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)
Show Hint
- $\sqrt{6}:1$
- $6:1$
- $36:1$
- $3:2$
The Correct Option is A
Solution and Explanation
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: \(\sqrt{6}:1\).
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