Question

Two thin lenses of power +3.5 D and -2.5 D are placed in contact. Find the focal length of the lens combination ?

• A. 2 m
• B. 4 m
• C. 1 m
• D. 0.5 m

Hint:

1. For lenses in contact, total power \(P_{total} = P_1 + P_2\). \(P_1 = +3.5\) D, \(P_2 = -2.5\) D. \(P_{total} = +3.5 + (-2.5) = 3.5 - 2.5 = +1.0\) D. 2. Focal length \(f\) (in meters) = \(1 / P\) (where P is in Diopters). \(f_{total} = 1 / P_{total} = 1 / (+1.0) = 1\) meter. The positive focal length means the combination is converging.

Answer 1

The Correct Option is C

Concept: (A) When thin lenses are placed in contact, the power of the combination (\(P_{comb}\)) is the algebraic sum of the powers of the individual lenses (\(P_1, P_2, \dots\)). \[ P_{comb} = P_1 + P_2 \] (B) The power (\(P\)) of a lens is the reciprocal of its focal length (\(f\)) in meters. \[ P = \frac{1}{f (\text{in meters})} \] Conversely, \(f (\text{in meters}) = \frac{1}{P}\). The unit of power is Diopter (D). Step 1: Identify the powers of the individual lenses Let \(P_1\) be the power of the first lens and \(P_2\) be the power of the second lens. Given:
\(P_1 = +3.5 \text{ D}\)
\(P_2 = -2.5 \text{ D}\) Step 2: Calculate the power of the lens combination (\(P_{comb}\)) \[ P_{comb} = P_1 + P_2 \] \[ P_{comb} = (+3.5 \text{ D}) + (-2.5 \text{ D}) \] \[ P_{comb} = 3.5 - 2.5 \text{ D} \] \[ P_{comb} = +1.0 \text{ D} \] The power of the lens combination is +1.0 Diopter. Step 3: Calculate the focal length of the lens combination (\(f_{comb}\)) The focal length is the reciprocal of the power (when focal length is in meters). \[ f_{comb} = \frac{1}{P_{comb}} \] \[ f_{comb} = \frac{1}{+1.0 \text{ D}} \] \[ f_{comb} = 1 \text{ meter (m)} \] Since the power is positive, the combination acts as a convex (converging) lens with a focal length of 1 meter. This matches option (3).