The refractive index of glass is `1.9`. If light travels through a glass slab of thickness d in time t and takes the same time to travel through a transparent beaker filled with water upto a level of `1.5` d, then the refractive index of water is

Options

A. `1.27`
B. `1.33`
C. `1.20`
D. `1.50`

Explanation

To solve the problem, we need to find the refractive index of water given the refractive index of glass and the thickness of the glass slab and water beaker.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- The refractive index of glass (μ_g) is given as 1.9.
- Light travels through a glass slab of thickness \( d \) in time \( t \).
- It also travels through a beaker filled with water up to a level of \( 1.5d \) in the same time \( t \).
- We need to find the refractive index of water (μ_w).

2. **Using the Formula for Time**:
- The time taken to travel through a medium can be expressed as:
\[ t = \frac{\text{Distance}}{\text{Velocity}} \]
- The velocity of light in a medium is given by:
\[ v = \frac{c}{\mu} \]
- Where \( c \) is the speed of light in vacuum and \( \mu \) is the refractive index of the medium.

3. **Time Taken in Glass**:
- For the glass slab:
\[ t = \frac{d}{v_g} = \frac{d}{\frac{c}{\mu_g}} = \frac{d \cdot \mu_g}{c} \]
- Substituting \( \mu_g = 1.9 \):
\[ t = \frac{d \cdot 1.9}{c} \]

4. **Time Taken in Water**:
- For the water beaker:
\[ t = \frac{1.5d}{v_w} = \frac{1.5d}{\frac{c}{\mu_w}} = \frac{1.5d \cdot \mu_w}{c} \]

5. **Setting the Times Equal**:
- Since both times are equal:
\[ \frac{d \cdot 1.9}{c} = \frac{1.5d \cdot \mu_w}{c} \]
- We can cancel \( c \) and \( d \) (assuming \( d \neq 0 \)):
\[ 1.9 = 1.5 \cdot \mu_w \]

6. **Solving for the Refractive Index of Water**:
- Rearranging the equation gives:
\[ \mu_w = \frac{1.9}{1.5} \]
- Calculating the value:
\[ \mu_w = \frac{19}{15} \approx 1.27 \]

### Final Result:
The refractive index of water (μ_w) is approximately **1.27**.

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