Question

When the object and the screen are 90 cm apart, it is observed that a clear image is formed on the screen when a convex lens is placed at two positions separated by 30 cm between the object and the screen. The focal length of the lens is: 

• A. 21.4 cm
• B. 20 cm
• C. 30 cm
• D. 30.8 cm

Hint:

For a convex lens forming a clear image at two positions, the focal length is calculated using: \[ f = \frac{d^2 - x^2}{4d} \] where \( d \) is the total object-screen distance, and \( x \) is the lens separation.

Answer 1

The Correct Option is B

Step 1: Use the Lens Formula The given data states that the object and screen are 90 cm apart, and the convex lens can be placed at two positions separated by 30 cm. Let \( d = 90 \) cm be the distance between the object and the screen, and let \( x = 30 \) cm be the separation between the two lens positions.

 Step 2: Use the Lens Formula for Two Positions The focal length of the convex lens is given by: \[ f = \frac{d^2 - x^2}{4d} \] Substituting the given values: \[ f = \frac{90^2 - 30^2}{4 \times 90} \] \[ f = \frac{8100 - 900}{360} \] \[ f = \frac{7200}{360} = 20 \text{ cm} \] Thus, the correct answer is: \[ \mathbf{20 \text{ cm}} \]