(a) From the differential form of Maxwell's equations with $\mathbf{J}=\mathbf{0}, \mathbf{B}=\mathbf{0}$ and a time-independent electric field, derive the integral form of Gauss's law.

(b) Derive an expression for the electric field $\mathbf{E}$ around an infinitely long line charge lying along the $z$-axis with charge per unit length $\mu$. Find the electrostatic potential $\phi$ up to an arbitrary constant.

(c) Now consider the line charge with an ideal earthed conductor filling the region $x>d$. State the boundary conditions satisfied by $\phi$ and $\mathbf{E}$ on the surface of the conductor.

(d) Show that the same boundary conditions at $x=d$ are satisfied if the conductor is replaced by a second line charge at $x=2 d, y=0$ with charge per unit length $-\mu$.

(e) Hence or otherwise, returning to the setup in (c), calculate the force per unit length acting on the line charge.

(f) What is the charge per unit area $\sigma(y, z)$ on the surface of the conductor?