A mark on the surface of a glass spehre `(mu=1.5)` is viewed from a diametrically opposite position. It appears to be at a distance 10 cm from its actual position. The radius of the sphere is

Options

A. 5 cm
B. 10 cm
C. 15 cm
D. 25 cm

Answer

To solve the problem step-by-step, we will use the concepts of geometrical optics, particularly the refraction at a spherical surface.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have a glass sphere with a refractive index \( \mu_1 = 1.5 \).
- The mark on the surface of the sphere is viewed from the diametrically opposite position.
- It appears to be at a distance of 10 cm from its actual position.

2. **Setting Up the Coordinates**:
- Let the radius of the sphere be \( r \).
- The object (mark) is located on the surface of the sphere, so its actual position can be considered as \( u = -2r \) (negative because it is on the same side as the incoming light).
- The image distance \( v \) is what we need to find.

3. **Applying the Refraction Formula**:
- The formula for refraction at a spherical surface is given by:
\[ \frac{\mu_2}{v} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{R} \]
- Here, \( \mu_2 = 1 \) (for air), \( \mu_1 = 1.5 \), and \( R = -r \) (the radius is negative for a convex surface).

4. **Substituting Values**:
- Substitute the values into the formula:
\[ \frac{1}{v} - \frac{1.5}{-2r} = \frac{1 - 1.5}{-r} \]
- This simplifies to:
\[ \frac{1}{v} + \frac{1.5}{2r} = \frac{-0.5}{-r} \]
- Further simplifying gives:
\[ \frac{1}{v} + \frac{0.75}{r} = \frac{0.5}{r} \]

5. **Finding \( \frac{1}{v} \)**:
- Rearranging the equation:
\[ \frac{1}{v} = \frac{0.5}{r} - \frac{0.75}{r} = \frac{-0.25}{r} \]
- Thus, we find:
\[ v = -\frac{4r}{1} \]

6. **Relating \( v \) to the Given Information**:
- The image appears to be at a distance of 10 cm from its actual position, which means:
\[ |v| = 2r + 10 \]
- Substituting \( v = -4r \):
\[ 4r = 2r + 10 \]

7. **Solving for \( r \)**:
- Rearranging gives:
\[ 4r - 2r = 10 \implies 2r = 10 \implies r = 5 \text{ cm} \]

8. **Conclusion**:
- The radius of the sphere is \( r = 5 \) cm.

### Final Answer:
The radius of the sphere is **5 cm**.