Show that the average value of radiant flux

Question:

Show that the average value of radiant flux density S over a single period T is given by S = 1/2cμ0 E02.

Solution:

Radiant flux density is given as

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

E = E0 cos (kx – ꞷt)

B = B0 cos (kx – ꞷt)

EB = c2 ε0 (E0B0) cos2 (kx – ꞷt)

Average value of the radiant flux density is

Sav = E02/2μ0c

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