Paper 2, Section I, $\mathbf{6 C}$

Write down Maxwell's equations for electromagnetic fields in a non-polarisable and non-magnetisable medium.

Show that the homogenous equations (those not involving charge or current densities) can be solved in terms of vector and scalar potentials $\mathbf{A}$ and $\phi$.

Then re-express the inhomogeneous equations in terms of $\mathbf{A}, \phi$ and $f=\nabla \cdot \mathbf{A}+c^{-2} \dot{\phi}$. Show that the potentials can be chosen so as to set $f=0$ and hence rewrite the inhomogeneous equations as wave equations for the potentials. [You may assume that the inhomogeneous wave equation $\nabla^{2} \psi-c^{-2} \ddot{\psi}=\sigma(\mathbf{x}, t)$ always has a solution $\psi(\mathbf{x}, t)$ for any given $\sigma(\mathbf{x}, t)$.]

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