A glass sphere (`mu` =1.5) of radius 20 cm has small air bubble 4 cm below its centre. The sphere is viewed from outside and along vertical line through the bubble. The apparent depth of the bubble below the surface of sphere is (in cm)

Options

A. 13.33
B. 26.67
C. 15
D. 30

Explanation

To find the apparent depth of the air bubble below the surface of the glass sphere, we can use the formula that relates the refractive indices and distances in optics. Here’s a step-by-step solution:

### Step 1: Identify the parameters
- Refractive index of air, \( \mu_2 = 1 \)
- Refractive index of glass, \( \mu_1 = 1.5 \)
- Radius of the glass sphere, \( R = 20 \, \text{cm} \)
- Depth of the bubble from the center of the sphere, \( d = 4 \, \text{cm} \)

### Step 2: Calculate the object distance (U)
The object distance \( U \) is measured from the surface of the sphere to the bubble. Since the bubble is 4 cm below the center of the sphere, we need to account for the radius of the sphere:

\[ U = - (R - d) = - (20 \, \text{cm} - 4 \, \text{cm}) = - 16 \, \text{cm} \]

### Step 3: Use the lens maker's formula
We use the formula for refraction at a spherical surface:

\[ \frac{\mu_2}{V} - \frac{\mu_1}{U} = \frac{\mu_2 - \mu_1}{R} \]

Substituting in the known values:

\[ \frac{1}{V} - \frac{1.5}{-16} = \frac{1 - 1.5}{-20} \]

### Step 4: Simplify the equation
Calculating the right side:

\[ \frac{1 - 1.5}{-20} = \frac{-0.5}{-20} = \frac{0.5}{20} = \frac{1}{40} \]

Now substituting back into the equation:

\[ \frac{1}{V} + \frac{1.5}{16} = \frac{1}{40} \]

### Step 5: Find a common denominator
The common denominator for the fractions is 80:

\[ \frac{1}{V} + \frac{1.5 \times 5}{80} = \frac{2}{80} \]

This simplifies to:

\[ \frac{1}{V} + \frac{7.5}{80} = \frac{2}{80} \]

### Step 6: Solve for \( \frac{1}{V} \)
Rearranging gives:

\[ \frac{1}{V} = \frac{2}{80} - \frac{7.5}{80} = \frac{-5.5}{80} \]

### Step 7: Calculate \( V \)
Taking the reciprocal:

\[ V = \frac{80}{-5.5} = -\frac{80}{5.5} \approx -14.55 \, \text{cm} \]

### Step 8: Calculate the apparent depth
The apparent depth of the bubble is given by the absolute value of \( V \):

\[ \text{Apparent Depth} = |V| \approx 14.55 \, \text{cm} \]

### Final Result
The apparent depth of the bubble below the surface of the sphere is approximately **14.55 cm**.

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