Use Maxwell's equations,

$$\boldsymbol{\nabla} \cdot \mathbf{E}=\rho, \quad \boldsymbol{\nabla} \cdot \mathbf{B}=0, \quad \boldsymbol{\nabla} \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}, \quad \boldsymbol{\nabla} \times \mathbf{B}=\mathbf{J}+\frac{\partial \mathbf{E}}{\partial t}$$

to derive expressions for $\frac{\partial^{2} \mathbf{E}}{\partial t^{2}}-\nabla^{2} \mathbf{E}$ and $\frac{\partial^{2} \mathbf{B}}{\partial t^{2}}-\nabla^{2} \mathbf{B}$ in terms of $\rho$ and $\mathbf{J}$.

Now suppose that there exists a scalar potential $\phi$ such that $\mathbf{E}=-\nabla \phi$, and $\phi \rightarrow 0$ as $r \rightarrow \infty$. If $\rho=\rho(r)$ is spherically symmetric, calculate $\mathbf{E}$ using Gauss's flux method, i.e. by integrating a suitable equation inside a sphere centred at the origin. Use your result to find $\mathbf{E}$ and $\phi$ in the case when $\rho=1$ for $r<a$ and $\rho=0$ otherwise.

For each integer $n \geqslant 0$, let $S_{n}$ be the sphere of radius $4^{-n}$ centred at the point $\left(1-4^{-n}, 0,0\right)$. Suppose that $\rho$ vanishes outside $S_{0}$, and has the constant value $2^{n}$ in the volume between $S_{n}$ and $S_{n+1}$ for $n \geqslant 0$. Calculate $\mathbf{E}$ and $\phi$ at the point $(1,0,0)$.