Introduction to Electrodynamics · Problem 11.28

Electrodynamics and Relativity — David J. Griffiths

Physics · radiation-from-surface-current

Problem

Suppose the (electrically neutral)

yz

plane carries a time-dependent but uniform surface current

K(t)\,\hat{\mathbf{z}}

. (a) Find the electric and magnetic fields at a height

x

above the plane if (i) a constant current is turned on at

t = 0

K(t) = \begin{cases} 0, & t \leq 0, \\ K_0, & t > 0. \end{cases}

(ii) a linearly increasing current is turned on at

t = 0

K(t) = \begin{cases} 0, & t \leq 0, \\ \alpha t, & t > 0. \end{cases}

(b) Show that the retarded vector potential can be written in the form

\mathbf{A}(x,t) = \frac{\mu_0 c}{2}\hat{\mathbf{z}} \int_0^\infty K\left(t - \frac{x}{c} - u\right)du,

and from this determine

\mathbf{E}

and

\mathbf{B}

. (c) Show that the total power radiated per unit area of surface is

\frac{\mu_0 c}{2}[K(t)]^2.

Explain what you mean by ``radiation,'' in this case, given that the source is not localized.