Question

The sun makes an angle of 0.5\(^\circ\) on earth's surface. Its image is made with convex lens of focal length 50 cm. The diameter of image will be

• A. 1.0 cm
• B. 5.0 cm
• C. 0.76 cm
• D. 0.43 cm

Hint:

A common mistake in problems involving angles in physics formulas (like in optics, simple harmonic motion, or circular motion) is forgetting to convert the angle from degrees to radians. Always check the requirements of the formula. For the small angle approximation (\(\tan \theta \approx \sin \theta \approx \theta\)), the angle \(\theta\) must be in radians.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
When a lens is used to view a very distant object like the sun, the parallel rays of light from the object converge to form an image at the focal plane of the lens. The size of this image is related to the focal length of the lens and the angular size of the object.

Step 2: Key Formula or Approach:
For small angles, the relationship between the angular size (\(\theta\)), the image diameter (\(d\)), and the focal length (\(f\)) can be approximated by: \[ \theta \approx \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{d}{f} \] This formula requires the angle \(\theta\) to be in radians.
The conversion from degrees to radians is: \( \text{angle in radians} = \text{angle in degrees} \times \frac{\pi}{180} \).

Step 3: Detailed Explanation:
We are given:
Angular size of the sun, \(\theta = 0.5^\circ\).
Focal length of the convex lens, \(f = 50\) cm.
First, we must convert the angular size from degrees to radians: \[ \theta_{\text{rad}} = 0.5 \times \frac{\pi}{180} \approx 0.5 \times \frac{3.14159}{180} \approx 0.008727 \text{ radians} \] Now, we can use the formula \(d = f \times \theta_{\text{rad}}\) to find the diameter of the image (\(d\)): \[ d = 50 \text{ cm} \times 0.008727 \] \[ d \approx 0.43635 \text{ cm} \]

Step 4: Final Answer:
The diameter of the image is approximately 0.43 cm. Comparing this with the given options, option (D) is the closest match.