The sun subtends asn angle of `(1//2)^(@)` on earth. The image of sun is obtained on the screen with the help of a convex lens of focal length 100 cm the diameter of the image obtained on the screen will be

Options

A. `18 cm`
B. `1mm`
C. `50 cm`
D. `8.73 mm`

Explanation

To solve the problem, we need to find the diameter of the image of the sun formed by a convex lens with a given focal length when the sun subtends an angle of \( \frac{1}{2}^\circ \) at the earth.

### Step-by-Step Solution:

1. **Understanding the Geometry**:
The sun subtends an angle \( \theta = \frac{1}{2}^\circ \) at the earth. This angle can be used to find the size of the image formed by the lens.

2. **Convert the Angle to Radians**:
To use the angle in calculations, we need to convert degrees to radians. The conversion formula is:
\[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \]
Therefore,
\[ \theta = \frac{1}{2} \times \frac{\pi}{180} = \frac{\pi}{360} \text{ radians} \]

3. **Using the Lens Formula**:
When the object (the sun) is at a very large distance (effectively at infinity), the image is formed at the focal point of the lens. The diameter of the image can be calculated using the formula:
\[ x = f \cdot \theta \]
where \( f \) is the focal length of the lens.

4. **Substituting the Values**:
Given that the focal length \( f = 100 \) cm, we can substitute the values:
\[ x = 100 \cdot \frac{\pi}{360} \]

5. **Calculating the Diameter**:
Now, we compute \( x \):
\[ x = 100 \cdot \frac{3.14}{360} \approx 8.73 \text{ cm} \]
Since the question asks for the diameter, we can directly use this value.

6. **Converting to Millimeters**:
To express the diameter in millimeters, we convert centimeters to millimeters:
\[ 8.73 \text{ cm} = 8.73 \times 10 = 87.3 \text{ mm} \]

### Final Answer:
The diameter of the image obtained on the screen is approximately **8.73 mm**.