If in a planoconvex lens, the radius of curvature of the convex surface is `10cm` and the focal length is `30 cm`, the refractive index of the material of the 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 planoconvex lens, we can use the lensmaker's formula, which relates the focal length of the lens, the radii of curvature, and the refractive index of the lens material.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Radius of curvature of the convex surface (R₁) = 10 cm
- Focal length (f) = 30 cm
- The plane surface (R₂) has an infinite radius of curvature (R₂ = ∞).

2. **Use the Lensmaker's Formula:**
The lensmaker's formula is given by:
\[ \frac{1}{f} = \mu - 1 \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
Here, \( \mu \) is the refractive index of the lens material.

3. **Substitute the Values into the Formula:**
Since R₂ is infinite, \( \frac{1}{R_2} = 0 \). Thus, the formula simplifies to:
\[ \frac{1}{f} = \mu - 1 \left( \frac{1}{R_1} \right) \]
Substituting the known values:
\[ \frac{1}{30} = \mu - 1 \left( \frac{1}{10} \right) \]

4. **Rearranging the Equation:**
This can be rewritten as:
\[ \frac{1}{30} = \mu - 1 \cdot \frac{1}{10} \]
Multiplying both sides by 30:
\[ 1 = 30(\mu - 1) \cdot \frac{1}{10} \]
Which simplifies to:
\[ 1 = 3(\mu - 1) \]

5. **Solve for \( \mu \):**
Rearranging gives:
\[ \mu - 1 = \frac{1}{3} \]
Therefore:
\[ \mu = \frac{1}{3} + 1 = \frac{4}{3} \]

6. **Final Result:**
The refractive index of the material of the lens is:
\[ \mu = \frac{4}{3} \approx 1.33 \]