Question

A plane electromagnetic wave propagating through a medium with $\epsilon_r = 8$ and $\mu_r = 2$ has $E = \frac{1}{2}e^{-\frac{z}{3}}\sin(10^8t-kz)\hat{i}$ V/m. [Consider $C = 3 \times 10^8$ m/s is wave velocity in free space]. Then-

• A. The value of $K = \frac{4}{3}$ rad/m
• B. The value of $K = \frac{3}{4}$ rad/m
• C. The wave velocity in the medium is $\frac{C}{4}$
• D. The wave velocity in the medium is $\frac{3C}{4}$

Answer 1

Answer: (A), (C)

A, C

Answer 2

The problem asks us to determine the wave number (K) and the wave velocity in a given medium for a plane electromagnetic wave.

1. Calculate the wave velocity (v) in the medium: The velocity of an electromagnetic wave in a medium is given by the formula: $v = \frac{c}{\sqrt{\mu_r \epsilon_r}}$ where $c$ is the speed of light in free space, $\mu_r$ is the relative permeability of the medium, and $\epsilon_r$ is the relative permittivity of the medium.

   Given: $c = 3 \times 10^8$ m/s $\epsilon_r = 8$ $\mu_r = 2$

   Substitute these values into the formula: $v = \frac{3 \times 10^8}{\sqrt{2 \times 8}} = \frac{3 \times 10^8}{\sqrt{16}} = \frac{3 \times 10^8}{4}$ Since $C$ is used to denote $3 \times 10^8$ m/s in the options: $v = \frac{C}{4}$ Therefore, option C is correct.

2. Calculate the wave number (K): From the given electric field equation, $E = \frac{1}{2}e^{-\frac{z}{3}}\sin(10^8t-kz)\hat{i}$ V/m, we can identify the angular frequency ($\omega$) and the wave number ($k$, which is denoted as K in the options). The angular frequency is $\omega = 10^8$ rad/s. The relationship between angular frequency, wave velocity, and wave number is: $K = \frac{\omega}{v}$

   Substitute the values of $\omega$ and the calculated $v$: $K = \frac{10^8}{\frac{3 \times 10^8}{4}} = \frac{10^8 \times 4}{3 \times 10^8} = \frac{4}{3} \text{ rad/m}$ Therefore, option A is correct.

Both options A and C are correct.