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 ?
- $\frac{\in_{r_1}}{\in_{r_2}} = 4 $
- $\frac{\in_{r_1}}{\in_{r_2}} = 2$
- $\frac{\in_{r_1}}{\in_{r_2}} = \frac{1}{4} $
- $\frac{\in_{r_1}}{\in_{r_2}} = \frac{1}{2} $
The Correct Option is C
Solution and Explanation
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:
- \(\frac{\varepsilon_{r_1}}{\varepsilon_{r_2}} = \frac{1}{4}\)
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Concepts Used:
Electromagnetic waves
The waves that are produced when an electric field comes into contact with a magnetic field are known as Electromagnetic Waves or EM waves. The constitution of an oscillating magnetic field and electric fields gives rise to electromagnetic waves.
Types of Electromagnetic Waves:
Electromagnetic waves can be grouped according to the direction of disturbance in them and according to the range of their frequency. Recall that a wave transfers energy from one point to another point in space. That means there are two things going on: the disturbance that defines a wave, and the propagation of wave. In this context the waves are grouped into the following two categories:
- Longitudinal waves: A wave is called a longitudinal wave when the disturbances in the wave are parallel to the direction of propagation of the wave. For example, sound waves are longitudinal waves because the change of pressure occurs parallel to the direction of wave propagation.
- Transverse waves: A wave is called a transverse wave when the disturbances in the wave are perpendicular (at right angles) to the direction of propagation of the wave.