxwell's Correction much simpler.
Now we can solve for A given J.
Before we talk about that, look at what happens to V. Let's plug in the time derivative of V for the divergence of A, and now we have an equation for V.
Now it's time to stop and look with wonder upon what we have discovered. We have two equations that look almost exactly alike, except we can change out A for V and J for rho.
This part on the left is the d'Almbertian. It is represented with a square box (squared), presumably because instead of 3 dimensions, we now have 4.
What's especially beautiful is now we have four equations where the Laplacian had 3. We can solve this. It's not as hard as you think.
Let's look at this more closely. The Laplacian on the vector potential V is simply the second derivative, the 'acceleration', if you will, of V compared with x, y, and z, added together. The fourth term is minus 1/c^2 the second derivative with respect to time, the actual acceleration. For A, we have actually 3 equations, one for each dimension. The x component of J must equal to the Laplacian of the X component of A minus the second time derivative of the x component of A. And the same for y, and z.
So we really have 4 equations here.
Anyway, we have what seemed to be an impossible problem to solve broken down to a set of problems we know how to solve. We're making a lot of progress.
7.4.4 Lorentz Force Law in Potential Form
Let's say we get our A and V figured out, and we want to get our force on the test charge moving through these potentials. We could convert them to E and B and run from there, or we can just rewrite the Lorentz Force Law in temrs of the potential.
Product Rule 4 gives us a nice symmetry we can exploit. Substitute these two terms for v cross the curl of A.
And we can move things around, to give us the derivatives with respect to time on one side and the potential on the other.
The stuff on the left is collected into a sort of momentum term we call the canonical momentum. Some of the momentum is bound up in the combination of the magnetic potential and v. On the right is the potential which unfortunately depends on velocity.
