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kq6up
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Homework Statement
In optics, the following expression needs to be evaluated in calculating the intensity
of light transmitted through a film after multiple reflections at the surfaces of the
film:
[tex]{\sum _{ n=0 }^{ \infty }{ { r }^{ 2n } } cos\quad n\theta })^{ 2 }+{ \sum _{ n=0 }^{ \infty }{ { r }^{ 2n } } sin\quad n\theta })^{ 2 }[/tex]
.
Show that this is equal to [tex]{ \left| \sum _{ n=0 }^{ \infty }{ { r }^{ 2n } } { e }^{ in\theta } \right| }^{ 2 } [/tex] and so evaluate it assuming |r| < 1 (r is
the fraction of light reflected each time).
Homework Equations
It looks like geometric series to me, so [tex]S=\frac{a}{1-r}[/tex] where S is the sum of the and r is some decimal number less than one.
The Attempt at a Solution
The text says the trick is to use only the imaginary part of the series (which is sign). I get a different answer than the book. I get [tex]S=\frac{1}{1-r^{2}sin^{2}\theta}[/tex] since I let replace r with the number that is in the geometric series. That is [tex]r^{2}e^{i\theta}[/tex]. The book's solution is [tex](1+r^{4}-2r^{2}cos\theta)^{-1}[/tex]. Not sure how they got that.
Thanks,
Chris Maness
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