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. `sqrt((epsilon_(0)mu_(0))/(epsilon mu))`
B. `sqrt((epsilon mu)/(epsilon_(0)mu_(0)))`
C. `sqrt((epsilon_(0)mu)/(epsilon mu_(0)))`
D. `sqrt((epsilon)/(epsilon_(0)))`

Answer

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

### Step-by-Step Solution:

1. **Understanding the Speed of Light**:
The speed of light in a vacuum (or free space) is given by the formula:
\[ c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \]
where \( \mu_0 \) is the magnetic permeability and \( \epsilon_0 \) is the electric permittivity of free space.

2. **Speed of Light in a Medium**:
The speed of light in a medium is given by:
\[ v = \frac{1}{\sqrt{\mu \epsilon}} \]
where \( \mu \) is the magnetic permeability and \( \epsilon \) is the electric permittivity of the medium.

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

4. **Substituting the Speeds**:
Substitute the expressions for \( c \) and \( v \) into the equation for \( n \):
\[ n = \frac{\frac{1}{\sqrt{\mu_0 \epsilon_0}}}{\frac{1}{\sqrt{\mu \epsilon}}} \]

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

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

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

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