A thin concavo-convex lens has two surfaces of radii of curvature R and 2R. The material of the lens has a refractive index p. When kept in air, the focal length of the lens

Options

A. will depend on the direction from which light is incident on it
B. will be the same, irrespective of the direction from which light is incident on it
C. will be equal to`R/(mu-1)`
D. will be equal to`2R/(mu-1)`

Answer

To find the focal length of a thin concavo-convex lens with radii of curvature R and 2R, and a refractive index \( \mu \), we can use the lens maker's formula.

### Step-by-Step Solution:

1. **Identify the Radii of Curvature**:
- For the concavo-convex lens, we have two surfaces:
- The first surface (convex) has a radius of curvature \( R_1 = +2R \) (positive for convex).
- The second surface (concave) has a radius of curvature \( R_2 = -R \) (negative for concave).

2. **Use the Lens Maker's Formula**:
The lens maker's formula 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.

3. **Substitute the Values**:
Substitute \( R_1 \) and \( R_2 \) into the formula:
\[ \frac{1}{f} = (\mu - 1) \left( \frac{1}{2R} - \frac{1}{-R} \right) \]

4. **Simplify the Equation**:
Simplifying the terms inside the parentheses:
\[ \frac{1}{f} = (\mu - 1) \left( \frac{1}{2R} + \frac{1}{R} \right) \]
\[ \frac{1}{f} = (\mu - 1) \left( \frac{1}{2R} + \frac{2}{2R} \right) \]
\[ \frac{1}{f} = (\mu - 1) \left( \frac{3}{2R} \right) \]

5. **Find the Focal Length**:
Taking the reciprocal to find \( f \):
\[ f = \frac{2R}{3(\mu - 1)} \]

6. **Conclusion**:
The focal length of the lens when kept in air is:
\[ f = \frac{2R}{3(\mu - 1)} \]