The ratio of thickness of plates of two transparent medium A and B is 6 : 4. If light takes equal time in passing through them, then refractive index of A with respect to B will be

Options

A. 1.4
B. 1.5
C. 1.75
D. 1.33

Explanation

To find the refractive index of medium A with respect to medium B given the ratio of thicknesses and the condition that light takes equal time to pass through both mediums, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information**:
- The ratio of thicknesses of the two mediums A and B is given as:
\[ \frac{d_A}{d_B} = \frac{6}{4} = \frac{3}{2} \]
- The time taken for light to pass through both mediums is equal:
\[ t_A = t_B \]

2. **Relate Speed, Distance, and Time**:
- The speed of light in a medium can be expressed as:
\[ v = \frac{d}{t} \]
- Therefore, for medium A:
\[ v_A = \frac{d_A}{t_A} \]
- And for medium B:
\[ v_B = \frac{d_B}{t_B} \]

3. **Since the times are equal**:
- We can write:
\[ v_A = \frac{d_A}{t} \quad \text{and} \quad v_B = \frac{d_B}{t} \]
- Here, \( t \) is the common time taken for both mediums.

4. **Express the Refractive Index**:
- The refractive index \( \mu \) of a medium is given by:
\[ \mu = \frac{c}{v} \]
- Therefore, the refractive indices of A and B can be expressed as:
\[ \mu_A = \frac{c}{v_A} \quad \text{and} \quad \mu_B = \frac{c}{v_B} \]

5. **Find the Ratio of Refractive Indices**:
- The ratio of refractive indices \( \frac{\mu_A}{\mu_B} \) can be expressed as:
\[ \frac{\mu_A}{\mu_B} = \frac{v_B}{v_A} \]

6. **Substituting the Speeds**:
- Substitute the expressions for \( v_A \) and \( v_B \):
\[ \frac{\mu_A}{\mu_B} = \frac{\frac{d_B}{t}}{\frac{d_A}{t}} = \frac{d_B}{d_A} \]

7. **Using the Thickness Ratio**:
- From the thickness ratio \( \frac{d_A}{d_B} = \frac{3}{2} \), we can find:
\[ \frac{d_B}{d_A} = \frac{2}{3} \]

8. **Final Calculation**:
- Thus, we have:
\[ \frac{\mu_A}{\mu_B} = \frac{d_B}{d_A} = \frac{2}{3} \]

9. **Conclusion**:
- Therefore, the refractive index of medium A with respect to medium B is:
\[ \mu_{A/B} = \frac{2}{3} \approx 0.666 \]