Chapter 16.5, Problem 38E

Maxwell’s equations relating the electric field E and magnetic field H as they vary with time in a region containing no charge and no current can be stated as follows:

$\begin{array}{ll}\text{div }E\text{ = 0}\hfill & \text{div }H\text{ = 0}\hfill \\ \text{curt }E\text{ =}-\frac{1}{c}\frac{\partial H}{\partial t}\hfill & \text{curl }H\text{ =}\frac{1}{c}\frac{\partial E}{\partial t}\hfill \end{array}$

where c is the speed of light. Use these equations to prove the following:

(a) $\nabla \times \text{(}\nabla \times E\text{)}=-\frac{1}{c^2}\frac{{\partial }^{2}E}{\partial t^2}$

(b) $\nabla \times \text{(}\nabla \times H\text{)}=-\frac{1}{c^2}\frac{{\partial }^{2}H}{\partial t^2}$

(c) ${\nabla }^{2}E=\frac{1}{c^2}\frac{{\partial }^{2}E}{\partial t^2}$ [Hint: Use Exercise 29.]

(d) ${\nabla }^{2}H=\frac{1}{c^2}\frac{{\partial }^{2}H}{\partial t^2}$