Three Mirrors, One Exhibition Stand
A science club has three teams preparing models for the district exhibition. Team A is given a concave mirror of radius of curvature 20 cm and must fix it on a stand so that a student looking into it sees her face erect and magnified three times. Team B has a convex mirror of focal length 15 cm and plans to fix it on the handlebar of a model car as a rear-view mirror. Team C must throw a clear image of a burning candle onto a screen fixed 40 cm in front of the same concave mirror used by Team A; the screen cannot be moved, so Team C must slide the mirror along the table until the image is sharp. The teacher asks every team to write a short justification of the mirror chosen, the exact position used, and the nature of the image, because the judge will ask why other positions on the table do not work.
1. Calculate where Team A must place the mirror relative to the face, and state the nature and position of the image. Use the mirror formula with a clear sign convention.
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For an erect, magnified image the face must lie between the pole and the focus, giving a virtual image behind the mirror. Concave mirror: R = 20 cm, so f = -10 cm (Cartesian convention: distances to the right positive; object on the left, so u is negative).
Magnification m = +3 = -v/u → v = -3u.
Substituting in 1/v + 1/u = 1/f: 1/(-3u) + 1/u = -1/10 (-1 + 3)/(3u) = -1/10 2/(3u) = -1/10 3u = -20 → u = -6.7 cm, and v = -3(-6.7) = +20 cm.
So the mirror is fixed about 6.7 cm from the face. The image is 20 cm behind the mirror, virtual, erect and three times as large. Check: 6.7 cm lies between f = 10 cm and the pole, exactly the region that gives an erect virtual image; if the mirror is pushed beyond 10 cm the image turns real and inverted and cannot be seen in the mirror by the student.
2. Team B wants to know whether their convex mirror can do Team A's job, and why a convex mirror is nevertheless the correct choice for a rear-view mirror. Advise them.
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No position of a real object gives a magnified image in a convex mirror. Rays from every point diverge after reflection, so the image is always virtual, erect and diminished, and it always lies behind the mirror between the pole and the focus. Team B therefore cannot get a three-times magnified view; to magnify they would need Team A's concave mirror with the object inside the focus.
For a rear-view mirror the diminished image is the whole point. The reflecting surface bulges outward, so rays arriving from a wide angle behind the vehicle are reflected into the driver's eye, giving a much larger field of view than a plane mirror of the same size. The price is that objects appear smaller and therefore farther away than they really are, which drivers must allow for.
The judgement is to choose the mirror by purpose, not by habit: wide field → convex; magnification of a nearby face → concave with object inside f; true size and no distortion → plane.
3. A student argues, "A concave mirror always gives a real image, so Team A's mirror must actually be convex." Evaluate this claim using the three teams' data.
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The claim is false, and Team A's own observation disproves it: an erect, magnified image seen in the mirror is possible only with a concave mirror and an object inside its focus, so the mirror is certainly concave.
The nature of the image depends on where the object is, not on the mirror alone. With f = 10 cm:
- u > 20 cm: real, inverted, diminished.
- u = 20 cm: real, inverted, same size.
- 10 cm < u < 20 cm: real, inverted, magnified (this is Team C's case at u = 13.3 cm, v = -40 cm).
- u = 10 cm: reflected rays are parallel; no image on a screen.
- u < 10 cm: virtual, erect, magnified - Team A's case.
A convex mirror, by contrast, gives a virtual image for every real object. A quick practical check settles it: hold the mirror in sunlight; the concave mirror forms a tiny bright spot on paper at 10 cm, the convex mirror never does.