= Solution
For a <plane wave> propagating along $\hat{\mathbf k}$ in <vacuum>, <Maxwell equations> give $\mathbf B=\hat{\mathbf k}\times\mathbf E/c$, with $\mathbf E\perp\hat{\mathbf k}$. Hence
$$
\mathbf S=\frac{\mathbf E\times\mathbf B}{\mu_0}
=\frac{E^2}{\mu_0c}\hat{\mathbf k}
=\epsilon_0cE^2\hat{\mathbf k}.
$$
Taking $E=E_0\cos(\mathbf k\cdot\mathbf r-\omega t)$, the period average of the squared <cosine> is $1/2$. Thus
$$
\boxed{\left\langle|\mathbf S|\right\rangle=\frac12\epsilon_0cE_0^2.}
$$
The fields are in phase, so the travelling-wave <energy flux> never reverses. The peak <amplitude> convention is essential; using a root-mean-square <electric field> would remove the factor $1/2$.
Back to article page
