A glass sphere `(mu=1.5)` of radius 20 cm has a small air bubble 4 cm below its centre. The sphere is viewed from outside and along a 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

Answer

To find the apparent depth of the air bubble located 4 cm below the center of a glass sphere with a refractive index of \( \mu = 1.5 \) and a radius of 20 cm, we can use the formula for apparent depth in a spherical medium.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Refractive index of air, \( \mu_1 = 1 \)
- Refractive index of glass, \( \mu_2 = 1.5 \)
- Radius of the glass sphere, \( R = 20 \) cm
- Depth of the bubble below the center of the sphere, \( d = 4 \) cm

2. **Calculate the Actual Depth of the Bubble from the Surface of the Sphere:**
- The distance from the center of the sphere to the surface is equal to the radius of the sphere. Therefore, the actual depth of the bubble from the surface of the sphere is:
\[ \text{Actual Depth} = R - d = 20 \text{ cm} - 4 \text{ cm} = 16 \text{ cm} \]

3. **Use the Formula for Apparent Depth:**
- The formula for apparent depth when viewing through a spherical medium is given by:
\[ \text{Apparent Depth} = \frac{\mu_1}{\mu_2} \times \text{Actual Depth} \]
- Substituting the values:
\[ \text{Apparent Depth} = \frac{1}{1.5} \times 16 \text{ cm} = \frac{16}{1.5} \text{ cm} = \frac{32}{3} \text{ cm} \approx 10.67 \text{ cm} \]

4. **Adjust for the Apparent Depth Below the Surface:**
- Since the bubble is viewed from outside the sphere, the apparent depth below the surface of the sphere is:
\[ \text{Apparent Depth Below Surface} = R - \text{Apparent Depth} = 20 \text{ cm} - 10.67 \text{ cm} \approx 9.33 \text{ cm} \]

5. **Final Calculation:**
- The apparent depth of the bubble below the surface of the sphere is approximately \( 9.33 \) cm. However, since we need to find the depth of the bubble as viewed from outside, we have to consider the total depth from the surface to the bubble, which is:
\[ \text{Total Apparent Depth} = 20 \text{ cm} - 9.33 \text{ cm} = 10.67 \text{ cm} \]

### Conclusion:
The apparent depth of the bubble below the surface of the sphere is approximately **10.67 cm**.