The $$\mathop E\limits^ \to $$ field in a rectangular waveguide of inner dimensions $$a\,\, \times \,\,b$$ is given by $$\mathop E\limits^ \to = {{\omega \,\mu } \over {{h^2}}}\,\left( {{\pi \over a}} \right)\,{H_0}\,\sin \,\left( {{{2\,\pi \,x} \over a}} \right)\,\,\sin \,(\omega \,t - \,\beta \,z)\hat y$$,

where $${H_0}$$ is a constant, a and b are the dimensions along the x-axis and the y-axis respectively. The mode of propagation in the waveguide is

An air-filled rectangular waveguide has inner dimensions of $$3\,cm\,\, \times \,\,2\,\,cm\,$$. The wave impedance of the $$T{E_{20}}$$ mode of propagation in the waveguide at a frequency of 30 GHz is (free space impedance $$\,{\eta _0} = \,377\,\,\Omega $$)

The $$\overrightarrow H $$ field (in A/m) of a plane wave propagating in free space is given by $$$\overrightarrow H = \widehat x{{5\sqrt 3 } \over {{\eta _0}}}\cos \left( {\omega \,t - \beta \,z} \right) + \widehat y{5 \over {{\eta _0}}}\sin \left( {\omega \,t - \beta \,z + {\pi \over 2}} \right)$$$

The time average power flow density in Watts is

If C is a closed curve enclosing a surface S, then the magnetic field intensity $$\overrightarrow H $$, the current density $$\overrightarrow J $$ and the electric flux density $$\overrightarrow D $$ are related by