State the four integral relationships between the electric field $\mathbf{E}$ and the magnetic field $\mathbf{B}$ and explain their physical significance. Derive Maxwell's equations from these relationships and show that $\mathbf{E}$ and $\mathbf{B}$ can be described by a scalar potential $\phi$ and a vector potential A which satisfy the inhomogeneous wave equations

$$\begin{gathered} \nabla^{2} \phi-\epsilon_{0} \mu_{0} \frac{\partial^{2} \phi}{\partial t^{2}}=-\frac{\rho}{\epsilon_{0}} \\ \nabla^{2} \mathbf{A}-\epsilon_{0} \mu_{0} \frac{\partial^{2} \mathbf{A}}{\partial t^{2}}=-\mu_{0} \mathbf{j} \end{gathered}$$

If the current $\mathbf{j}$ satisfies Ohm's law and the charge density $\rho=0$, show that plane waves of the form

$$\mathbf{A}=A(z, t) e^{i \omega t} \hat{\mathbf{x}},$$

where $\hat{\mathbf{x}}$ is a unit vector in the $x$-direction of cartesian axes $(x, y, z)$, are damped. Find an approximate expression for $A(z, t)$ when $\omega \ll \sigma / \epsilon_{0}$, where $\sigma$ is the electrical conductivity.