An object is placed at a distance of 15 cm from a convex lens of focal length 10 cm. On the other side of the lens, a convex mirror is placed such that its distance, from the lens, equals the focal length of the lens. The image formed by this combination is observed to coincide with the object itself. Find the focal length of the convex mirror.

Solution

In the case of convex lens,

Object distance, u = -15 cm
Focal length, f = 10 cm

Now, using the lens formula,

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

$\frac{1}{v} = \frac{1}{f} + \frac{1}{u}$
   $= \frac{1}{10} + \frac{1}{- 15}$

= $\frac{3 - 2}{30} = \frac{1}{30}$

i.e., the image is formed at a distance of 30 cm.

For convex mirror,

Object distance, $u = 30 - 10 = 20 cm$
Image distance, $v = 10 + 15 = 25 cm$

Now, using mirror formula,

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

   $= \frac{1}{25} + \frac{1}{- 20}$

   $= \frac{4 - 5}{100} = \frac{- 1}{100}$
i.e., $f = - 100 cm .$

Therefore, focal length of the convex mirror is 100 cm.

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