A point object is placed at a distance of 12 cm on the axis of a convex lens of focal length 10 cm. On the other side of the lens, a convex mirror is placed at the distance of 10 cm from the lens such that the image formed by the combination coincides with the object itself. What is the focal length of the mirror?

Solution

For refraction at convex lens, we have

Object distance, u = –12 cm
Image distance, v = ?
Focal length, f = + 10 cm

Using lens formula, we have

\frac{1}{v} - \frac{1}{(- 12)} = \frac{1}{10}

i.e.,,

v = + 60 cm

Thus, convex lens will form the image I₁, at a distance of 60 cm behind the lens.
As the mirror is at a distance of 10 cm from the lens, image I₁ will be at a distance of (60 – 10) = 50 cm from the mirror, i.e., MI₁ = 50 cm.

Now, as the final image I₂ is formed at the object itself, the rays after reflection from the mirror retraces its path, i.e., the rays on the mirror are incident normally, i.e., I₁ is the centre of the mirror so that
$R = {MI}_{1} = + 50 cm$

and $f = \frac{R}{2} = \frac{50}{2} = 25 cm .$

which is the focal length of the mirror.

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