For a plano convex lens, the radius of curvature of convex surface is 10 cm and the focal length is 30 cm. The refractive index of the material of the lens is

Options

A. 1.5
B. 1.66
C. 1.33
D. 2.5

Answer

To find the refractive index of the material of a plano-convex lens, we can use the lens maker's formula, which relates the focal length of the lens to its radii of curvature and the refractive index of the material.

### Step-by-Step Solution:

1. **Understand the given values:**
- Radius of curvature of the convex surface (R1) = 10 cm
- Radius of curvature of the plane surface (R2) = ∞ (since it is a plane surface)
- Focal length (f) = 30 cm

2. **Apply the lens maker's formula:**
The lens maker's formula for a lens in air is given by:
\[ \frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
where:
- \( n \) is the refractive index of the lens material,
- \( R_1 \) is the radius of curvature of the first surface (convex),
- \( R_2 \) is the radius of curvature of the second surface (plane).

3. **Substitute the known values into the formula:**
- Since \( R_2 = \infty \), \( \frac{1}{R_2} = 0 \).
- Substitute \( R_1 = 10 \, \text{cm} \) and \( f = 30 \, \text{cm} \):
\[ \frac{1}{30} = (n - 1) \left( \frac{1}{10} - 0 \right) \]

4. **Simplify the equation:**
\[ \frac{1}{30} = (n - 1) \left( \frac{1}{10} \right) \]
Multiply both sides by 10:
\[ \frac{10}{30} = n - 1 \]
Simplifying gives:
\[ \frac{1}{3} = n - 1 \]

5. **Solve for the refractive index (n):**
\[ n = 1 + \frac{1}{3} = \frac{4}{3} \]

6. **Convert to decimal form:**
\[ n = 1.33 \]

### Final Answer:
The refractive index of the material of the lens is \( n = 1.33 \).