Question

An $EM$ wave from air enters a medium. The electric fields are $\vec{E_1} = E_{01} \hat{x} \cos \left[ 2 \pi v\left( \frac{z}{c} - t \right) \right]$ in air and $\vec{E}_2 = E_{02} \hat{x} \cos [k (2 z - ct)]$ in medium, where the wave number $k$ and frequency $?$ refer to their values in air. The medium is non-magnetic . If $\in_{r_1} $ and $\in_{r_2} $ refer to relative permittivities of air and medium respectively, which of the following options is correct ?

• A. $\frac{\in_{r_1}}{\in_{r_2}} = 4 $
• B. $\frac{\in_{r_1}}{\in_{r_2}} = 2$
• C. $\frac{\in_{r_1}}{\in_{r_2}} = \frac{1}{4} $
• D. $\frac{\in_{r_1}}{\in_{r_2}} = \frac{1}{2} $

Answer 1

The Correct Option is C

To solve this problem, we need to determine the relationship between the relative permittivities of air (\(\varepsilon_{r_1}\)) and the medium (\(\varepsilon_{r_2}\)) given the electromagnetic (EM) wave characteristics.

An electromagnetic wave undergoes a change in speed when it enters a different medium. The speed of the EM wave in a medium is given by:

\(v = \frac{c}{\sqrt{\varepsilon_r}}\)

where \(c\) is the speed of light in a vacuum, and \(\varepsilon_r\) is the relative permittivity of the medium.

From the problem, we are given the electric field equations:

• In air: \(\vec{E_1} = E_{01} \hat{x} \cos \left[ 2 \pi v\left( \frac{z}{c} - t \right) \right]\), with wave number in air as \(\frac{2\pi}{\lambda_1} = \frac{2\pi v}{c}\).
• In the medium: \(\vec{E_2} = E_{02} \hat{x} \cos [k (2 z - ct)]\), with wave number as \(\frac{2\pi}{\lambda_2} = k\).

By comparing the wave numbers in both media, we have:

• In air: \(\lambda_1 = \frac{c}{v} = \frac{c}{f}\).
• In the medium: \(\lambda_2 = \frac{\lambda_1}{\sqrt{\varepsilon_{r_2}}}\) (as speed decreases).

Given that the wave number is maintained, \(\frac{2\pi}{\lambda_2} = 2k\), we have:

\(\lambda_2 = \frac{\lambda_1}{2}\)

Thus, by using the formula for the change in wavelength, we get:

\(\frac{\lambda_1}{\lambda_2} = \sqrt{\frac{\varepsilon_{r_2}}{\varepsilon_{r_1}}}\)

Substitute the given condition of \(\lambda_2 = \frac{\lambda_1}{2}\) into the formula:

\(\frac{\varepsilon_{r_2}}{\varepsilon_{r_1}} = \left(2\right)^2\)

This indicates:

\(\frac{\varepsilon_{r_1}}{\varepsilon_{r_2}} = \frac{1}{4}\)

Hence, the correct option is:

• \(\frac{\varepsilon_{r_1}}{\varepsilon_{r_2}} = \frac{1}{4}\)