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Zorluk: ZorReflection of Light at Plane and Curved Mirrors

A convex security mirror installed in a store has a focal length of magnitude 20 cm20\text{ cm}. If an upright image of a customer is formed with a linear magnification of 0.250.25, at what distance from the mirror is the customer standing?

  1. 60 cm60\text{ cm}Cevap
  2. B
    100 cm100\text{ cm}
  3. C
    15 cm15\text{ cm}
  4. D
    25 cm25\text{ cm}

Cevap

The customer is standing at a distance of 60 cm60\text{ cm} from the convex mirror.
By sign convention, a convex mirror has a negative focal length (f=−20 cmf = -20\text{ cm}). An upright image formed by a mirror has a positive magnification m=+0.25m = +0.25. Using m=−v/um = -v/u, we get v=−u/4v = -u/4. Substituting these into the mirror equation 1f=1u+1v\frac{1}{f} = \frac{1}{u} + \frac{1}{v} gives −120=1u−4u=−3u-\frac{1}{20} = \frac{1}{u} - \frac{4}{u} = -\frac{3}{u}, which yields u=60 cmu = 60\text{ cm}.

Adım Adım Çözüm

1
Identify the optical properties and apply sign conventions for a convex mirror
Focal length f=−20 cmf = -20\text{ cm} (convex mirror focal length is virtual/negative). Magnification m=+0.25m = +0.25 (upright image).
Convex mirrors always form virtual, upright, diminished images behind the mirror.
2
Relate image distance vv to object distance uu using the linear magnification formula
Since m=−v/u=1/4m = -v/u = 1/4, we obtain v=−u/4v = -u/4.
The negative sign in the magnification definition accounts for virtual image distance.
3
Substitute f=−20 cmf = -20\text{ cm} and v=−u/4v = -u/4 into the mirror formula 1f=1u+1v\frac{1}{f} = \frac{1}{u} + \frac{1}{v}
1−20=1u+1−u/4  ⟹  −120=1u−4u=−3u\frac{1}{-20} = \frac{1}{u} + \frac{1}{-u/4} \implies -\frac{1}{20} = \frac{1}{u} - \frac{4}{u} = -\frac{3}{u}.
Combines fractions with a common denominator uu to solve for the unknown object distance.
4
Solve for the object distance uu
120=3u  ⟹  u=60 cm\frac{1}{20} = \frac{3}{u} \implies u = 60\text{ cm}.
Cross-multiplying yields the real object distance in front of the mirror.

Anahtar Kavram

Reflection at Convex Mirrors and Optical Sign Conventions
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Konular

Reflection of Light at Plane and Curved Mirrors

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