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Electromagnetic Waves. Radiation

Problem 4.200

A standing electromagnetic wave E=Emcos⁡kx⋅cos⁡ωt\mathbf{E}=\mathbf{E}_ {m} \cos k x \cdot \cos \omega t is sustained along the xx axis in vacuum. Find the projection of the Poynting vector on the xx axis Sx(x,t)S_ {x}(x, t) and the mean value of that projection averaged over an oscillation period.

Reveal Answer
Sx=1/4ε0cEm2sin⁡2kx⋅sin⁡2ωt,⟨Sx⟩=0S_{x}=1 / 4 \varepsilon_{0} c E_{m}^{2} \sin 2 k x \cdot \sin 2 \omega t, \quad\left\langle S_{x}\right\rangle=0
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