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 medium compared to their speed in a vacuum.

The speed of electromagnetic waves in a vacuum, denoted as \(c\), can be expressed as:

\(c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}\) 

where:

• \(\mu_0\) is the permeability of free space (vacuum).
• \(\varepsilon_0\) is the permittivity of free space (vacuum).

The speed of electromagnetic waves in a medium, \(v\), is given by:

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

where:

• \(\mu = 2\mu_0\) is the permeability of the medium.
• \(\varepsilon = k \varepsilon_0\); here, \(k\) is the dielectric constant of the medium, given as 3.

Substituting the given values into the expression for the speed in the medium:

\(v = \frac{1}{\sqrt{(2\mu_0)(3\varepsilon_0)}}\)

This can be simplified to:

\(v = \frac{1}{\sqrt{6\mu_0 \varepsilon_0}}\)

Now, the ratio of the speed of electromagnetic waves in a vacuum to that in the medium is:

\(\text{Ratio} = \frac{c}{v} = \frac{1/\sqrt{\mu_0 \varepsilon_0}}{1/\sqrt{6\mu_0 \varepsilon_0}} = \sqrt{6}\)

Therefore, the ratio of the speeds is \(\sqrt{6}:1\).

Thus, the correct answer is: $\sqrt{6}:1$