Question

In an experiment a convex lens of focal length 15cm is placed coaxially on an optical bench in front of a convex mirror at a distance of 5cm from it. It is found that an object and it’s image coincide, if the object is placed at a distance of 20cm from the lens. The focal length of the convex mirror is:
A. 27.5 cm
B. 20.0 cm
C. 25.0 cm
D. 30.5 cm

Answer 1

Hint: Use Lens formula to find image distance from object distance and focal length of lens. Then subtract the distance of the convex lens from image distance. This will give a radius of curvature. Now, use the relation between radius of curvature and focal length to get focal length of the convex mirror.

Formulas used:
${\displaystyle \frac{1}{v}}-{\displaystyle \frac{1}{u}}={\displaystyle \frac{1}{f}}$ …(1)

Complete step by step answer:
Lens Formula is given by,
${\displaystyle \frac{1}{v}}-{\displaystyle \frac{1}{u}}={\displaystyle \frac{1}{f}}$ …(1)
where, v: Image distance from the lens
            u: Object distance from the lens
            f: Focal length of the lens

Given: u= -20cm
            f= 15 cm
            distance (d) of convex lens= 5cm
By substituting the values in equation.(1) we get,
${\displaystyle \frac{1}{v}}+{\displaystyle \frac{1}{20}}={\displaystyle \frac{1}{15}}$
$\Rightarrow {\displaystyle \frac{1}{v}}={\displaystyle \frac{1}{15}}-{\displaystyle \frac{1}{20}}$
$\Rightarrow {\displaystyle \frac{1}{v}}={\displaystyle \frac{5}{300}}$
$\Rightarrow v={\displaystyle \frac{300}{5}}$
$\Rightarrow v=60cm$
Thus, the image will be formed at 60cm to the right of the lens and it will be inverted.
Therefore, distance between lens and mirror will be
d = image distance – radius of curvature of convex mirror
$d=v-2f$
By substituting the values in above equation we get,
$5=60-R$
$\Rightarrow R=55$
But, the radius of curvature (R) is given by,
$R=2f$
$\Rightarrow f={\displaystyle \frac{55}{2}}$
$\Rightarrow f=27.5$
Therefore, the focal length of the convex mirror is 27.5cm.
Hence, the correct answer is option A i.e. 27.5cm.

Note:
You must take care of the sign conventions of the image, object and focal length. In this problem, we take object distance to be negative as the object is kept on the left side of the lens. If it was on the right side then the object distance would be positive.
The image formed due to the convex lens at the right side of the convex lens acts as a virtual image for the convex mirror.