(i) Write down the two Maxwell equations that govern steady magnetic fields. Show that the boundary conditions satisfied by the magnetic field on either side of a current sheet, $\mathbf{J}$, with unit normal to the sheet $\mathbf{n}$, are

$$\mathbf{n} \wedge \mathbf{B}_{2}-\mathbf{n} \wedge \mathbf{B}_{1}=\mu_{0} \mathbf{J}$$

State without proof the force per unit area on $\mathbf{J}$.

(ii) Conducting gas occupies the infinite slab $0 \leqslant x \leqslant a$. It carries a steady current $\mathbf{j}=(0,0, j)$ and a magnetic field $\mathbf{B}=(0, B, 0)$ where $\mathbf{j}$, $\mathbf{B}$ depend only on $x$. The pressure is $p(x)$. The equation of hydrostatic equilibrium is $\nabla p=\mathbf{j} \wedge \mathbf{B}$. Write down the equations to be solved in this case. Show that $p+\left(1 / 2 \mu_{0}\right) B^{2}$ is independent of $x$. Using the suffixes 1,2 to denote values at $x=0, a$, respectively, verify that your results are in agreement with those of Part (i) in the case of $a \rightarrow 0$.

Suppose that

$$j(x)=\frac{\pi j_{0}}{2 a} \sin \left(\frac{\pi x}{a}\right), \quad B_{1}=0, \quad p_{2}=0$$

Find $B(x)$ everywhere in the slab.