Assume that a plane wave in air with an electric field
$$\overrightarrow E = 10\cos \left( {\omega t - 3x - \sqrt {3z} } \right){\widehat a_{_y}}\,\,\,V/m$$ is incident
on a non-magnetic dielectric slab of relative permittivity $$3$$ which covers the region $$z > 0$$ . The angle of transmission in the dielectric slab is _______ degrees.

A region shown below contains a perfect conducting half-space and air. The surface current $${\overrightarrow K _{_s}}$$ on the surface of the perfect conductor is $${\overrightarrow K _{_s}} = \widehat x\,2$$ amperes per meter. The tangential $$\overrightarrow H $$ field in the air just above the perfect conductor is

If the electric field of a plane wave is $$$\overrightarrow E \left( {z,t} \right) = \widehat x3\cos \left( {\omega t - kz + {{30}^ \circ }} \right) - \widehat y4\sin \left( {\omega t - kz + {{45}^ \circ }} \right)\left( {mV/m} \right)$$$

The polarization state of the plane wave is

In spherical coordinates, let $${{{\widehat a}_{_0}},\,{{\widehat a}_{_\phi }}}$$ denote unit vectors along the $$\theta ,\,\,\phi $$directions. $$$E = {{100} \over r}\sin \,\theta \cos \left( {\omega t - \beta r} \right){\widehat a_{_\theta }}\,\,V/m$$$
and $$$H = {{0.265} \over r}\sin \,\theta \cos \left( {\omega t - \beta r} \right){\widehat a_{_\phi }}\,\,A/m$$$

Represent the electric and magnetic field components of the EM wave at large distances $$r$$ from a dipole antenna, in free space. The average power $$(W)$$ crossing the hemispherical shell located at $$r = 1\,\,km$$, $$0 \le \theta \le \pi /2$$ _______.