A biconvex lens of focal length f forms a circular image of radius r of sun is focal plane. Then, which option is correct?

Options

A. `pir^(2)propf`
B. `pir^(2)propf^(2)`
C. If lower half part is covered by black sheet, then area of the image is equal to `pir^(2)//2`
D. If f is double, intensity will increase

Explanation

To solve the problem, we need to analyze the relationship between the radius of the circular image formed by a biconvex lens and its focal length. Let's go through the solution step by step.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a biconvex lens with a focal length \( f \).
- The lens forms a circular image of the sun at its focal plane, which means the image is formed at a distance \( f \) from the lens.

2. **Drawing the Diagram**:
- Draw the biconvex lens and its principal axis.
- Mark the focal point \( F \) on the principal axis at a distance \( f \) from the lens.
- The rays of light coming from the sun can be considered parallel due to the large distance of the sun.

3. **Identifying the Image Formation**:
- Light rays from the sun pass through the lens and converge at the focal plane.
- The radius of the circular image formed at the focal plane is given as \( r \).

4. **Using Trigonometry**:
- Consider one of the rays passing through the center of the lens and forming an angle \( \theta \) with the principal axis.
- The relationship between the radius \( r \) of the image and the focal length \( f \) can be expressed using the tangent function:
\[ \tan(\theta) = \frac{r}{f} \]
- Rearranging gives:
\[ r = f \cdot \tan(\theta) \]

5. **Calculating the Area**:
- The area \( A \) of the circular image is given by:
\[ A = \pi r^2 \]
- Substituting \( r = f \cdot \tan(\theta) \) into the area formula:
\[ A = \pi (f \cdot \tan(\theta))^2 = \pi f^2 \tan^2(\theta) \]

6. **Determining the Proportionality**:
- From the area expression, we see that:
\[ A \propto f^2 \tan^2(\theta) \]
- Since \( \tan(\theta) \) is a constant for a given setup, we can conclude that:
\[ \pi r^2 \propto f^2 \]

7. **Final Conclusion**:
- Therefore, the correct relationship is:
\[ \pi r^2 \text{ is proportional to } f^2 \]
- Thus, the correct option is **B**: \( \pi r^2 \propto f^2 \).