A glass sphere of diameter 50 cm and `mu=1.5` has a small air bubble. Looking from outside the diameter, the bubble appears to be at a distance 10 cm the surface. Find the apparent position of the bubble when it is viewed from the diametrically opposite position.

Options

A. real image, at the pole of 2 nd face
B. real image, at the pole of 1st face
C. virtual image, at the pole of 1st face
D. virtual image, at the centre of the sphere

Answer

To solve the problem, we need to find the apparent position of the air bubble when viewed from the diametrically opposite position of the glass sphere. We will use the lens maker's formula and the concept of refraction to find the position.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Diameter of the glass sphere (D) = 50 cm
- Radius of the sphere (R) = D/2 = 25 cm
- Refractive index of glass (μ) = 1.5
- Apparent distance of the bubble from the surface (v1) = 10 cm

2. **Determine the Object Distance:**
- The object (air bubble) is inside the sphere. The distance from the center of the sphere to the bubble can be calculated as:
\[ u = R - v_1 = 25 \text{ cm} - 10 \text{ cm} = 15 \text{ cm} \]
- Here, u is taken as negative because the object is inside the sphere.

3. **Use the Lens Formula:**
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
where:
- f = focal length
- v = image distance (which we need to find)
- u = object distance

4. **Calculate the Focal Length:**
The focal length (f) for a spherical interface can be calculated using the formula:
\[ f = \frac{R}{2(\mu - 1)} \]
Substituting the values:
\[ f = \frac{25 \text{ cm}}{2(1.5 - 1)} = \frac{25 \text{ cm}}{2 \times 0.5} = \frac{25 \text{ cm}}{1} = 25 \text{ cm} \]

5. **Apply the Lens Formula:**
Now we can apply the lens formula:
\[ \frac{1}{25} = \frac{1}{v} - \frac{1}{-15} \]
Rearranging gives:
\[ \frac{1}{v} = \frac{1}{25} - \frac{1}{15} \]
Finding a common denominator (75):
\[ \frac{1}{v} = \frac{3}{75} - \frac{5}{75} = -\frac{2}{75} \]
Thus:
\[ v = -\frac{75}{2} = -37.5 \text{ cm} \]

6. **Interpret the Result:**
The negative sign indicates that the image is virtual and located on the same side as the object (inside the sphere). The apparent position of the bubble when viewed from the diametrically opposite position is 37.5 cm from the center of the sphere.

### Final Result:
The apparent position of the bubble when viewed from the diametrically opposite position is **37.5 cm** from the center of the sphere.