A boy stands in front of a mirror at a distance of 40 cm from it. He sees his erect image whose height is ⅕ th of his real height. What is the nature of the mirror? Calculate its focal length.
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A boy stands in front of a mirror at a distance of 40 cm from it. He sees his erect image whose height is ⅕ th of his real height. What is the nature of the mirror? Calculate its focal length.
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Explanation:
Only a convex mirror can produce an erect, virtual and diminished image of an object because in question it is given height of image is 51th of real height.
u−v=M<1
⟹v<−u
⟹v1>−u1
v1+u1=f1
⟹f1>−u1+u1>0
⟹f>0
⟹ Convex mirror
The nature mirror is a convex mirror and the focal length of the convex mirror is 10cm
Given:
Object distance = - 40cm
m = 1/5
To Find:
The nature of the mirror and the focal length of the mirror
Solution:
Since the convex mirror produces images that are diminished and erect from the size of the object, the given mirror is a convex mirror.
The image is 1/5th of the boy's height. Therefore,
m = -v/u
m = (v/40)
1/5 = v/40
v = 8cm
To find the focal length,
[tex]\frac{1}{f} = \frac{1}{v} + \frac{1}{u}[/tex]
[tex]\frac{1}{f} = \frac{1}{8} - \frac{1}{40} \\\frac{1}{f} = \frac{40-8}{320} \\\frac{1}{f} = \frac{32}{320}\\ \frac{1}{f} = \frac{1}{10}\\[/tex]
f = 10cm
Therefore, the focal length of the convex mirror is 10cm.
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