(i) Show that the Lorentz force corresponds to a curvature force and the gradient of a magnetic pressure, and that it can be written as the divergence of a second rank tensor, the Maxwell stress tensor.

Consider the potential field $\mathbf{B}$ given by $\mathbf{B}=-\nabla \Phi$, where

$$\Phi(x, y)=\left(\frac{B_{0}}{k}\right) \cos k x e^{-k y}$$

referred to cartesian coordinates $(x, y, z)$. Obtain the Maxwell stress tensor and verify that its divergence vanishes.

(ii) The magnetic field in a stellar atmosphere is maintained by steady currents and the Lorentz force vanishes. Show that there is a scalar field $\alpha$ such that $\nabla \wedge \mathbf{B}=\alpha \mathbf{B}$ and $\mathbf{B} \cdot \nabla \alpha=0$. Show further that if $\alpha$ is constant, then $\nabla^{2} \mathbf{B}+\alpha^{2} \mathbf{B}=0$. Obtain a solution in the form $\mathbf{B}=\left(B_{1}(z), B_{2}(z), 0\right)$; describe the structure of this field and sketch its variation in the $z$-direction.