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Question: A man has a concave shaving mirror of focal length 0.2m. How far should the mirror be held from his ...

A man has a concave shaving mirror of focal length 0.2m. How far should the mirror be held from his face in order to give an image of two fold magnification?
A. 0.1 m
B. 0.2 m
C. 0.3 m
D. 0.4 m

Explanation

Solution

Given magnification is twofold, means 2. Magnification is the ratio of image distance (v) and object distance (u). Find the image distance in terms of object distance using magnification. Substitute the values of image and object distances which are in terms of u in the Mirror’s formula to get the value of object distance.

Complete step by step answer:
We are given that a man has a concave shaving mirror of focal length 0.2m.
We have to find the object distance when the magnification can be made twice.
The focal length of the given concave mirror is 0.2 m. Given a mirror is concave so consider the values as negative.
Magnification is the ratio of v and u.
m=−vum = \dfrac{{ - v}}{u}
Magnification= 2.
m=−vu 2=−vu v=−2u  m = \dfrac{{ - v}}{u} \\\ 2 = \dfrac{{ - v}}{u} \\\ v = - 2u \\\
We got the image distance in terms of object distance (u).
We know that the mirror’s formula is 1f=1v+1u\dfrac{1}{f} = \dfrac{1}{v} + \dfrac{1}{u}, where f is the focal length, u and v are object and image distances respectively.
1f=1v+1u f=−0.2m v=−2u ⇒1−0.2=1−2u+1u ⇒−102=−1+22u ⇒−5=12u ⇒2u=1−5 ⇒u=12×5=−110 ⇒u=−0.1m  \dfrac{1}{f} = \dfrac{1}{v} + \dfrac{1}{u} \\\ f = - 0.2m \\\ v = - 2u \\\ \Rightarrow \dfrac{1}{{ - 0.2}} = \dfrac{1}{{ - 2u}} + \dfrac{1}{u} \\\ \Rightarrow \dfrac{{ - 10}}{2} = \dfrac{{ - 1 + 2}}{{2u}} \\\ \Rightarrow - 5 = \dfrac{1}{{2u}} \\\ \Rightarrow 2u = \dfrac{1}{{ - 5}} \\\ \Rightarrow u = \dfrac{1}{{2 \times 5}} = - \dfrac{1}{{10}} \\\ \Rightarrow u = - 0.1m \\\
Therefore, the object distance is 0.1 m.
The mirror must be held 0.1 m from his face in order to give an image of two fold magnification.
The correct option is Option A, 0.1 m.

Note: The distance between the object and the pole of the mirror is called the object distance (u). The distance between the image and the pole of the mirror is called the Image distance (v). The distance between the Principal focus and pole of the mirror is called Focal Length (f).