Radius of curvature of first surface of double convex lens is three times that of the other. If focal length of the lens is 30 cm and refractive index of the lens is 3/2, then radius of curvature of the first surface is

Options

A. 20 cm
B. 40 cm
C. 60 cm
D. 80 cm

Answer

To solve the problem, we will use the lens maker's formula, which relates the focal length of a lens to its radii of curvature and refractive index. Let's go through the solution step by step.

### Step-by-Step Solution:

1. **Understanding the Problem:**
- We have a double convex lens.
- Let the radius of curvature of the first surface be \( r_1 \) and the radius of curvature of the second surface be \( r_2 \).
- According to the problem, \( r_1 = r \) and \( r_2 = -3r \) (the negative sign indicates that the second surface is concave relative to the incoming light).

2. **Given Data:**
- Focal length \( f = 30 \) cm
- Refractive index \( \mu = \frac{3}{2} \)

3. **Lens Maker's Formula:**
The lens maker's formula is given by:
\[ \frac{1}{f} = (\mu - 1) \left( \frac{1}{r_1} - \frac{1}{r_2} \right) \]

4. **Substituting Values:**
Substitute the values of \( r_1 \) and \( r_2 \) into the lens maker's formula:
\[ \frac{1}{30} = \left( \frac{3}{2} - 1 \right) \left( \frac{1}{r} - \frac{1}{-3r} \right) \]
Simplifying \( \mu - 1 \):
\[ \frac{3}{2} - 1 = \frac{1}{2} \]
Now substitute this back:
\[ \frac{1}{30} = \frac{1}{2} \left( \frac{1}{r} + \frac{1}{3r} \right) \]

5. **Combining Terms:**
Combine the terms inside the parentheses:
\[ \frac{1}{r} + \frac{1}{3r} = \frac{3}{3r} + \frac{1}{3r} = \frac{4}{3r} \]
Thus, we have:
\[ \frac{1}{30} = \frac{1}{2} \cdot \frac{4}{3r} \]
Simplifying this gives:
\[ \frac{1}{30} = \frac{2}{3r} \]

6. **Cross-Multiplying:**
Cross-multiply to solve for \( r \):
\[ 3r = 2 \times 30 \]
\[ 3r = 60 \]
\[ r = \frac{60}{3} = 20 \text{ cm} \]

7. **Finding the Radius of Curvature of the First Surface:**
Since \( r_1 = r \), the radius of curvature of the first surface is:
\[ r_1 = 20 \text{ cm} \]

### Final Answer:
The radius of curvature of the first surface is **20 cm**.