If a ray of light takes `t_(1)` and `t_(2)` times in two media of absolute refractive indices `mu_(1)` and `mu_(2)` respectively to travel same distance, then

Options

A. `mu_(1)t_(1)=mu_(2)t_(2)`
B. `mu_(1)t_(2)=mu_(2)t_(1)`
C. `t_(1)sqrt(mu_(1))=t_(2)sqrt(mu_(2))`
D. `t_(1)sqrt(mu_(2))=t_(2)sqrt(mu_(1))`

Answer

To solve the problem, we need to establish the relationship between the refractive indices of two media and the times taken by light to travel the same distance in those media.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have two media with absolute refractive indices \( \mu_1 \) and \( \mu_2 \). A ray of light takes time \( t_1 \) to travel a distance \( d \) in the first medium and time \( t_2 \) to travel the same distance \( d \) in the second medium.

2. **Using the Formula for Refractive Index**:
The refractive index \( \mu \) of a medium is defined as:
\[ \mu = \frac{c}{v} \]
where \( c \) is the speed of light in vacuum and \( v \) is the speed of light in the medium.

3. **Calculating Speed in the First Medium**:
The speed of light in the first medium can be expressed as:
\[ v_1 = \frac{d}{t_1} \]
Substituting this into the refractive index formula gives:
\[ \mu_1 = \frac{c}{v_1} = \frac{c}{\frac{d}{t_1}} = \frac{c \cdot t_1}{d} \]
Rearranging this, we get:
\[ \frac{\mu_1}{t_1} = \frac{c}{d} \quad \text{(Equation 1)} \]

4. **Calculating Speed in the Second Medium**:
Similarly, for the second medium, the speed of light can be expressed as:
\[ v_2 = \frac{d}{t_2} \]
Thus, the refractive index for the second medium is:
\[ \mu_2 = \frac{c}{v_2} = \frac{c}{\frac{d}{t_2}} = \frac{c \cdot t_2}{d} \]
Rearranging this gives:
\[ \frac{\mu_2}{t_2} = \frac{c}{d} \quad \text{(Equation 2)} \]

5. **Equating the Two Equations**:
From Equations 1 and 2, we have:
\[ \frac{\mu_1}{t_1} = \frac{\mu_2}{t_2} \]
This can be rearranged to give:
\[ \mu_1 \cdot t_2 = \mu_2 \cdot t_1 \]

6. **Final Result**:
Thus, the relationship between the refractive indices and the times taken is:
\[ \frac{\mu_1}{\mu_2} = \frac{t_1}{t_2} \]