Electricity and Magnetism · Problem 9.23

ELECTRIC FIELDS IN MATTER — Edward Purcell & David Morin

Physics — E&M · em-waves-in-box

Problem

Show that the electromagnetic field described by

\mathbf{E} = E_0\hat{\mathbf{z}}\cos kx\cos ky\cos\omega t,\quad \mathbf{B} = B_0(\hat{\mathbf{x}}\cos kx\sin ky - \hat{\mathbf{y}}\sin kx\cos ky)\sin\omega t\tag{9.63}

will satisfy the empty-space Maxwell equations in Eq. (9.18) if

E_0 = \sqrt{2}\,cB_0

and

\omega = \sqrt{2}\,ck

. This field can exist inside a square metal box enclosing the region

-\pi/2k < x < \pi/2k

and

-\pi/2k < y < \pi/2k

, with arbitrary height in the

z

direction. Roughly what do the electric and magnetic fields look like?