Poynting theorem — UPSC IFoS 2010 Physics PYQ

The electromagnetic field inside a device is given by \vec{E} = \hat{y} E_0 \sin(k_x x) e^{-j k_z z}, \quad\text{where } k_x = \frac{n\pi}{a} \vec{H} = E_0 [\hat{x} \frac{-k_z}{\omega\mu} \sin(k_x x) + \hat{z} \frac{j k_x}{\omega\mu} \cos(k_x x)] e^{-j k_z z} Obtain an expression for the z-component of time-averaged Poynting vector \vec{S}.