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Here's an item that stumped me for a while. It's based on a question
from the April 2003 Physics Challenges in The Physics Teacher. I'll
loosely adapt to bring out the issue of interest.
Suppose you have a lens with known index and radii of curvatures of
the two spherical surfaces. An object (in air) at known distance
forms an image by reflection off the *back* surface of the lens
(relative to the object). Find the image location.
/snip/ find the focal length f for transmission
through the lens. Also let f' = R/2 be the focal length of the back
surface of radius R treated as a mirror. Then 1/object-distance +
1/image-distance = 1/f + 1/f' + 1/f because you go through the lens,
reflect off the back, then go back through the lens.
What stumped me for a while is that the second solution would seem at
first glance to be wrong because it includes a forward and reverse
transmission through the back surface, which don't actually occur.
Nevertheless, if you invent and substitute numerical values, both
methods agree. It may amuse you (or frustrate you, as it did me for a
while) to understand intuitively why they agree. Happy trails for
those who enjoy such puzzles, Carl
--
Carl E. Mungan, Asst. Prof. of Physics 410-293-6680 (O) -3729 (F)