Using the Maxwell equations

$$\begin{gathered} \nabla \cdot \mathbf{E}=\frac{\rho}{\epsilon_{0}}, \quad \nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t} \\ \nabla \cdot \mathbf{B}=0, \quad \boldsymbol{\nabla} \times \mathbf{B}-\epsilon_{0} \mu_{0} \frac{\partial \mathbf{E}}{\partial t}=\mu_{0} \mathbf{j} \end{gathered}$$

show that in vacuum, E satisfies the wave equation

$$\frac{1}{c^{2}} \frac{\partial^{2} \mathbf{E}}{\partial t^{2}}-\nabla^{2} \mathbf{E}=0$$

where $c^{2}=\left(\epsilon_{0} \mu_{0}\right)^{-1}$, as well as $\nabla \cdot \mathbf{E}=0$. Also show that at a planar boundary between two media, $\mathbf{E}_{t}$ (the tangential component of $\mathbf{E}$ ) is continuous. Deduce that if one medium is of negligible resistance, $\mathbf{E}_{t}=0$.

Consider an empty cubic box with walls of negligible resistance on the planes $x=0$, $x=a, y=0, y=a, z=0, z=a$, where $a>0$. Show that an electric field in the interior of the form

$$\begin{aligned} &E_{x}=f(x) \sin \left(\frac{m \pi y}{a}\right) \sin \left(\frac{n \pi z}{a}\right) e^{-i \omega t} \\ &E_{y}=g(y) \sin \left(\frac{l \pi x}{a}\right) \sin \left(\frac{n \pi z}{a}\right) e^{-i \omega t} \\ &E_{z}=h(z) \sin \left(\frac{l \pi x}{a}\right) \sin \left(\frac{m \pi y}{a}\right) e^{-i \omega t} \end{aligned}$$

with $l, m$ and $n$ positive integers, satisfies the boundary conditions on all six walls. Now suppose that

$$f(x)=f_{0} \cos \left(\frac{l \pi x}{a}\right), \quad g(y)=g_{0} \cos \left(\frac{m \pi y}{a}\right), \quad h(z)=h_{0} \cos \left(\frac{n \pi z}{a}\right)$$

where $f_{0}, g_{0}$ and $h_{0}$ are constants. Show that the wave equation $(*)$ is satisfied, and determine the frequency $\omega$. Find the further constraint on $f_{0}, g_{0}$ and $h_{0}$ ?