A convex lens forms a real and inverted image of a needle at a distance of 50 cm from it. Where is the needle placed in front of the convex lens if the image is equal to the size of the object? Also, find the power of the lens.
The needle is placed at a distance of 50 cm from the convex lens. The power of the lens is +4 D.
Explanation
This question tests understanding of image formation by convex lenses. From Table 9.4 in the textbook, when an object is placed 'At 2F₁', the image is formed 'At 2F₂' with 'Same size' and is 'Real and inverted'. This matches the given conditions. Since the image distance equals 50 cm, and for same-size real image, object distance must equal image distance (both at 2F), the object is also at 50 cm. The focal length becomes half of this distance (25 cm), and power is calculated using P = 1/f (in metres).
Solution Steps
Step 1: Identify the condition - The image is real, inverted, and equal to the size of the object. From Table 9.4, this occurs when the object is placed at 2F₁, and the image is formed at 2F₂.
Step 2: Determine object distance - Since image is formed at 2F₂ at distance 50 cm, and object at 2F₁ must be at same distance from lens. Therefore, object distance u = -50 cm (negative by sign convention).
Step 3: Find focal length - At 2F position, distance = 2f. So 2f = 50 cm, giving f = 25 cm = 0.25 m.
Step 4: Calculate power - Power P = 1/f = 1/0.25 = +4 D (positive for convex lens).