If in a plano-convex lens, the radius of curvature of the convex surface is 10 cm and the focal length of the lens is 30 cm , then the refractive index of the material of lens will be

Options

A. 20.5 cm
B. 10 cm
C. 15.5 cm
D. 5 cm

Explanation

To find the refractive index (μ) of a plano-convex lens given the radius of curvature (R) and the focal length (F), we can use the lens maker's formula:

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

### Step-by-Step Solution:

1. **Identify the parameters:**
- The radius of curvature of the convex surface, \( R_1 = 10 \, \text{cm} \)
- The radius of curvature of the plano surface, \( R_2 = \infty \) (since it is flat)
- The focal length of the lens, \( F = 30 \, \text{cm} \)

2. **Substitute the values into the lens maker's formula:**
\[ \frac{1}{F} = (\mu - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
Since \( R_2 = \infty \), we have:
\[ \frac{1}{R_2} = 0 \]
Thus, the equation simplifies to:
\[ \frac{1}{30} = (\mu - 1) \left( \frac{1}{10} - 0 \right) \]
This simplifies to:
\[ \frac{1}{30} = (\mu - 1) \frac{1}{10} \]

3. **Rearranging the equation:**
Multiply both sides by 10:
\[ \frac{10}{30} = \mu - 1 \]
Simplifying gives:
\[ \frac{1}{3} = \mu - 1 \]

4. **Solve for μ:**
Add 1 to both sides:
\[ \mu = 1 + \frac{1}{3} = \frac{4}{3} \]

5. **Convert to decimal:**
\[ \mu = 1.33 \]

### Conclusion:
The refractive index of the material of the lens is approximately \( \mu = 1.33 \).