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A rod of length L lies along the principal axis of a concave mirror of focal length f. The near end of the rod is at a distance L from the mirror (with L > f). Find the length of the image of the rod formed by the mirror.

1. L * f² / ((L - f)(2L - f))
2. L² * f / ((L - f)(2L - f))
3. 2L * f² / ((L - f)(2L - f))
4. 2L² * f / ((L - f)(2L - f))

Correct answer: L * f² / ((L - f)(2L - f))

Solution

Using the mirror formula with real-positive sign convention for a concave mirror: 1/v + 1/u = 1/f. Near end at u1 = L: v1 = fL/(L - f). Far end at u2 = 2L: v2 = 2fL/(2L - f). Image length = v1 - v2 = fL/(L-f) - 2fL/(2L-f) = fL[(2L-f) - 2(L-f)] / [(L-f)(2L-f)] = fL[f] / [(L-f)(2L-f)] = Lf² / [(L-f)(2L-f)].

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