(a) A surface current $\mathbf{K}=K \mathbf{e}_{x}$, with $K$ a constant and $\mathbf{e}_{x}$ the unit vector in the $x$-direction, lies in the plane $z=0$. Use Ampère's law to determine the magnetic field above and below the plane. Confirm that the magnetic field is discontinuous across the surface, with the discontinuity given by

$$\lim _{z \rightarrow 0^{+}} \mathbf{e}_{z} \times \mathbf{B}-\lim _{z \rightarrow 0^{-}} \mathbf{e}_{z} \times \mathbf{B}=\mu_{0} \mathbf{K}$$

where $\mathbf{e}_{z}$ is the unit vector in the $z$-direction.

(b) A surface current $\mathbf{K}$ flows radially in the $z=0$ plane, resulting in a pile-up of charge $Q$ at the origin, with $d Q / d t=I$, where $I$ is a constant.

Write down the electric field $\mathbf{E}$ due to the charge at the origin, and hence the displacement current $\epsilon_{0} \partial \mathbf{E} / \partial t$.

Confirm that, away from the plane and for $\theta<\pi / 2$, the magnetic field due to the displacement current is given by

$$\mathbf{B}(r, \theta)=\frac{\mu_{0} I}{4 \pi r} \tan \left(\frac{\theta}{2}\right) \mathbf{e}_{\phi}$$

where $(r, \theta, \phi)$ are the usual spherical polar coordinates. [Hint: Use Stokes' theorem applied to a spherical cap that subtends an angle $\theta$.]