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. `2.5`
C. `1.6`
D. `1.5`

Answer

To find the refractive index of the material of the cube, we can follow these steps:

### Step 1: Understand the Problem
We have a transparent cube with a side length of 15 cm containing an air bubble. The apparent depths when viewed through one face are given as 6 cm and 4 cm when viewed through the opposite face.

### Step 2: Define Variables
Let:
- \( x \) = actual depth of the bubble from the first face
- \( 15 - x \) = actual depth of the bubble from the opposite face

### Step 3: Apply the Formula for Apparent Depth
The formula for apparent depth is given by:
\[ \text{Apparent Depth} = \frac{\text{Actual Depth}}{\mu} \]
where \( \mu \) is the refractive index of the medium.

### Step 4: Set Up Equations
From the first face (where the apparent depth is 6 cm):
\[ 6 = \frac{x}{\mu} \quad \text{(1)} \]

From the opposite face (where the apparent depth is 4 cm):
\[ 4 = \frac{15 - x}{\mu} \quad \text{(2)} \]

### Step 5: Solve for \( x \) in Terms of \( \mu \)
From equation (1):
\[ x = 6\mu \quad \text{(3)} \]

From equation (2):
\[ 15 - x = 4\mu \quad \text{(4)} \]

### Step 6: Substitute Equation (3) into Equation (4)
Substituting \( x \) from equation (3) into equation (4):
\[ 15 - 6\mu = 4\mu \]

### Step 7: Rearrange the Equation
Rearranging gives:
\[ 15 = 6\mu + 4\mu \]
\[ 15 = 10\mu \]

### Step 8: Solve for \( \mu \)
Dividing both sides by 10:
\[ \mu = \frac{15}{10} = 1.5 \]

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