The focal length of a convex lens is 10 cm and its refractive index is 1.5 . If the radius of curvature of one surface is 7.5 cm, the radius of curvature of the second surface will be :

Options

A. `7. 5 cm`
B. `15 cm`
C. `75 cm`
D. `5 cm`

Explanation

To find the radius of curvature of the second surface of the convex lens, we can use the lens maker's formula, which relates the focal length (f), refractive index (μ), and the radii of curvature (R1 and R2) of the lens surfaces. The formula is given by:

\[ \frac{1}{f} = (\mu - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]

### Step-by-Step Solution:

1. **Identify the given values:**
- Focal length (f) = 10 cm
- Refractive index (μ) = 1.5
- Radius of curvature of the first surface (R1) = 7.5 cm

2. **Substitute the known values into the lens maker's formula:**
\[ \frac{1}{10} = (1.5 - 1) \left( \frac{1}{7.5} - \frac{1}{R_2} \right) \]

3. **Simplify the equation:**
- Calculate \( \mu - 1 \):
\[ 1.5 - 1 = 0.5 \]
- Substitute this back into the equation:
\[ \frac{1}{10} = 0.5 \left( \frac{1}{7.5} - \frac{1}{R_2} \right) \]

4. **Multiply both sides by 2 to eliminate the fraction on the right:**
\[ \frac{2}{10} = \frac{1}{7.5} - \frac{1}{R_2} \]
\[ \frac{1}{5} = \frac{1}{7.5} - \frac{1}{R_2} \]

5. **Convert \( \frac{1}{5} \) to a common denominator with \( \frac{1}{7.5} \):**
- The common denominator of 5 and 7.5 is 15.
\[ \frac{1}{5} = \frac{3}{15}, \quad \frac{1}{7.5} = \frac{2}{15} \]
- Substitute these values into the equation:
\[ \frac{3}{15} = \frac{2}{15} - \frac{1}{R_2} \]

6. **Rearrange the equation to solve for \( \frac{1}{R_2} \):**
\[ \frac{1}{R_2} = \frac{2}{15} - \frac{3}{15} = -\frac{1}{15} \]

7. **Take the reciprocal to find \( R_2 \):**
\[ R_2 = -15 \, \text{cm} \]

### Final Answer:
The radius of curvature of the second surface (R2) is -15 cm.