# NCERT Physics Class 12 Exemplar Ch 9 Ray Optics and Optical Instruments Part 5

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Q 32. (i) Consider a thin lens placed between a source (S) and an observer (O) (Fig.). Let the thickness of the lens vary as, where b is the vertical distance from the pole. is a constant. Using Fermat’s principle i.e. the time of transitfor 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

Show that an observer will see an image of a point object as a ring about the center of the lens with an angular radius

Ans:

(i) The time required to travel from S to is

The time required to travel from to O is

The time required to travel through the lens is

where *n* is the refractive index.

Thus the total time is

Then

Fermet’s principle gives

Thus a convergent lens is formed if . This is independent of b and hence all paraxial rays from S will converge at O (i.e. for rays

Since , the focal length is D.

ii) In this case

Thus all rays passing at a height *b* shall contribute to the image. The ray paths make an angle