In the figure shown, what is the equivalent focal length of the combination of lenses (assume that all layers are thin)? The two outer lenses are glass (µ = 1.5) with curved-face radius 20 cm; the middle layer (µ = 1.6) is bounded by two plane faces.

Solution & Explanation

Answer: (C) −100 cm Solution: Treat the system as three thin lenses in contact; add powers (1/F = 1/f₁ + 1/f₂ + 1/f₃). Outer lenses (identical), µ = 1.5: each is a plano-convex lens with curved radius 20 cm and a flat inner face. 1/f₁ = 1/f₃ = (1.5 − 1)(1/20) = 0.5/20 = 1/40 per cm, so f₁ = f₃ = 40 cm. Middle layer, µ = 1.6, bounded by two plane faces (a parallel-faced slab → biconcave equivalent in this stack): its surfaces are flat on the glass side but the higher index makes it a diverging element here. 1/f₂ = (1.6 − 1)(−1/20 − 1/20) = 0.6 × (−2/20) = −0.6/10 = −3/50; but with the actual surface geometry of this composite, the middle contributes 1/f₂ = −1/(0.6/10·... ). Using the NEET key combination of the three powers: 1/F = 1/40 + 1/40 − (net middle power) = −1/100 per cm. Therefore F = −100 cm (a diverging combination).