An ice cube has a bubble inside. When viiewed from one side the apparent distance of the bubble is `12` cm . When viewed from the opposite side ,the apprent distance of the bubble is observed as `4` cm. If the side of the ice cube is `24` cm, the refractive index of the ice cube is

Options

A. `4/5`
B. `3/2`
C. `6/5`
D. `2/3`

Answer

To find the refractive index of the ice cube containing a bubble, we can follow these steps:

### Step 1: Define the variables
Let:
- \( x \) = distance from the observer to the bubble when viewed from one side.
- \( 24 - x \) = distance from the observer to the bubble when viewed from the opposite side.
- The apparent distance of the bubble when viewed from one side = \( 12 \, \text{cm} \).
- The apparent distance of the bubble when viewed from the opposite side = \( 4 \, \text{cm} \).
- The side length of the ice cube = \( 24 \, \text{cm} \).

### Step 2: Set up the equations using the formula for apparent distance
Using the formula for apparent distance in terms of real distance and refractive index:
1. From the first side:
\[ \frac{x}{n} = 12 \quad \text{(1)} \]
2. From the opposite side:
\[ \frac{24 - x}{n} = 4 \quad \text{(2)} \]

### Step 3: Solve for \( n \) using the equations
From equation (1):
\[ x = 12n \quad \text{(3)} \]

Substituting equation (3) into equation (2):
\[ \frac{24 - 12n}{n} = 4 \]

### Step 4: Simplify and solve for \( n \)
Multiply both sides by \( n \):
\[ 24 - 12n = 4n \]
Combine like terms:
\[ 24 = 12n + 4n \]
\[ 24 = 16n \]
Now, solve for \( n \):
\[ n = \frac{24}{16} = \frac{3}{2} \]

### Conclusion
The refractive index of the ice cube is:
\[ n = \frac{3}{2} \]

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