Chapter 33, Problem 28P

A large, flat sheet carries a uniformly distributed electric current with current per unit width J_s. This current creates a magnetic field on both sides of the sheet, parallel to the sheet and perpendicular to the current, with magnitude $B=\frac{1}{2}{\mu }_0{J}_s$. If the current is in the y direction and oscillates in time according to

$J_{\max }(\cos \omega t)\hat{j}=J_{\max }[\cos (-\omega t)]\hat{j}$

the sheet radiates an electromagnetic wave. Figure P33.28 shows such a wave emitted from one point on the sheet chosen to be the origin. Such electromagnetic waves arc emitted from all points on the sheet. The magnetic field of the wave to the right of the sheet is described by the wave function

$\stackrel{\to }{B}=\frac{1}{2}{\mu }_0{J}_{\max }[\cos (kx-\omega t)]\hat{k}$

(a) Find the wave function for the electric field of the wave to the right of the sheet. (b) Find the Poynting vector as a function of x and t. (c) Find the intensity of the wave. (d) What If? If the sheet is to emit radiation in each direction (normal to the plane of the sheet) with intensity 570 W/m², what maximum value of sinusoidal current density is required?

Figure P33.28