Refraction at Concave Surface

Let O be a point object in a medium of refractive index separated from a medium of refraction index by a concave refracting surface of radius of curvature R. P is the pole of the surface and OPQ is its axis.

Here, . The image appears to be formed at I. Considering the area of the image-object Δ OAI,

Δ OAI = Δ OAC – Δ IAC

½ AO.AI sin (i – r) = ½ AO.AC sin i – ½ AI.AC sin r

Taking, AO = m, AI = n and AC = l

½ m.n sin (i – r) = ½ m.l sin i – ½ nl sin r

mn(sin i cos r – cos i sin r) = ml sin i – nl sin r

From Snell’s law, sin i = sin r



Dividing by mnl



For paraxial rays, m = PO = u, n = PI = v, l = PC = R and for small angles cos i = cos r = 1



Applying the sign convention, u is –ve, v is –ve and R is –ve



If the first medium is air and the second medium has a refractive index , then = 1, and = .

∴ μ/v – 1/μ = (μ-1)/R                                      (iii)

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