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 determine the equivalent distance that light would travel in a vacuum in the same time it takes to travel 5 meters in a medium with a refractive index \( \mu \).

### Step-by-Step Solution:

1. **Understand the relationship between speed, distance, and time**:
The formula for distance is given by:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]

2. **Identify the speed of light in the medium**:
The speed of light in a medium with refractive index \( \mu \) is given by:
\[ v = \frac{c}{\mu} \]
where \( c \) is the speed of light in vacuum (approximately \( 3 \times 10^8 \, \text{m/s} \)).

3. **Calculate the time taken to travel 5 meters in the medium**:
Using the distance formula:
\[ d = v \times t \implies t = \frac{d}{v} \]
Substituting the values:
\[ t = \frac{5 \, \text{m}}{\frac{c}{\mu}} = \frac{5 \mu}{c} \]

4. **Determine the distance light would travel in vacuum in the same time**:
In vacuum, the speed of light is \( c \). Therefore, the distance \( d_0 \) traveled in vacuum in the same time \( t \) is:
\[ d_0 = c \times t = c \times \frac{5 \mu}{c} = 5 \mu \]

5. **Conclusion**:
The equivalent distance that light would travel in vacuum in the same time it takes to travel 5 meters in the medium is:
\[ d_0 = 5 \mu \, \text{meters} \]

### Final Answer:
The equivalent path in vacuum is \( 5 \mu \) meters.

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