The time averaged Poynting vector, in W/m², for a wave with $$\vec E = 24{e^{j\left( {\omega t + \beta z} \right)}}{\mkern 1mu} {\overrightarrow a _y}$$ V/m in free space is

A plane wave with $$\overrightarrow E = 10\,{e^{j\left( {\omega t - \beta z} \right)\,}}\,\,{\overrightarrow a _{_y}}$$ is incident normally on a thick plane conductor lying in the $$X - Y$$ plane. Its conductivity is $$6 \times {10^6}\,\,\,S/m\,\,\,$$ and surface impedance is $$5 \times {0^{ - 4}}\,\angle {45^0}\Omega $$. Determine the propagation constant and the skin depth in the conductor.

The electric field vector of a wave is given as $$$\vec E = {E_0}{\mkern 1mu} {e^{j\left( {\omega t + 3x - 4y} \right)}}{\mkern 1mu} {{8{{\vec a}_x} + 6{{\vec a}_y} + 5{{\vec a}_z}} \over {\sqrt {125} }}\,\,V/m$$$

Its frequency is 10 GHz.

(i) Investigate if this wave is a plane wave.
(ii) Determine its propagation constant, and
(iii) Calculate the phase velocity in $$y$$-direction.

An antenna in free space receives 2$$\mu $$ W of power when the incident electric field is 20 m V/m rms. The effective aperture of the antenna is