A cube of side 15 cm is having an air bubble. The bubble appears at 6 cm from one face and at 4 cm from opposite face. The refractive index of cube is

Options

A. `5/2`
B. `3/2`
C. `2/3`
D. `2/5`

Explanation

To find the refractive index of the cube with an air bubble, we can follow these steps:

### Step 1: Understand the problem
We have a cube of side 15 cm with an air bubble inside it. The bubble appears at a distance of 6 cm from one face of the cube and at a distance of 4 cm from the opposite face. We need to find the refractive index (μ) of the cube.

### Step 2: Define the variables
Let:
- \( x_1 \) = actual distance of the bubble from the face where it appears at 6 cm
- \( x_2 \) = actual distance of the bubble from the opposite face where it appears at 4 cm

### Step 3: Relate apparent depth to actual depth
The apparent depth (the distance at which the bubble appears) is related to the actual depth by the formula:
\[ \text{Apparent Depth} = \frac{\text{Actual Depth}}{\mu} \]
From the problem, we have:
1. \( 6 \, \text{cm} = \frac{x_1}{\mu} \)
2. \( 4 \, \text{cm} = \frac{x_2}{\mu} \)

### Step 4: Solve for actual depths
From the equations above, we can express \( x_1 \) and \( x_2 \) in terms of μ:
1. \( x_1 = 6\mu \)
2. \( x_2 = 4\mu \)

### Step 5: Use the total distance
Since the total distance from one face to the opposite face is equal to the side of the cube, we have:
\[ x_1 + x_2 = 15 \, \text{cm} \]
Substituting the expressions for \( x_1 \) and \( x_2 \):
\[ 6\mu + 4\mu = 15 \]
\[ 10\mu = 15 \]

### Step 6: Solve for μ
Now, we can solve for μ:
\[ \mu = \frac{15}{10} = 1.5 \]

### Step 7: Conclusion
The refractive index of the cube is:
\[ \mu = 1.5 \]

### Final Answer
Thus, the refractive index of the cube is \( \frac{3}{2} \).

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