Two spherical mirrors, one convex and the other concave, each of same radius of curvature R are arranged coaxially at a distance of 2R from each other. A small circle of radius a is drawn on the convex mirror as shown in figure. What is the radii of first two images of the circle ?

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$a / 3, a / 11$

$a / 6, a / 2$

$a / 3, a / 4$

$a / 12, a / 4$

answer is A.

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Detailed Solution

Question Image
When I^st reflection takes place on concave, 2^nd on convex mirror
For concave: $u = - 2 R$ , $f = \frac{- R}{2}$
$\frac{1}{v} + \frac{1}{u} = \frac{1}{f} \Rightarrow \frac{1}{v} + \frac{1}{- 2 R} = \frac{- 2}{R}$
$\frac{1}{v} = \frac{- 2}{R} + \frac{1}{2 R} \Rightarrow v = \frac{- 2 R}{3}$
$m = - \frac{v}{u} = - (\frac{- 2 R / 3}{2 R}) = \frac{- 1}{3}$
Size of image = $- \frac{a}{3}$ (laterally inverted)
For convex:
$u = \frac{4 R}{3}, f = - \frac{R}{2}$
$\frac{1}{v} + \frac{1}{u} = \frac{1}{f} \Rightarrow \frac{1}{v} = - \frac{2}{R} - \frac{3}{4 R}$
$v = - \frac{4 R}{11}$
$m = - \frac{v}{u} = - (\frac{- 4 R / 11}{4 R / 3}) = \frac{3}{11}$
Size of image $= \frac{3}{11} \frac{a}{3} = \frac{a}{11}$ (erect) (most appropriate)