If `mu_(0)` be the permeability and `epsilon_(0)` be the permittivity of a medium, then its refractive index is given by

Options

A. `(1)/(sqrt(mu_(0)epsilon_(0)))`
B. `(1)/(mu_(0)epsilon_(0))`
C. `sqrt(mu_(0)epsilon_(0))`
D. `mu_(0)epsilon_(0)`

Explanation

To find the refractive index of a medium in terms of its permeability (\( \mu_0 \)) and permittivity (\( \epsilon_0 \)), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definition of 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 permeability and permittivity as:
\[ c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \]

3. **Speed of Light in a Medium**:
The speed of light in a medium can be expressed as:
\[ v = \frac{1}{\sqrt{\mu \epsilon}} \]
where \( \mu \) and \( \epsilon \) are the permeability and permittivity of the medium, respectively.

4. **Relate Medium Properties to Vacuum Properties**:
The permeability and permittivity of the medium can be related to those of vacuum by:
\[ \mu = \mu_r \mu_0 \quad \text{and} \quad \epsilon = \epsilon_r \epsilon_0 \]
where \( \mu_r \) and \( \epsilon_r \) are the relative permeability and permittivity of the medium.

5. **Substituting into the Speed of Light in Medium**:
Substitute the expressions for \( \mu \) and \( \epsilon \) into the equation for \( v \):
\[ v = \frac{1}{\sqrt{\mu_r \mu_0 \epsilon_r \epsilon_0}} \]

6. **Finding the Refractive Index**:
Now, substitute \( c \) and \( v \) into the refractive index formula:
\[ n = \frac{c}{v} = \frac{\frac{1}{\sqrt{\mu_0 \epsilon_0}}}{\frac{1}{\sqrt{\mu_r \mu_0 \epsilon_r \epsilon_0}}} \]
Simplifying this gives:
\[ n = \sqrt{\frac{\mu_r \epsilon_r}{1}} = \sqrt{\mu_r \epsilon_r} \]

7. **Final Expression**:
If we consider the case where \( \mu_r = 1 \) and \( \epsilon_r = 1 \) (which is often the case for vacuum), we can conclude:
\[ n = \sqrt{\frac{\mu_0}{\epsilon_0}} \]

### Final Result:
Thus, the refractive index of the medium in terms of its permeability and permittivity is:
\[ n = \sqrt{\mu_0 \epsilon_0} \]