3 01/29 F

1. 1st Ed: p80, q84, 2nd Ed: p93, q84.

2. 1st Ed: p80, q87, 2nd Ed: p93, q87. Include the divergence in the discussion.

3. 1st Ed: p80, q102, 2nd Ed: p94, q102.

4. 1st Ed: p81, q107, 2nd Ed: p94, q107. (20 points). You need to show that any solution of Maxwell’s equations is given by scalar and vector potentials as shown. Hints: Recall that if the divergence of a vector is zero, the vector is the curl of some other vector $\vec A$. Also, you can certaintly define $\vec E_\phi$ by setting
   
   $$\vec E = - \frac{1}{c} \frac{\partial \vec A}{\partial t} + \vec E_\phi$$
   
   
   but it is not automatic that $\vec E_\phi$ is the gradient of some scalar.

   Next, show that you can still choose the divergence of $\vec A$ whatever you want. To show this, assume that you have some $\vec A_0$ that has the right curl, but the wrong divergence. Then show that if you define
   

   $$\vec A = \vec A_0 + \nabla \psi$$
   
   
   where the function $\psi$ satisfies the Poisson equation
   
   $$\nabla^2 \psi = \mbox{correct-div} A - {\rm div} \vec A_0$$
   
   
   then $A$ has both the right curl and the right divergence.

   Use this freedom in the choice of the divergence of $A$ to derive the asked equations for $\vec A$ and $\phi$.

   The final two asked equations are wave equations, whose solutions consist of waves propagating with speed $c$, the speed of light. Thus Maxwell concluded that light is electromagnetic waves.

   For unknown reasons, the book has listed Maxwell’s equations in inverse order.

5. 1st Ed: p102, q32, 2nd Ed: p122, q32.

6. 1st Ed: p103, q44, 2nd Ed: p123, q44. Do it without using Stokes.