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Difficulty: MediumReflection of Light at Plane and Curved Mirrors

A side-view convex mirror on a bus has a radius of curvature of 40 cm40\text{ cm}. If a motorcycle is located 30 cm30\text{ cm} in front of the mirror, what is the location of the image formed relative to the mirror?

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

Answer

The image is formed 12 cm12\text{ cm} behind the mirror.
For a convex mirror, the focal length is virtual, so f=−20 cmf = -20\text{ cm}. With an object distance of u=+30 cmu = +30\text{ cm}, applying the mirror equation 1f=1u+1v\frac{1}{f} = \frac{1}{u} + \frac{1}{v} gives 1v=−120−130=−112\frac{1}{v} = -\frac{1}{20} - \frac{1}{30} = -\frac{1}{12}, leading to v=−12 cmv = -12\text{ cm}. The negative sign specifies that the virtual image is located 12 cm12\text{ cm} behind the mirror.

Step-by-Step Solution

1
Determine the focal length of the mirror from its radius of curvature
f=−R2=−40 cm2=−20 cmf = -\frac{R}{2} = -\frac{40\text{ cm}}{2} = -20\text{ cm}
For spherical mirrors, focal length is half the radius of curvature. Convex mirrors have a negative focal length by sign convention.
2
Set up the mirror formula using the given object distance u=+30 cmu = +30\text{ cm}
1f=1u+1v  ⟹  −120=130+1v\frac{1}{f} = \frac{1}{u} + \frac{1}{v} \implies -\frac{1}{20} = \frac{1}{30} + \frac{1}{v}
The mirror equation relates focal length, object distance, and image distance.
3
Solve for the image distance vv
1v=−120−130=−3+260=−560=−112  ⟹  v=−12 cm\frac{1}{v} = -\frac{1}{20} - \frac{1}{30} = -\frac{3 + 2}{60} = -\frac{5}{60} = -\frac{1}{12} \implies v = -12\text{ cm}
Algebraic manipulation yields a negative image distance.
4
Interpret the physical meaning of the calculated value
The negative sign indicates a virtual image located 12 cm12\text{ cm} behind the mirror.
Under standard optical sign conventions, negative image distances correspond to virtual images formed behind the mirror.

Key Concept

Mirror equation and sign conventions for convex spherical mirrors
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Topics

Reflection of Light at Plane and Curved Mirrors

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