The same colour of light takes `t_(1)` sec and `t_(2)` sec to travel the same distance ‘x’ in two media ‘A’ and ‘B’ respectively. Refractive index of medium ‘A’ w.r.t to ‘B’ is

Options

A. `(xT_(1))/t_(2)`
B. `t_(2)/(xt_(1))`
C. `t_(2)/t_(1)`
D. `t_(1)/t_(2)`

Explanation

To find the refractive index of medium A with respect to medium B, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**: We know that light takes `t1` seconds to travel a distance `x` in medium A and `t2` seconds to travel the same distance `x` in medium B. We need to find the refractive index of medium A with respect to medium B.

2. **Define the Speed of Light in Each Medium**:
- The speed of light in a vacuum is denoted as `c`.
- The speed of light in medium A, denoted as `vA`, is given by:
\[ vA = \frac{c}{\mu_A} \]
- The speed of light in medium B, denoted as `vB`, is given by:
\[ vB = \frac{c}{\mu_B} \]

3. **Using the Formula for Distance**:
- The distance traveled in each medium can be expressed using the formula:
\[ x = v \cdot t \]
- For medium A:
\[ x = vA \cdot t1 \implies x = \left(\frac{c}{\mu_A}\right) \cdot t1 \]
- For medium B:
\[ x = vB \cdot t2 \implies x = \left(\frac{c}{\mu_B}\right) \cdot t2 \]

4. **Setting the Distances Equal**:
- Since both expressions equal `x`, we can set them equal to each other:
\[ \frac{c}{\mu_A} \cdot t1 = \frac{c}{\mu_B} \cdot t2 \]

5. **Canceling `c` from Both Sides**:
- We can cancel `c` from both sides (assuming `c` is not zero):
\[ \frac{t1}{\mu_A} = \frac{t2}{\mu_B} \]

6. **Rearranging to Find the Refractive Index**:
- Rearranging the equation gives us:
\[ \frac{\mu_A}{\mu_B} = \frac{t1}{t2} \]

7. **Conclusion**:
- Therefore, the refractive index of medium A with respect to medium B is:
\[ \mu_A = \frac{t1}{t2} \cdot \mu_B \]

### Final Answer:
The refractive index of medium A with respect to medium B is:
\[ \mu_A = \frac{t1}{t2} \cdot \mu_B \]