曲面鏡成像公式的證明

Study

2011-11-07 2:32 am

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。

回答 (1)

天同

2011-11-07 4:48 am

✔ 最佳答案

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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