A convex lens is cut into two parts in different ways that are arranged in four manners, as shown. Which arrangement will give maximum optical power?

Options

A.
B.
C.
D.

Explanation

To determine which arrangement of the cut convex lens gives the maximum optical power, we will analyze the focal lengths of each arrangement and calculate their respective powers. The power \( P \) of a lens is given by the formula:

\[ P = \frac{1}{f} \]

where \( f \) is the focal length of the lens. The goal is to find the arrangement with the minimum focal length, as this will correspond to the maximum optical power.

### Step-by-Step Solution:

1. **Understanding the Lens Arrangement**:
The convex lens is cut into two parts in different ways, and we need to analyze four different arrangements.

2. **Power and Focal Length Relationship**:
Recall that the power of a lens is inversely proportional to its focal length. Thus, to maximize the power, we need to minimize the focal length.

3. **Analyzing Each Arrangement**:
- **Arrangement 1**:
- The two parts of the lens are arranged such that they work together. The effective focal length \( f_1 \) can be calculated as:
\[ \frac{1}{f_1} = \frac{1}{f_1'} + \frac{1}{f_2'} \quad \text{(where \( f_1' \) and \( f_2' \) are the focal lengths of each half)} \]
- For equal halves of a lens, if the original lens had a focal length \( f \), then each half will have a focal length of \( \frac{f}{2} \). Thus:
\[ \frac{1}{f_1} = \frac{2}{f/2} = \frac{4}{f} \quad \Rightarrow \quad f_1 = \frac{f}{4} \]

- **Arrangement 2**:
- In this arrangement, one part acts like a mirror. Thus, the effective focal length \( f_2 \) becomes infinite, leading to:
\[ P_2 = 0 \quad \text{(since \( f_2 = \infty \))} \]

- **Arrangement 3**:
- The two parts are arranged such that they are still effective. The effective focal length \( f_3 \) can be calculated similarly:
\[ \frac{1}{f_3} = \frac{1}{f/2} + \frac{1}{f/2} = \frac{4}{f} \quad \Rightarrow \quad f_3 = \frac{f}{4} \]

- **Arrangement 4**:
- The two parts are arranged oppositely. The effective focal length \( f_4 \) can be calculated as:
\[ \frac{1}{f_4} = \frac{1}{f/2} - \frac{1}{f/2} = 0 \quad \Rightarrow \quad f_4 = \infty \]

4. **Calculating Powers**:
- For Arrangement 1 and 3, \( f = \frac{f}{4} \):
\[ P_1 = P_3 = \frac{4}{f} \]
- For Arrangement 2 and 4, \( f = \infty \):
\[ P_2 = P_4 = 0 \]

5. **Conclusion**:
The arrangements that yield the maximum optical power are Arrangement 1 and Arrangement 3, both having the same focal length and thus the same power. Therefore, the arrangement that gives maximum optical power is Arrangement 1 (or 3).

### Final Answer:
**Arrangement 1 gives the maximum optical power.**