Question

Dielectric constant of a medium is 3 and its magnetic permeability \(\mu = 2\mu_0\). Find ratio of velocity of light in vacuum to velocity of light in medium :

• A. \(\sqrt{5}\)
• B. \(\sqrt{6}\)
• C. 2
• D. 3

Hint:

The refractive index of a material is not just determined by its dielectric properties but also its magnetic properties.
For non-magnetic materials, \(\mu_r \approx 1\), and the formula simplifies to the more common \(n = \sqrt{\epsilon_r}\). Always check if the material is magnetic.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are asked to find the ratio of the speed of light in a vacuum (\(c\)) to the speed of light in a given medium (\(v\)). This ratio is, by definition, the refractive index (\(n\)) of the medium.
Step 2: Key Formula or Approach:
The speed of light in a vacuum is given by \(c = \frac{1}{\sqrt{\epsilon_0 \mu_0}}\), where \(\epsilon_0\) is the permittivity and \(\mu_0\) is the permeability of free space.
The speed of light in a medium is given by \(v = \frac{1}{\sqrt{\epsilon \mu}}\), where \(\epsilon\) is the permittivity and \(\mu\) is the permeability of the medium.
The refractive index \(n\) is given by \(n = \frac{c}{v} = \sqrt{\frac{\epsilon \mu}{\epsilon_0 \mu_0}} = \sqrt{\epsilon_r \mu_r}\), where \(\epsilon_r\) is the relative permittivity (dielectric constant) and \(\mu_r\) is the relative permeability.
Step 3: Detailed Explanation:
We are given:
- Dielectric constant (relative permittivity), \(\epsilon_r = \frac{\epsilon}{\epsilon_0} = 3\).
- Magnetic permeability, \(\mu = 2\mu_0\). This means the relative permeability is \(\mu_r = \frac{\mu}{\mu_0} = 2\).
Now, we can calculate the refractive index \(n\), which is the desired ratio:
\[ n = \frac{c}{v} = \sqrt{\epsilon_r \mu_r} \] \[ n = \sqrt{3 \times 2} \] \[ n = \sqrt{6} \] Step 4: Final Answer:
The ratio of the velocity of light in vacuum to the velocity of light in the medium is \(\sqrt{6}\).