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.
