A thin concavo-convex lens has two surfaces of radii of curvature R and 2R. The material of the lens has a refractive index n. 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/n-1
D. Will be equal to 2R/n-1

Answer

To find the focal length of a thin concavo-convex lens with radii of curvature R and 2R, and a refractive index n when kept in air, we can use the Lensmaker's formula. The formula is given by:

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

### Step-by-Step Solution:

1. **Identify the Radii of Curvature**:
- Let \( R_1 = -R \) (the concave surface) and \( R_2 = +2R \) (the convex surface). The negative sign for \( R_1 \) is due to the sign convention for concave surfaces.

2. **Apply the Lensmaker's Formula**:
- Substitute \( R_1 \) and \( R_2 \) into the Lensmaker's formula:
\[ \frac{1}{f_1} = (n - 1) \left( \frac{1}{-R} - \frac{1}{2R} \right) \]

3. **Simplify the Expression**:
- Calculate the right-hand side:
\[ \frac{1}{f_1} = (n - 1) \left( -\frac{1}{R} - \frac{1}{2R} \right) = (n - 1) \left( -\frac{2}{2R} - \frac{1}{2R} \right) = (n - 1) \left( -\frac{3}{2R} \right) \]
- Therefore, we have:
\[ \frac{1}{f_1} = -\frac{3(n - 1)}{2R} \]

4. **Calculate the Focal Length**:
- Taking the reciprocal gives:
\[ f_1 = -\frac{2R}{3(n - 1)} \]

5. **Consider the Second Case**:
- If the lens is flipped, we have:
\[ R_1 = +2R \quad \text{and} \quad R_2 = -R \]
- Applying the formula again:
\[ \frac{1}{f_2} = (n - 1) \left( \frac{1}{2R} - \frac{1}{-R} \right) = (n - 1) \left( \frac{1}{2R} + \frac{1}{R} \right) = (n - 1) \left( \frac{1 + 2}{2R} \right) = (n - 1) \left( \frac{3}{2R} \right) \]
- Thus, we find:
\[ \frac{1}{f_2} = \frac{3(n - 1)}{2R} \]
- Taking the reciprocal gives:
\[ f_2 = \frac{2R}{3(n - 1)} \]

6. **Conclusion**:
- Since \( f_1 \) and \( f_2 \) have the same magnitude but opposite signs, the focal lengths are equal in magnitude but differ in sign depending on the direction of light incidence. Thus, the focal length of the lens when kept in air is:
\[ f = -\frac{2R}{3(n - 1)} \]