Question

The ratio of speeds of electromagnetic waves in vacuum and a medium, having dielectric constant k = 3 and permeability of $\mu = 2\mu_0$, is ($\mu_0$ = permeability of vacuum)

• A. $\sqrt{6}:1$
• B. $6:1$
• C. $36:1$
• D. $3:2$

Hint:

Refractive index $n = \sqrt{\epsilon_r \mu_r}$.

Answer 1

The Correct Option is A

To solve this problem, we need to understand the relationship between the speed of electromagnetic waves in a vacuum and in a medium. The speed of electromagnetic waves is given by the formula:

\(v = \frac{1}{\sqrt{\epsilon \mu}}\) 

where \(\epsilon\) is the permittivity and \(\mu\) is the permeability of the medium.

The speed of light (an electromagnetic wave) in a vacuum is denoted by \(c\) and is given by:

\(c = \frac{1}{\sqrt{\epsilon_0 \mu_0}}\)

In a medium, the speed becomes:

\(v_m = \frac{1}{\sqrt{\epsilon \mu}}\)

Given:

• Dielectric constant (relative permittivity) \(k = \frac{\epsilon}{\epsilon_0} = 3\)
• Permeability \(\mu = 2\mu_0\)

 

Thus, \(\epsilon = 3\epsilon_0\) and \(\mu = 2\mu_0\).

The speed of electromagnetic waves in the medium is:

\(v_m = \frac{1}{\sqrt{3\epsilon_0 \cdot 2\mu_0}} = \frac{1}{\sqrt{6\epsilon_0 \mu_0}}\)

Now, to find the ratio of \(c\) to \(v_m\):

\(\text{Ratio} = \frac{c}{v_m} = \frac{1/\sqrt{\epsilon_0 \mu_0}}{1/\sqrt{6\epsilon_0 \mu_0}} = \sqrt{6}\)

So, the ratio of the speed of electromagnetic waves in a vacuum to that in the medium is \(\sqrt{6}:1\).

Thus, the correct answer is: \(\sqrt{6}:1\).