Question

An object lies 10cm in front of a convex mirror of radius of curvature 8cm. If a convex lens is placed 4cm in front of the convex mirror, the final image coincides with the object. Find the focal length of this lens. How to calculate???

Answer 1

The focal length of this lens is `f=4` cm

Explanation:

The above figure represents the ray diagram of the phenomenon as described in the problem .

It is given that the position of the final image is same as the position of the object `O`. Thr divergent beam of light emerging from O after refraction through convex lens `(L)` on subsequent reflection on convex mirror `(M)` and refraction again through the same lens coincides with the position of the object `O`.

Hence we can say that retracing of rays occurs here as shown in the ray diagram. This retracing is only possible, if the convergent beam produced by first refraction through lens is directed to converge at the center of curvature of the convex mirror.
The object is placed at a distance of 10 cm from the mirror and the lens is placed at 4 cm distance in front of the mirror. The lens converges beam after refraction at the center of curvature of the mirror i.e. at 8 cm back of the mirror . So image distance of the object due to first refraction will be `v=(4+8)=12`cm

So for the lens

`"object distance "u=-(10-4)=-6cm`

`"image distance "v=+12cm`

Inserting these in the conjugate foci relation of lens

`1/v-1/u=1/f`, where `f` is the focal lens of the lens,

`=>1/12-1/(-6)=1/f`

`=>1/f=1/12+1/6==(1+2)/12=1/4`

`=>f=4` cm