A mark is made on the surface of a glass sphere of diameter 10 cm and refractive index 1.5. It is viewed through the glass from a portion directly opposite. The distance of the image of the mark from the centre of the sphere will be

Options

A. 15 cm
B. 17.5 cm
C. 20 cm
D. 22.5 cm

Explanation

To solve the problem of finding the distance of the image of the mark from the center of the glass sphere, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Data:**
- Diameter of the glass sphere, \( D = 10 \, \text{cm} \)
- Radius of the glass sphere, \( R = \frac{D}{2} = 5 \, \text{cm} \)
- Refractive index of glass, \( \mu_2 = 1.5 \)
- Refractive index of air, \( \mu_1 = 1 \)

2. **Set Up the Sign Convention:**
- The object (mark) is on the surface of the sphere. Since we are measuring distances from the center of the sphere and the mark is on the surface, we take the object distance \( u = -5 \, \text{cm} \) (negative as per the sign convention for real objects).

3. **Use the Lens Maker's Formula:**
The formula relating the object distance \( u \), image distance \( v \), and the radius of curvature \( R \) is given by:
\[ \frac{\mu_2}{v} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{R} \]
Substituting the known values:
\[ \frac{1.5}{v} - \frac{1}{-5} = \frac{1.5 - 1}{5} \]

4. **Simplify the Equation:**
\[ \frac{1.5}{v} + \frac{1}{5} = \frac{0.5}{5} \]
\[ \frac{1.5}{v} + \frac{1}{5} = \frac{0.1}{1} \]

5. **Multiply through by \( 5v \) to eliminate fractions:**
\[ 5 \cdot 1.5 = 0.5v + v \]
\[ 7.5 = 1.5v \]

6. **Solve for \( v \):**
\[ v = \frac{7.5}{1.5} = 5 \, \text{cm} \]

7. **Determine the Distance from the Center:**
Since the image distance \( v \) is measured from the center of the sphere, and the mark was at the surface, the distance of the image from the center of the sphere is \( 5 \, \text{cm} \).

### Final Answer:
The distance of the image of the mark from the center of the sphere is \( 5 \, \text{cm} \).

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