The optical path of a monochromatic light is same if it goes through 4.0 cm of glass or 4.5 cm of water. If the refractive index of glass is 1.53, the refractive index of the water is

Options

A. 1.3
B. 1.36
C. 1.42
D. 1.46

Explanation

To solve the problem, we need to find the refractive index of water given that the optical path length is the same for light traveling through two different media: glass and water.

### Step-by-Step Solution:

1. **Understand the Concept of Optical Path Length**:
The optical path length (OPL) is defined as the product of the refractive index (μ) of the medium and the physical length (x) of the path traveled in that medium. Mathematically, it is given by:
\[ \text{OPL} = \mu \cdot x \]

2. **Set Up the Equation**:
According to the problem, the optical path length through glass is equal to the optical path length through water:
\[ \mu_g \cdot x_g = \mu_w \cdot x_w \]
where:
- \( \mu_g = 1.53 \) (refractive index of glass)
- \( x_g = 4.0 \, \text{cm} \) (length of glass)
- \( x_w = 4.5 \, \text{cm} \) (length of water)
- \( \mu_w \) is the refractive index of water, which we need to find.

3. **Substitute Known Values**:
Plugging in the known values into the equation:
\[ 1.53 \cdot 4.0 = \mu_w \cdot 4.5 \]

4. **Calculate the Left Side**:
Calculate the left side of the equation:
\[ 1.53 \cdot 4.0 = 6.12 \]

5. **Rearrange to Solve for \( \mu_w \)**:
Now, rearranging the equation to solve for \( \mu_w \):
\[ \mu_w = \frac{6.12}{4.5} \]

6. **Perform the Division**:
Calculate \( \mu_w \):
\[ \mu_w = 1.36 \]

7. **Conclusion**:
Thus, the refractive index of water is:
\[ \mu_w = 1.36 \]

### Final Answer:
The refractive index of water is **1.36**.

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