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Question:

(a) A point object 'O' is kept in a medium of refractive index n1 in front of a convex spherical surface of radius of curvature R which separates the second medium of refractive index n2 from the first one as shown in the figure. Draw the ray diagram showing the image formation and deduce the relationship between the object distance and the image distance in terms of n1, n2 and R. (b) When the image formed above acts as a virtual object for a concave spherical surface separating the media n2 and n1 (n2 > n1), draw this ray diagram and write the similar (similar to (a)) relation. Hence obtain the expression for the lens maker's formula.

Solution:

(a)tan∠NOM=MN/OM
tan∠NCM=MN/MC
tan∠NMI=MN/NI
For △NOC, i is the exterior angle.
i=∠NOM+∠NCM=MN/OM+MN/MC
Similarly, r=MN/MC−MN/MI
By Snell's Law, n1sin i = n2sin r
For small angles, n1i = n2r
n1(MN/OM+MN/MC) = n2(MN/MC−MN/MI)
n1/OM+n2/MI=(n2−n1)/MC
Putting OM=−u, MI=+v, MC=+R
n2/v−n1/u=(n2−n1)/R (i)
(b)For the second surface, n1/v′−n2/v=(n1−n2)/R′ (ii)
Adding (i) and (ii), n1/v′−n1/u=(n2−n1)/R−(n2−n1)/R′
1/v−1/u=(n2−n1)/n1[1/R−1/R′]