Introduction to Electrodynamics · Problem 9.12

Electromagnetic Waves — David J. Griffiths

Physics · energy-momentum-electromagnetic-waves

Problem

In the complex notation there is a clever device for finding the time average of a product. Suppose

f(\mathbf{r},t) = A\cos(\mathbf{k}\cdot\mathbf{r} - \omega t + \delta_a)

and

g(\mathbf{r},t) = B\cos(\mathbf{k}\cdot\mathbf{r} - \omega t + \delta_b)

. Show that

\langle fg\rangle = (1/2)\mathrm{Re}(\tilde{f}\tilde{g}^*)

, where the star denotes complex conjugation. [Note that this only works if the two waves have the same

\mathbf{k}

and

\omega

, but they need not have the same amplitude or phase.] For example,

\langle u\rangle = \frac{1}{4}\mathrm{Re}\left(\epsilon_0 \tilde{\mathbf{E}}\cdot\tilde{\mathbf{E}}^* + \frac{1}{\mu_0}\tilde{\mathbf{B}}\cdot\tilde{\mathbf{B}}^*\right) \quad \text{and} \quad \langle\mathbf{S}\rangle = \frac{1}{2\mu_0}\mathrm{Re}\left(\tilde{\mathbf{E}}\times\tilde{\mathbf{B}}^*\right).