Question

(a) Draw a ray diagram to show the image formation by a combination of two thin convex
lenses in contact . Obtain the expression for the power of this combination in terms of the
focal lengths of the lenses.
(b) A ray of light passing from air through an equilateral glass prism undergoes minimum
deviation when the angle of incidence is `(3)/(4)^(th)` of the angle of prism . Calculate the speed of ltbRgt light in the prism.

Answer 1

(a) Consider lenes A and B with focal length `f_(1) and f_(2)` placed in contact . Object is placed at O
beyond focus of A . First lens produces image at I, (its real) and serve as virtual object for
lens B producing image I. As the lenses are lenses are thin , so optc centres of lenses are closer to
each other . Let this centre point be denoted by P.
Image formation by A `(1)/(v_(1)) - (1)/(u) = (1) /(f_(1)) ` and ...(i)
Image formation by B `(1)/(v) - (1)/(v_(1)) = (1)/(f_(2)) ` ...(ii)
Adding (i) and (ii) we get , ` (1)/(v) - (1)/(u) = (1)/(f_(1)) + (1)/(f_(2))` ...(iii)

If two lenses system is taken as one lens equivalent system then, eq . (iii) can be ` (1)/(v)- (1)/(u) = (1)/(f)`
where, ` (1)/(f) = (1) /(f_(1)) + (1)/(f_(2))`. Then is valid for any number of lens `f_(1) , f_(2) , f_(3)`... are the focal lengths
of lenses in contact
`(1)/(f) = (1)/(f_(1) ) + (1)/(f_(2)) + (1)/(f_(3)) +.....u`
` P = P _(1) + P_(2) + P _(3) + ...`
(b) Given, ` i = (3)/(4) A`
Also, `mu = ("sin" [(A+ delta m)/(2)])/("sin" (A)/(2)) or , delta _(m) to ` Angle of min . deviation .
or `(C_(1))/(C_(2)) = ("sin" [(A+ delta m)/(2)])/("sin" (A)/(2)) or , delta _(m) = 2i - A = 2xx (3A)/(4) - A = (A)/(2)`
`delta _(m) = (60^(@))/(2) = 30^(@)`
`C_(2) = (3xx10^(8))/(sqrt(2)) ms^(-1) = 1.5 sqrt(2)ms^(-1) xx10^(8)`
`rArr C_(2) 2.12 xx10^(8)` ms^(-1)`.