Electricity and Magnetism · Problem 9.7

ELECTRIC FIELDS IN MATTER — Edward Purcell & David Morin

Physics — E&M · em-waves-superposition

Problem

Consider the two oppositely traveling electric-field waves,

\mathbf{E}_1 = \hat{\mathbf{x}} E_0\cos(kz-\omega t) \quad \text{and} \quad \mathbf{E}_2 = \hat{\mathbf{x}} E_0\cos(kz+\omega t).\tag{9.60}

The sum of these two waves is the standing wave,

2\hat{\mathbf{x}}E_0\cos kz\cos\omega t

. (a) Find the magnetic field associated with this standing electric wave by finding the

\mathbf{B}

fields associated with each of the above traveling

\mathbf{E}

fields, and then adding them. (b) Find the magnetic field by instead using Maxwell's equations to find the

\mathbf{B}

field associated with the standing electric wave,

2\hat{\mathbf{x}}E_0\cos kz\cos\omega t