The power of a double convex lens of radius of curvature R each is Y. The power of a plano convex lens of same material and of radius of curvature 2R is

Options

A. `Y/4`
B. `Y/2`
C. `2Y`
D. 4Y

Explanation

To solve the problem, we need to find the power of a plano-convex lens given the power of a double convex lens of the same material. We will use the Lensmaker's formula to derive the solution step by step.

### Step-by-Step Solution:

1. **Understanding the Power of a Lens**:
The power \( P \) of a lens is given by the formula:
\[ P = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
where \( n \) is the refractive index of the lens material, and \( R_1 \) and \( R_2 \) are the radii of curvature of the two lens surfaces.

2. **Power of the Double Convex Lens**:
For a double convex lens with both surfaces having radius of curvature \( R \):
- \( R_1 = R \) (for the first surface)
- \( R_2 = -R \) (for the second surface, taken as negative for convention)

Therefore, the power \( P_1 \) is:
\[ P_1 = (n - 1) \left( \frac{1}{R} - \frac{1}{-R} \right) = (n - 1) \left( \frac{1}{R} + \frac{1}{R} \right) = (n - 1) \left( \frac{2}{R} \right) \]
Given that \( P_1 = Y \), we have:
\[ Y = (n - 1) \left( \frac{2}{R} \right) \]

3. **Power of the Plano-Convex Lens**:
For a plano-convex lens with one flat surface and one convex surface with radius of curvature \( 2R \):
- \( R_1 = 2R \) (for the convex surface)
- \( R_2 = \infty \) (for the flat surface, since the radius of curvature is infinite)

Therefore, the power \( P_2 \) is:
\[ P_2 = (n - 1) \left( \frac{1}{2R} - \frac{1}{\infty} \right) = (n - 1) \left( \frac{1}{2R} \right) \]

4. **Finding the Relationship Between \( P_1 \) and \( P_2 \)**:
We can express \( P_2 \) in terms of \( Y \):
\[ P_2 = (n - 1) \left( \frac{1}{2R} \right) = \frac{1}{2} \left( (n - 1) \left( \frac{2}{R} \right) \right) = \frac{1}{2} Y \]

5. **Conclusion**:
Thus, the power of the plano-convex lens is:
\[ P_2 = \frac{Y}{2} \]

### Final Answer:
The power of the plano-convex lens is \( \frac{Y}{2} \).