Question

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 the relation between the times and refractive indices is:

• 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. \(\sqrt{\mu_1} t_1 = \sqrt{\mu_2} t_2\)

Hint:

For light traveling the same distance in different media, the product of the refractive index and the time taken is a constant, illustrating the inverse relationship between speed and refractive index.

Answer 1

The Correct Option is B

Step 1: Relate the times to the velocities in the media. \[ v = \frac{c}{\mu} \] \[ t = \frac{d}{v} = \frac{d \mu}{c} \] where \(d\) is the distance, \(c\) is the speed of light, and \(v\) is the velocity of light in the medium. 
Step 2: Derive the relationship using the indices and times. Since \( v_1 = \frac{c}{\mu_1} \) and \( v_2 = \frac{c}{\mu_2} \), \[ t_1 = \frac{d}{v_1} = \frac{d \mu_1}{c} \] \[ t_2 = \frac{d}{v_2} = \frac{d \mu_2}{c} \] Equating and rearranging gives: \[ \mu_1 t_2 = \mu_2 t_1 \]

Answer 2

To solve the problem, we need to derive the correct relation between the times taken by a ray of light to travel the same distance in two media with different refractive indices.

1. Understanding the Problem:
The time taken for light to travel a certain distance in a medium is related to the speed of light in that medium. The speed of light \(v\) in any medium is given by:

\[ v = \frac{c}{\mu} \] where: - \(c\) is the speed of light in vacuum, - \(\mu\) is the refractive index of the medium. Since the light travels the same distance in both media, the time taken in each medium is inversely proportional to the speed of light in that medium. Hence, we have: \[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{\text{Distance}}{c / \mu} = \mu \cdot \frac{\text{Distance}}{c} \] Thus, the time taken in medium 1 is: \[ t_1 = \mu_1 \cdot \frac{\text{Distance}}{c} \] And the time taken in medium 2 is: \[ t_2 = \mu_2 \cdot \frac{\text{Distance}}{c} \] Since the distance is the same in both media, the ratio of times is: \[ \frac{t_1}{t_2} = \frac{\mu_1}{\mu_2} \]

2. Conclusion:
The correct relation between the times \(t_1\) and \(t_2\) is: \( \mu_1 t_2 = \mu_2 t_1 \)

Final Answer:
The correct answer is (B) \( \mu_1 t_2 = \mu_2 t_1 \).