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標題:

曲面鏡成像公式的證明

發問:

如何用數學方法(如相似三角形)證明曲面鏡(包括凸面鏡和凹面鏡)中 1/f = 1/u + 1/v(即1 / 焦距 = 1 / 物距 + 1 / 像距)？請留意，問題是曲面鏡，不是透鏡。

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Refer to the diagram in the following web-page http://www.physicstutorials.org/home/optics/reflection-of-light/curved-mirrors/mirror-equations-of-curved-mirrors Let Ho be the object height Hi be the image height u be the object distance v be the image distance f be the focal length of the mirror k is the distance between object and focus By similar trianges (th two shaded triangles), we have Ho/k = Hi/f i.e. Hi/Ho =f/k ------------------- (1) But Hi/Ho = magnification = v/u ( simialr triangle relationship) ----------- (2) Equating (1) and (2): f/k = v/u but f + k = u i.e. k = u - f hence, f/(u-f) = v/u fu = uv - fv divide both sides by uvf 1/v = 1/f - 1/u hence, 1/u + 1/v = 1/f

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