Prob. $5.$ A plane wave

$E_i=$hat$(y)100cos(t-x-1\mathrm{.}73z)$

is incident in air upon a plane soil surface at $z=0.$ The soil medium may be assumed to be lossless dielectric with a relative permittivity of $4.$

$($a$)$ Determine the incident angle,

$($b$)$ Obtain expressions for the total electric fields in air and in the soil medium,

$1$

$($c$)$ Determine the average power density carried by the wave traveling in the soil medium.

Prob. $6.$ A dielectric rod or fiber of a transparent material can be used to guide light or an electromagnetic wave under the conditions of total internal reflection. Determine the minimum relative permittivity, $lon=lo{n}_{0}lo{n}_r,$ of the guiding medium so that a wave incident on one end at any angle will be confined within the rod until it emerges from the other end. Assume that the permeability is the same as the air. Namely, $={}_0.$

Prob. $7.$ For preventing interference of waves in neighboring fibers and for mechanical protection, individual optical fibers are usually cladded by a material of a lower refractive index, as shown in the figure, where ${}_a{n}_0,n_1{n}_2{}_{a}sin{}_a{}_a{n}_1=2,n_2=1\mathrm{.}74n_0=1n_2.$

$(a)$ Express the maximum angle $of$ incidence ${}_{a}in$ terms $of{n}_0,n_1$ and $n_2$ for meridional rays incident $on$ the core's end face $tobe$ trapped inside the core $by$ total internal reflection. $(Meridional$ rays are those that pass through the fiber axis. The angle ${}_{a}is$ called the accepted angle, and $sin{}_a$ the numerical aperture $(N.A.)of$ the fiber.$)$

$(b)$ Find ${}_a$ and $N.A.if{n}_1=2,n_2=1\mathrm{.}74$ and $n_0=1.$