A plano convex lens is made of glass of refractive index 1.5. The radius of curvature of its convex surface is R. Its focal length is

Options

A. `R/2`
B. R
C. 2R
D. 1.5R

Explanation

To find the focal length of a plano-convex lens made of glass with a refractive index of 1.5 and a radius of curvature \( R \), we can use the lens maker's formula. The lens maker's formula is given by:

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

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.

### Step-by-Step Solution:

1. **Identify the surfaces of the lens**:
- For a plano-convex lens, one surface is flat (plane) and the other is convex.
- The radius of curvature for the convex surface \( R_1 = R \) (positive since it is convex).
- The radius of curvature for the plane surface \( R_2 = \infty \) (since it is flat, we consider it as infinite).

2. **Substitute the values into the lens maker's formula**:
- Given \( \mu = 1.5 \), \( R_1 = R \), and \( R_2 = \infty \), we can substitute these into the formula:
\[ \frac{1}{f} = (1.5 - 1) \left( \frac{1}{R} - \frac{1}{\infty} \right) \]

3. **Simplify the equation**:
- Since \( \frac{1}{\infty} = 0 \), the equation simplifies to:
\[ \frac{1}{f} = 0.5 \left( \frac{1}{R} - 0 \right) \]
- Thus, we have:
\[ \frac{1}{f} = \frac{0.5}{R} \]

4. **Solve for the focal length \( f \)**:
- Taking the reciprocal gives:
\[ f = \frac{R}{0.5} = 2R \]

### Final Result:
The focal length \( f \) of the plano-convex lens is:
\[ f = 2R \]