Question

A sinusoidal electromagnetic wave is given by \( \vec{E} = 20 \sin \left( \frac{2}{300} x - 10^6 t \right) \) propagating in a non-magnetic material. Dielectric constant of the material is.

• A. \( 9 \times 10^4 \)
• B. \( 3 \times 10^2 \)
• C. 4
• D. 2

Hint:

In non-magnetic materials, the dielectric constant can be found by comparing the wave number and angular frequency in the wave equation.

Answer 1

The Correct Option is C

The problem involves finding out the dielectric constant of a non-magnetic material through which a given sinusoidal electromagnetic wave propagates. The electric field of the wave is given by:

\(\vec{E} = 20 \sin \left( \frac{2}{300} x - 10^6 t \right)\). 

To solve this, we can use the formula for the wave's wave number, \(\boldsymbol{k}\), and angular frequency, \(\boldsymbol{\omega}\), which are essential for determining the speed of propagation of this wave in the medium:

• \(k = \frac{2}{300}\)
• \(\omega = 10^6\)

The speed of the wave, \(v\), is given by the relation:

\(v = \frac{\omega}{k}\)

Substituting the values of \(\omega\) and \(k\):

\(v = \frac{10^6}{\frac{2}{300}} = \frac{10^6 \times 300}{2} = 1.5 \times 10^8 \, \text{m/s}\)

In a vacuum, the speed of light \(c\) is \(3 \times 10^8 \, \text{m/s}\). The speed of light in a material is related to the dielectric constant \((\epsilon_r)\) by:

\(v = \frac{c}{\sqrt{\epsilon_r}} \implies \epsilon_r = \left(\frac{c}{v}\right)^2\)

Substituting the values of \(c\) and \(v\):

\(\epsilon_r = \left(\frac{3 \times 10^8}{1.5 \times 10^8}\right)^2 = (2)^2 = 4\)

Therefore, the dielectric constant of the material is 4. This matches the correct answer provided.