If in a plano-convex lens, the radius curvature of the convex surface is 10 cm and the focal length of the lens is 20 cm , then the refractive index of the material of lens will be

Options

A. `1.5`
B. `1.66`
C. `1.33`
D. 3

Explanation

To find the refractive index of the material of a plano-convex lens using the given information, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Radius of curvature of the convex surface (R1) = 10 cm
- Radius of curvature of the plano surface (R2) = ∞ (since it is flat)
- Focal length (f) = 20 cm

2. **Use the Lensmaker's Formula**:
The Lensmaker's formula for a lens is given by:
\[ \frac{1}{f} = \mu - 1 \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
where:
- \( f \) is the focal length of the lens,
- \( \mu \) is the refractive index of the lens material,
- \( R_1 \) is the radius of curvature of the first surface,
- \( R_2 \) is the radius of curvature of the second surface.

3. **Substitute the Known Values**:
Substitute \( R_1 = 10 \, \text{cm} \) and \( R_2 = \infty \) into the formula:
\[ \frac{1}{20} = \mu - 1 \left( \frac{1}{10} - \frac{1}{\infty} \right) \]
Since \( \frac{1}{\infty} = 0 \), this simplifies to:
\[ \frac{1}{20} = \mu - 1 \left( \frac{1}{10} \right) \]

4. **Simplify the Equation**:
This can be rewritten as:
\[ \frac{1}{20} = \frac{\mu - 1}{10} \]

5. **Cross-Multiply to Solve for \( \mu - 1 \)**:
Cross-multiplying gives:
\[ 10 \cdot \frac{1}{20} = \mu - 1 \]
Simplifying this:
\[ \frac{10}{20} = \mu - 1 \quad \Rightarrow \quad \frac{1}{2} = \mu - 1 \]

6. **Solve for \( \mu \)**:
Adding 1 to both sides:
\[ \mu = 1 + \frac{1}{2} = 1.5 \]

7. **Conclusion**:
The refractive index of the material of the lens is \( \mu = 1.5 \).