(i) Consider a thin lens placed between

(i) Consider a thin lens placed between a source (S) and an observer (O). Let the thickness of the lens vary as 2 0 () – α = b wb w, where b is the verticle

distance from the pole. w0 is a constant. Using Fermat’s principle i.e. the time of transit for a ray between the source and observer is an extremum, find the

condition that all paraxial rays starting from the source will converge at a point O on the axis. Find the focal length.

(ii) A gravitational lens may be assumed to have a varying width of the form w(b) = k1 ln (k2/b) = k1 ln (k2/bmin). Show that an observer will see an image of a

point object as a ring about the centre of the lens with an angular radius.

β = √(n-1)k1 u/v / u + v


(i) Time taken by the ray to travel from S to P1 is = t1 = √u2 + b2/c

Time taken by the ray to travel from P1 to O is = t2 = v/c (1+ ½ b2/v2)

Time taken to travel through the lens is = tl = (n – 1)w(b)/c

Total time is = t = 1/c (u + v + ½ b2/D + (n – 1)(wo + b2/a))

D is the focal length as 1/D = 1/u + 1/v

(ii) Differentiating the expression for time, we get

β = b/v = √(n-1)k1u/(u + v)v


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