Question

A small convex mirror is placed 60 cm from the pole and on the axis of a large concave mirror of radius of curvature 200 cm.The position of the convex mirror is such that a real image of distant object is formed on the plane of a hole drilled through the concave mirror at its pole. Calculate ‘R’ of convex mirror and height of the real image if distant object forms 0.5 degree angle at the pole of the concave mirror.

Answer 1

Let the first image formed by the concave mirror is at v,

focal length of the concave mirror $f=\frac{R}{2}$

$\frac{1}{v}+\frac{1}{u}=\frac{1}{f}$

As object is at the long distance, hence $u=\infty$

$v=\frac{-200}{2}=-100cm$

distance between the concave and convex lens is 60cm

Hence first image of the object will act as the object for the convex mirror,

$u=100-60=40cm$

$\frac{1}{v}+\frac{1}{40}=\frac{1}{f_1}$

let final image is at the position x from the concave lens,

$\tan 0.5=\frac{h_i}{x}$

$x =\frac{h_i}{\tan0.5}$

$\tan 0.5\sim0.5$

Hence $x =2h_i$

$h_i=\frac{x}{2}$

as comparison to 60 cm, x will be negligible,

$\frac{}{}$ $\frac{1}{-60}+\frac{1}{40}=\frac{1}{f_1}$

$\Rightarrow \frac{1}{f_1}=\frac{1}{120}$

hence focal length of the convex mirror$120cm$ hence radius of the concave mirror is 240cm