A screen is placed 90cm from an object. The image of the object on the screen is formed by a convex lens at two different locations separated by 20cm. Determine the focal length of the lens.
The focal length of the convex lens is 21.4 cm (approximately 21.39 cm).
Explanation
This is a displacement method problem from Ray Optics. When a convex lens forms a real image on a screen, and the distance between object and screen (D) is greater than 4f, there exist two positions of the lens that produce sharp images. The separation between these positions is given as 'd'. The focal length can be calculated using the formula: f = (D² - d²)/(4D). This formula is derived from the lens formula and the principle of reversibility of light, where at one position, the object distance equals the image distance at the other position.
Solution Steps
Step 1: Identify given values: Distance between object and screen, D = 90 cm; Separation between two lens positions, d = 20 cm.
Step 2: Apply the displacement method formula: f = (D² - d²)/(4D)
Step 3: Substitute values: f = (90² - 20²)/() = (8100 - 400)/360
Step 4: Calculate: f = 7700/360 = 21.39 cm 21.4 cm