5.
We have that the instantaneous magnetic field \textbf{H} of a wave propagating in a
lossless linear, homogeneous, isotropic, (LHI) material that is charge free, non-magnetic ($\mu = \mu_0$)
and has a dielectric relative permittivity of $\epsilon_r$. The instantaneous representation of the
magnetic field intensity is given as,
$\textbf{H} = 75 \cos(\pi \times 10^7 t + 0.5 \pi y) \hat{a}_x$ $\mu A/m$
(a) Is this wave a uniform plane wave? If so, explain which plane, and why, or why not, is the
wave uniform. Which direction and with what velocity is this plane wave travelling in?
(b) Calculate the relative permittivity $\epsilon_r$, and the complex permittivity $\epsilon_c$ of this medium that
this wave is propagating in.
(c) Calculate the phasor magnetic field intensity $\tilde{\textbf{H}}$ based on the phasor transform.
(d) State whose law that you will use, write down the expression, and then calculate the phasor
electric field intensity $\tilde{\textbf{E}}$ of this wave. Show the proper units.
(e) For this to be a TEM plane wave what electric and magnetic field components
($E_x$, $E_y$, $E_z$, $H_x$, $H_y$, and $H_z$) must be equal to zero? For this to be a uniform plane wave,
what derivatives ($\partial/\partial x$, $\partial/\partial y$, $\partial/\partial z$) of the remaining field components must be equal
to zero? Calculate the instantaneous time electric field intensity \textbf{E} of this wave and state
what the amplitude and phase relationship between the electric field \textbf{E} and magnetic field
\textbf{H} is.
