Question

A 10 mm long awl pin is placed vertically in front of a concave mirror. A 5 mm long image of the awl pin is formed at 30 cm in front of the mirror. What is the focal length of the concave mirror?
A. 30 cm
B. 20 cm
C. 40 cm
D. 60 cm

Answer 1

Hint: Here, we will proceed by first calculating the distance of the object from the pole with the help of the relation between the magnifying ratio with the image and object distance. After this, the focal length of the mirror can be calculated with the help of the relation between the image and the object distance from the pole.

Complete Step-by-Step solution:
The focal length of the concave mirror can be formed by the formula
$\dfrac{1}{f} = \dfrac{1}{v} + \dfrac{1}{u}$
Where f= focal length of the mirror
v=distance of the image from the pole
u=Distance of the object from the pole
In the above question, it is given that the image is formed in front of the mirror, so the image is real and is at a distance of 30 cm from the pole
Therefore, $v = - 30{\text{ }}cm$
Also, we are given with the image height as well as the object height.
$\therefore $ We will calculate the magnifying factor as
$
   \Rightarrow m = - \dfrac{y}{x} \\
   \Rightarrow m = - \dfrac{5}{{10}} \\
$
Also,$m = \dfrac{v}{u}$
$
  \therefore m = \dfrac{{( - 30)}}{u} = - \dfrac{1}{2} \\
   \Rightarrow u = 60{\text{ cm}} \\
$
It should be noted that the object distance is always negative, so $u = - 60{\text{ }}cm$
So, the focal length of the concave mirror is calculated by the formula
$ \Rightarrow \dfrac{1}{f} = \dfrac{1}{v} + \dfrac{1}{u}$
$
   \Rightarrow \dfrac{1}{f} = - \dfrac{1}{{30}} - \dfrac{1}{{60}} \\
   \Rightarrow \dfrac{1}{f} = - \dfrac{{90}}{{1800}} \\
   \Rightarrow f = - 20{\text{ cm}} \\
$
The focal length of the concave mirror is always negative as it is clear from the value obtained above.
So, the magnitude of the focal length of the mirror is 20 cm.
$\therefore $ Option B is the correct option.

Note- The thing that we should know about the concave mirror is that it can form both real as well as virtual images. The real image is formed on the left side of the mirror and its distance is taken as negative while the virtual image is formed on the right side of the mirror and its distance is taken as positive.