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Difficulty: MediumReflection of Light at Plane and Curved Mirrors

An object placed 30 cm30\text{ cm} in front of a concave mirror forms a sharp real image on a screen located 60 cm60\text{ cm} in front of the mirror. If the concave mirror is replaced by a convex mirror having the same magnitude of focal length, where will the image of the object be formed?

  1. 12 cm12\text{ cm} behind the mirrorAnswer
  2. B
    60 cm60\text{ cm} in front of the mirror
  3. C
    60 cm60\text{ cm} behind the mirror
  4. D
    30 cm30\text{ cm} behind the mirror

Answer

The image is formed 12 cm12\text{ cm} behind the mirror.
First, calculate the focal length of the concave mirror using the mirror equation 1f=1u+1v\frac{1}{f} = \frac{1}{u} + \frac{1}{v}. Substituting u=+30 cmu = +30\text{ cm} and v=+60 cmv = +60\text{ cm} gives f=20 cmf = 20\text{ cm}. Next, for a convex mirror of the same focal length magnitude, the focal length is negative (f=−20 cmf = -20\text{ cm}). Substituting u=+30 cmu = +30\text{ cm} into the mirror equation gives 1−20=130+1v′\frac{1}{-20} = \frac{1}{30} + \frac{1}{v'}, which yields v′=−12 cmv' = -12\text{ cm}. The negative sign confirms the image is virtual and located 12 cm12\text{ cm} behind the mirror.

Step-by-Step Solution

1
Determine the focal length of the concave mirror using the mirror formula.
1f=1u+1v=130+160=360  ⟹  f=20 cm\frac{1}{f} = \frac{1}{u} + \frac{1}{v} = \frac{1}{30} + \frac{1}{60} = \frac{3}{60} \implies f = 20\text{ cm}.
Both object distance u=+30 cmu = +30\text{ cm} and real image distance v=+60 cmv = +60\text{ cm} are positive in front of the mirror.
2
Assign the appropriate sign for the convex mirror's focal length.
f′=−20 cmf' = -20\text{ cm}.
Convex mirrors have a virtual focus behind the mirror surface, requiring a negative sign convention.
3
Calculate the image position for the object in front of the convex mirror.
1−20=130+1v′  ⟹  1v′=−120−130=−560=−112  ⟹  v′=−12 cm\frac{1}{-20} = \frac{1}{30} + \frac{1}{v'} \implies \frac{1}{v'} = -\frac{1}{20} - \frac{1}{30} = -\frac{5}{60} = -\frac{1}{12} \implies v' = -12\text{ cm}.
Solving for v′v' yields a negative value, indicating a virtual image formed 12 cm12\text{ cm} behind the convex mirror.

Key Concept

Reflection at spherical mirrors and application of mirror sign conventions
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Topics

Reflection of Light at Plane and Curved Mirrors

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