A thin plano-convex lens and a thin equi-concave lens are kept coaxially in contact as shown in the figure. Assuming both the lenses are made of glass of refractive index \(\mu\), and \(R\) is the radius of curvature of each curved surface, the focal length of the combination is :
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When combining lenses, adding their individual powers algebraically (\(P = P_1 + P_2\)) prevents reciprocal errors. A plano-convex lens has a power of \(P_1 = \frac{\mu-1}{R}\), while an equi-concave lens has a power of \(P_2 = \frac{-2(\mu-1)}{R}\). Adding them gives \(P_{\text{net}} = \frac{-(\mu-1)}{R}\), leading directly to \(F = -\frac{R}{\mu-1}\).
