Question

6. Write the phasor expressions for the incident waves described in problems 4 and 5 if the wave has a frequency of 100 MHz.

A perpendicularly polarized wave has an angle of incidence of 30 degrees. The dielectric boundary between air and a medium with relative dielectric permittivity of 4 is at z 4. 0. a. Find the angle of the refracted wave b. Find the reflection coefficient If the wave in problem 4 had parallel polarization, what would be the Brewster angle? Write the phasor expressions for the incident waves described in problems 4 and 5 if the wave has a frequency of 100 MHz. 5. 6.

Answer 1

4.

For a perpendicularly polarized wave electric field is perpendicular to the plane. here xz plane is the plane of incidence.

a. To calculate the refracted angle we will use here snell's laaw which states that

$n_{1}sin\theta _{i} = n_{2}sin\theta _{t}$

$sin \theta _{t} = \frac{n_{1}sin\theta _{i}}{ n_{2}}$

here incident angle is 30 degree

$sin \theta _{t} = \frac{\sqrt{\epsilon _{1}}sin\theta _{i}}{\sqrt{\epsilon _{2}}}$

$sin \theta _{t} = \frac{1\times sin\theta _{i}}{\sqrt{4}}$

$sin \theta _{t} = \frac{1\times sin 30}{2}$

$sin \theta _{t} = \frac{1}{4}$

$\theta _{t} =14.47^{\circ}$

b. Since wave is perpendicularly polarized hence incident angle will be equal to reflected angle

$\theta _{r} =30^{\circ}$

5. To calculate brewster angle we follow below formula

$tan \theta _{B} =\sqrt{ \frac{ \epsilon _{2}}{ \epsilon _{1}}}$

$tan \theta _{B} = \sqrt{ \frac{4}{ 1}}$

$tan \theta _{B} = 2$

$\theta _{B} = 63.43 ^{\circ}$

6. To write the phasor expression of a wave we need propagation constant of the wave which can be calculated from below

formula

$k = \omega \sqrt{\mu_{o} \epsilon_{o} }$

here frequency is given as 100MHz

hence

$k =2 \pi f \sqrt{\mu_{o} \epsilon_{o} }$

$k =2 \pi \times 100\times 10^{6} \frac{1}{c}$

where c is velocity of wave

$k =2 \pi \times 100\times 10^{6} \times \frac{1}{3 \times 10^{9}}$

here k is the magnitude of the propagation constant

Electric field for incident wave can be represented in phasor form as given below

$\vec{E} = E_{o} e^{jwt} e^{j\vec{k}.r}$

where r is the direction in which wave is travelling

$\vec{E} = E_{o} e^{j 2 \pi \times 10^{8} t} e^{j \vec{k} \cdot r}$

similarly for magnetic field

$\vec{H} = \frac{1}{ \mu \epsilon } \vec{k} \times E$