The figure shows three arrangements, P, Q and R, of two lenses each. The radii of curvature of all the curved surfaces are the same. The ratio of the equivalent focal lengths of the combinations P, Q and R is .

Question

Accepted Answer

(1) 1:1:1. Given: three arrangements, each of two thin lenses in contact, all curved surfaces of the same radius R and all of the same material. Idea: each lens in the figure is plano-convex with one face of radius R and one plane face, so by the lens maker's formula each has the same power P = (n-1)/R. Reversing a lens does not change that: it swaps R₁ and R₂ and their signs at the same time, leaving (1/(R₁)-1/(R₂)) alone. For thin lenses in contact the powers simply add, P_eq = P + P = (2(n-1))/R, in every arrangement. So every combination has the same equivalent focal length f_eq = R/(2(n-1)); how the lenses are turned or ordered makes no difference. The ratio is therefore 1:1:1.

The figure shows three arrangements, P, Q and R, of two lenses each. The radii of curvature of all the curved surfaces are the same. The ratio of the equivalent focal lengths of the combinations P, Q and R is ______.

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