Show that the average value of radiant flux density S over a single period $T$ is given by $S=\frac{(E_{0})^{2}}{2c\mu_{0}}$

Radiant flux density is given as

$\vec{S}=\frac{1}{\mu_{0}}\left(\vec{E} \times \vec B\right)=c^{2} \epsilon_{0}\left({\vec{E}} \times \vec B\right)$

$\mathrm{E}=\mathrm{E}_{0} \cos (\mathrm{kx}-\omega \mathrm{t})$

$B=B_{0} \cos (k x-\omega t)$

$E_ B=c^{2} \varepsilon_{0}\left(E_{0} B_{0}\right) \cos ^{2}(k x-\omega t)$

So, the average value of the radiant flux density will be

$S_{av}=\frac{(E_{0})^{2}}{2c\mu_{0}}$