A transparent cube of `15 cm` edge contains a small air bubble. Its apparent depth when viewed through one face is `6 cm` and when viewed through the opposite face is `4 cm`. Then the refractive index of the material of the cube is

Options

A. `2.0`
B. `1.5`
C. `1.6`
D. `2.5`

Answer

To solve the problem of finding the refractive index of the material of the cube, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**:
We have a transparent cube with an edge length of 15 cm. An air bubble inside the cube has an apparent depth of 6 cm when viewed from one face and 4 cm when viewed from the opposite face.

2. **Define Variables**:
- Let \( x \) be the actual depth of the bubble from the face where the apparent depth is 6 cm.
- The remaining depth from the opposite face will then be \( 15 - x \).

3. **Use the Formula for Apparent Depth**:
The relationship between the actual depth (\( d \)), apparent depth (\( d' \)), and the refractive index (\( \mu \)) is given by:
\[ d' = \frac{d}{\mu} \]
From this, we can express the refractive index as:
\[ \mu = \frac{d}{d'} \]

4. **Set Up Equations**:
- For the face where the apparent depth is 6 cm:
\[ \mu = \frac{x}{6} \quad \text{(Equation 1)} \]
- For the opposite face where the apparent depth is 4 cm:
\[ \mu = \frac{15 - x}{4} \quad \text{(Equation 2)} \]

5. **Equate the Two Expressions for Refractive Index**:
Since both expressions equal \( \mu \), we can set them equal to each other:
\[ \frac{x}{6} = \frac{15 - x}{4} \]

6. **Cross-Multiply to Solve for \( x \)**:
\[ 4x = 6(15 - x) \]
Expanding the right side:
\[ 4x = 90 - 6x \]
Rearranging gives:
\[ 4x + 6x = 90 \]
\[ 10x = 90 \]
\[ x = 9 \text{ cm} \]

7. **Calculate the Refractive Index**:
Substitute \( x \) back into either Equation 1 or Equation 2 to find \( \mu \). Using Equation 1:
\[ \mu = \frac{9}{6} = 1.5 \]

### Final Answer:
The refractive index of the material of the cube is \( \mu = 1.5 \).