3 01/31 F

1. 1st Ed: p81, q107, 2nd Ed: p94, q107. (20 points). You need to show that any solution $\vec E,\vec H$ of Maxwell’s equations is given by scalar and vector potentials $\phi,\vec A$ as shown.

   Procedure to follow:

   1. Recall that if the divergence of a vector is zero, the vector is the curl of some other vector $\vec A_0$. Apply that to the appropriate physical vector (like the electric or magnetic field, say).
   2. Next define a vector $\vec E_\phi$ by setting
      
      $$\vec E = - \frac{1}{c} \frac{\partial \vec A_0}{\partial t} + \vec E_\phi$$
      
      

   3. Prove that the $\vec E_\phi$ defined this way is minus the gradient of some scalar function $\phi_0$.
   4. Unfortunately, $\vec A_0$ and $\phi_0$ are not unique and do not normally satisfy (1) in the book. The potentials you need are of the form
      
      $$\vec A = \vec A_0 + \nabla \psi \qquad \phi = \phi_0 - \frac{1}{c} \frac{\partial \psi}{\partial t}$$
      
      
      Show that in those terms,
      
      $$\vec E = - \frac{1}{c} \frac{\partial \vec A}{\partial t} -\nabla\phi \qquad \vec H = \nabla \times \vec A$$
      
      
      regardless of what you take for $\psi$. That is the famous gauge property of the electromagnetic field. It is central to quantum field theory.
   5. Since you can take $\psi$ whatever you like, you can choose it to simplify the mathematics. The way that you want to take $\psi$ here is so that equation (1), the famous “Lorenz condition,” in the book is satisfied. (No, there is no t in Lorenz. That is another Lorentz.) Show that this leads to a partial differential equation for $\psi$. (This equation is called an inhomogeneous wave equation. The properties of this equation will be discussed in the second part of the class.)
   6. Now substitute what you got so far into the four Maxwell equations and so find the requirements that $\vec A$ and $\phi$ must satisfy. (I.e. get rid of the electric and magnetic fields in favor of the vector and scalar potentials $\vec A$ and $\phi$.)
   7. How come only one vector equation and one scalar equation are left?
   8. Clean up! You should have decoupled equations for the two potentials.
   9. Show directly from Maxwell's first and last equation that the charge density must not vary in time. (That is because the current density in the last equation was left out. There should be a $4\pi\vec j/c$ in the last equation, as well as in (3). That would give the full Maxwell equations.)

2. 1st Ed: p103, q44, 2nd Ed: p123, q44. Do it without using Stokes. Then redo it using Stokes.

3. 1st Ed: p104, q62, 2nd Ed: p124, q62. Do the surface integrals both directly and using the divergence theorem. Make sure to include the base of the cone. Note: in doing the surface integrals directly, you are required to write them down in Cartesian coordinates using the expression for $\vec{n} {\rm d}{S}$ given in class. After that, switch to polar coordinates to actually do the integration.

4. 1st Ed: p132, q50, 2nd Ed: p154, q50 MODIFIED. Given
   
   $$\vec v = \frac{(-y,x)}{x^2+y^2}$$
   
   
   1. Evaluate $\nabla\times\vec{v}$.
   2. Also evaluate, presumably using polar coordinates,
      
      $$\oint_{\rm I} \vec v \cdot {\rm d}\vec r \qquad \oint_{\rm II} \vec v \cdot {\rm d}\vec r$$
      
      
      where path I is the semi circle of radius $r$ going clockwise from $(r,0)$ to $(-r,0)$, and path II is the semi circle of radius $r$ going counter-clockwise from $(r,0)$ to $(-r,0)$.
   3. Explain why the integral over II minus the integral over I is the integral over the closed circle.
   4. Explain why Stokes implies that the closed contour integral should be the integral of the $z$-component of $\nabla\times\vec{v}$ over the inside of the circle.
   5. Then explain why you would then normally expect the contour integral to be zero. That means that the two integrals I and II should be equal, but they are not.
   6. Explain what the problem is.
   7. Do you expect integrals over closed circles of different radii to be equal? Why?
   8. Are they actually equal?
   Now assume that you allow singular functions to be OK, like Heaviside step functions and Dirac delta functions say. Then figure out in what part of the interior of the circle, $\int\!\int\nabla\times\vec{v}\cdot{\hat k} {\rm d}{x}{\rm d}{y}$ is not zero. So how would you describe $\nabla\times\vec{v}$ for this vector field in terms of singular functions?

5. 1st Ed: p133, q56, 2nd Ed: p155, q56.