If light travels a distance of `5` m in a medium of refractive index `mu`, then equivalent path in vaccum it would travel in same time is

Options

A. `(5/mu)` m
B. `(5 mu)` m
C. `(25 mu)` m
D. `(25/mu)` m

Answer

To solve the problem, we need to find the equivalent distance that light would travel in a vacuum in the same time it takes to travel 5 m in a medium with a refractive index \( \mu \).

### Step-by-Step Solution:

1. **Understand the Relationship of Refractive Index**:
The refractive index \( \mu \) of a medium is defined as the ratio of the speed of light in vacuum \( c \) (or \( V_1 \)) to the speed of light in the medium \( V_2 \):
\[ \mu = \frac{V_1}{V_2} \]

2. **Express Time in Terms of Distance and Velocity**:
The time taken to travel a distance in any medium can be expressed as:
\[ t = \frac{d}{v} \]
where \( d \) is the distance and \( v \) is the speed of light in that medium.

3. **Set Up the Equation for Both Mediums**:
For the medium:
- Distance traveled \( D_2 = 5 \, \text{m} \)
- Speed in the medium \( V_2 \)
- Time taken \( t_2 = \frac{D_2}{V_2} = \frac{5}{V_2} \)

For vacuum:
- Distance traveled \( D_1 \)
- Speed in vacuum \( V_1 \)
- Time taken \( t_1 = \frac{D_1}{V_1} \)

Since the time taken in both cases is the same:
\[ t_1 = t_2 \]
Therefore:
\[ \frac{D_1}{V_1} = \frac{5}{V_2} \]

4. **Substitute the Refractive Index**:
From the refractive index definition, we know:
\[ V_2 = \frac{V_1}{\mu} \]
Substitute \( V_2 \) in the time equation:
\[ \frac{D_1}{V_1} = \frac{5}{\frac{V_1}{\mu}} \]

5. **Simplify the Equation**:
Cross-multiplying gives:
\[ D_1 \cdot \frac{V_1}{\mu} = 5 \cdot V_1 \]
Dividing both sides by \( V_1 \) (assuming \( V_1 \neq 0 \)):
\[ \frac{D_1}{\mu} = 5 \]
Thus:
\[ D_1 = 5 \mu \]

6. **Final Answer**:
The equivalent distance in vacuum that light would travel in the same time is:
\[ D_1 = 5 \mu \, \text{meters} \]