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. 1.5
B. 1.66
C. 1.33
D. 3

Answer

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

\[ \frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]

Where:
- \( f \) = focal length of the lens
- \( n \) = refractive index of the lens material
- \( R_1 \) = radius of curvature of the first surface (convex surface)
- \( R_2 \) = radius of curvature of the second surface (flat surface)

For a plano-convex lens:
- The convex surface has a radius of curvature \( R_1 = +10 \, \text{cm} \) (positive because it is convex).
- The flat surface has a radius of curvature \( R_2 = \infty \) (since it is flat, the radius is considered infinite).

Given:
- \( f = 30 \, \text{cm} \)

Now, substituting the values into the lens maker's formula:

1. **Substituting values into the formula:**

\[ \frac{1}{30} = (n - 1) \left( \frac{1}{10} - \frac{1}{\infty} \right) \]

Since \( \frac{1}{\infty} = 0 \), we can simplify this to:

\[ \frac{1}{30} = (n - 1) \left( \frac{1}{10} \right) \]

2. **Solving for \( n - 1 \):**

\[ \frac{1}{30} = \frac{n - 1}{10} \]

Multiplying both sides by 10:

\[ \frac{10}{30} = n - 1 \]

This simplifies to:

\[ \frac{1}{3} = n - 1 \]

3. **Finding \( n \):**

Adding 1 to both sides:

\[ n = 1 + \frac{1}{3} \]

This gives:

\[ n = \frac{3}{3} + \frac{1}{3} = \frac{4}{3} \]

Thus, the refractive index of the material of the lens is:

\[ \boxed{\frac{4}{3}} \]

---