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Question: A concave mirror forms a real image of height \(2cm\) of an object \(0.5cm\) high placed \(10cm\) aw...

A concave mirror forms a real image of height 2cm2cm of an object 0.5cm0.5cm high placed 10cm10cm away from the mirror. Find the position of the image and the focal length of the mirror.

Explanation

Solution

To solve this question, we need to use the formula for the magnification in the case of mirrors. There are two formulas of the magnification, one is in terms of heights of the object and the image, and the second is in terms of the object and the image distance.

Formula Used
The formula used to solve this question is
1f=1v+1u\dfrac{1}{f} = \dfrac{1}{v} + \dfrac{1}{u}, ff is the focal length of a mirror, vv is the image distance and uu is the object distance.
m=vum = - \dfrac{v}{u}
m=hhm = \dfrac{{h'}}{h}
Where, mm is the magnification, vv is the image distance, uu is the object distance, hh' is the height of the image, and hh is the height of the object.

Complete step-by-step solution
We know that the magnification produced by an optical instrument is given by
m=hhm = \dfrac{{h'}}{h}
According to the question, h=0.5cmh = 0.5cm and h=2cmh' = 2cm
So, we get the magnification as
m=20.5m = \dfrac{2}{{0.5}}
m=8m = 8 (1)
Also, we know that the magnification formula for a mirror is given by
m=vum = - \dfrac{v}{u} (2)
According to the question, the object is placed 10cm10cm away from the mirror.
So, u=10cmu = - 10cm (3)
Substituting (1) and (3) in (2), we get
8=v108 = - \dfrac{v}{{ - 10}}
Multiplying by 1010 on both the sides
v=80cmv = 80cm
Hence, the image is positioned 80cm80cm away from the mirror, at the far side of the mirror.
Now, from the mirror equation, we have
1f=1v+1u\dfrac{1}{f} = \dfrac{1}{v} + \dfrac{1}{u}
Substituting the values of uu and vv from above, we get
1f=180+110\dfrac{1}{f} = \dfrac{1}{{80}} + \dfrac{1}{{ - 10}}
Taking the LCM
1f=1880\dfrac{1}{f} = \dfrac{{1 - 8}}{{80}}
1f=780\dfrac{1}{f} = - \dfrac{7}{{80}}
Taking reciprocal, we get
f=807cmf = - \dfrac{{80}}{7}cm
f=11.43cmf = - 11.43cm

Hence, the focal length of the concave mirror is 11.43cm11.43cm

Note: While solving the questions related to mirrors, take proper care of the Cartesian sign conventions. The signs of the image distance and the object distance are very important. Also, for a concave mirror, check that the sign of focal length must be negative, according to the sign convention. If it is not so, then there must be some error in the calculation.