The Maxwell stress tensor $\sigma$ of the electromagnetic fields is a two-index Cartesian tensor with components

$$\sigma_{i j}=-\epsilon_{0}\left(E_{i} E_{j}-\frac{1}{2}|\mathbf{E}|^{2} \delta_{i j}\right)-\frac{1}{\mu_{0}}\left(B_{i} B_{j}-\frac{1}{2}|\mathbf{B}|^{2} \delta_{i j}\right)$$

where $i, j=1,2,3$, and $E_{i}$ and $B_{i}$ denote the Cartesian components of the electric and magnetic fields $\mathbf{E}(\mathbf{x}, t)$ and $\mathbf{B}(\mathbf{x}, t)$ respectively.

(i) Consider an electromagnetic field sourced by charge and current densities denoted by $\rho(\mathbf{x}, t)$ and $\mathbf{J}(\mathbf{x}, t)$ respectively. Using Maxwell's equations and the Lorentz force law, show that the components of $\sigma$ obey the equation

$$\sum_{j=1}^{3} \frac{\partial \sigma_{i j}}{\partial x_{j}}+\frac{\partial g_{i}}{\partial t}=-(\rho \mathbf{E}+\mathbf{J} \times \mathbf{B})_{i}$$

where $g_{i}$, for $i=1,2,3$, are the components of a vector field $\mathbf{g}(\mathbf{x}, t)$ which you should give explicitly in terms of $\mathbf{E}$ and $\mathbf{B}$. Explain the physical interpretation of this equation and of the quantities $\sigma$ and $\mathbf{g}$.

(ii) A localised source near the origin, $\mathbf{x}=0$, emits electromagnetic radiation. Far from the source, the resulting electric and magnetic fields can be approximated as

$$\mathbf{B}(\mathbf{x}, t) \simeq \mathbf{B}_{0}(\mathbf{x}) \sin (\omega t-\mathbf{k} \cdot \mathbf{x}), \quad \mathbf{E}(\mathbf{x}, t) \simeq \mathbf{E}_{0}(\mathbf{x}) \sin (\omega t-\mathbf{k} \cdot \mathbf{x})$$

where $\mathbf{B}_{0}(\mathbf{x})=\frac{\mu_{0} \omega^{2}}{4 \pi r c} \hat{\mathbf{x}} \times \mathbf{p}_{0}$ and $\mathbf{E}_{0}(\mathbf{x})=-c \hat{\mathbf{x}} \times \mathbf{B}_{0}(\mathbf{x})$ with $r=|\mathbf{x}|$ and $\hat{\mathbf{x}}=\mathbf{x} / r$. Here, $\mathbf{k}=(\omega / c) \hat{\mathbf{x}}$ and $\mathbf{p}_{0}$ is a constant vector.

Calculate the pressure exerted by these fields on a spherical shell of very large radius $R$ centred on the origin. [You may assume that $\mathbf{E}$ and $\mathbf{B}$ vanish for $r>R$ and that the shell material is absorbant, i.e. no reflected wave is generated.]