If `epsilon_0` and `mu_0` are respectively the electric permittivity and the magnetic permeability of free space and `epsilon` and `mu` the corresponding quantities in a medium, the refractive index of the medium is

Options

A. (a) `sqrt(((muepsilon)/(mu_0epsilon_0)))`
B. (b) `(muepsilon)/(mu_0epsilon_0)`
C. (c) `sqrt(((mu_0epsilon_0)/(muepsilon)))`
D. (d) `sqrt(((mumu_0)/(epsilonepsilon_0)))`

Explanation

To find the refractive index of a medium in terms of the electric permittivity and magnetic permeability of free space and the corresponding quantities in the medium, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Refractive Index**:
The refractive index \( n \) of a medium is defined as the ratio of the speed of light in vacuum \( c \) to the speed of light in the medium \( v \):
\[ n = \frac{c}{v} \]

2. **Speed of Light in Vacuum**:
The speed of light in vacuum \( c \) can be expressed in terms of the electric permittivity \( \epsilon_0 \) and magnetic permeability \( \mu_0 \) of free space:
\[ c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \]

3. **Speed of Light in a Medium**:
The speed of light in a medium can be expressed in terms of the electric permittivity \( \epsilon \) and magnetic permeability \( \mu \) of that medium:
\[ v = \frac{1}{\sqrt{\mu \epsilon}} \]

4. **Substituting into the Refractive Index Formula**:
Now, substituting the expressions for \( c \) and \( v \) into the refractive index formula:
\[ n = \frac{c}{v} = \frac{\frac{1}{\sqrt{\mu_0 \epsilon_0}}}{\frac{1}{\sqrt{\mu \epsilon}}} \]

5. **Simplifying the Expression**:
This simplifies to:
\[ n = \frac{\sqrt{\mu \epsilon}}{\sqrt{\mu_0 \epsilon_0}} = \sqrt{\frac{\mu \epsilon}{\mu_0 \epsilon_0}} \]

6. **Final Expression for the Refractive Index**:
Therefore, the refractive index \( n \) of the medium can be expressed as:
\[ n = \sqrt{\frac{\mu}{\mu_0} \cdot \frac{\epsilon}{\epsilon_0}} \]

### Final Answer:
The refractive index of the medium is given by:
\[ n = \sqrt{\frac{\mu}{\mu_0} \cdot \frac{\epsilon}{\epsilon_0}} \]