The electric field intensity of a plane wave traveling in free space is given by the following expression

$$E\left( {x,t} \right) = {\widehat a_{_y}}24\pi \,\,\cos \left( {\omega t - {k_0}x} \right)\,\,\,\left( {V/m} \right)$$. In this field, consider a square area $$10 cm$$ $$ \times $$ $$10 cm$$ on a plane $$x + y = 1$$. The total time-averaged power $$(in mW)$$ passing through the square area is ________.

Consider a uniform plane wave with amplitude $$\left( {{E_0}} \right)$$ of $$10\,\,\,V/m$$ and $$1.1 GHz$$ frequency travelling in air, and incident normally on a dielectric medium with complex relative permittivity $$\left( {{\varepsilon _r}} \right)$$ and permeability $$\left( {{\mu _r}} \right)$$ as shown in the figure.

The magnitude of the transmitted electric field component (in V/m) after it has travelled a distance of $$10$$ cm inside the dielectric region 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 $$r = 1\,\,km$$, $$0 \le \theta \le \pi /2$$ _______.

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.