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Snell's law and inverse function

  1. Nov 22, 2016 #1
    1. The problem statement, all variables and given/known data
    Snap1.jpg Snell's law is:
    $$\frac{\sin\theta_1}{c_1}=\frac{\sin\theta_2}{c_2}$$
    $$\frac{c_1}{c_2}=n_{12}$$
    Express ##\theta_2## as a function of ##\theta_1##
    Find the largest value of ##\theta_1## for which the expression for ##\theta_2## that you just found is defined (for larger values of ##\theta_1## than this the incoming light will be reflected).

    2. Relevant equations
    Inverse sine: ##y=\sin^{-1}(x)~\rightarrow~\sin(y)=x##

    3. The attempt at a solution
    $$\sin ( \theta_2 )=\frac{\sin( \theta_1 ) }{n_{12}}~\rightarrow~\sin^{-1}\left( \frac{\sin( \theta_1 )}{n_{12}} \right)=\theta_2$$
    ##\theta_2## can be ##\frac{\pi}{2}## at the max:
    $$\sin^{-1}\left( \frac{\sin( \theta_1 )}{n_{12}} \right)=\frac{\pi}{2}~~\rightarrow~~\sin\left( \frac{\pi}{2} \right)=\frac{\sin(\theta_1)}{n_{12}}$$
    $$\Rightarrow~\sin(\theta_1)=n_{12}\sin\left( \frac{\pi}{2} \right)=n_{12}$$
    $$\theta_1<\arcsin(n_{12})$$
    I didn't use at all the definition of inverse function in the second question, i feel what i have done isn't what it's meant from the chapter of inverse functions
     
  2. jcsd
  3. Nov 22, 2016 #2

    haruspex

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    Not sure what you mean by that. The answer you gave involved an inverse function, and your final step involved inverting the sine function. Can you be more specific about what it is that you feel you have not used?
    By the way, the diagram doesn't match the text. It clearly shows a case with c2>c1, whereas the text assumes the reverse.
     
  4. Nov 22, 2016 #3
    How does the text show anything about the diagram? only the last formula: ##\theta_1<\arcsin(n_{12})## to my opinion, may show something in that direction.
    if ##c_2>c_1~~\rightarrow~\sin(\theta_2)>\sin(\theta_2)~\rightarrow~\theta_2>\theta_1## and the diagram shows the inverse.
    The diagram for ##c_2>c_1## must be:
    Snap1.jpg
    for ##c_2>c_1~\rightarrow~n_{12}<1## and ##\theta_1## has a definite value and that's correct.
     
  5. Nov 23, 2016 #4

    haruspex

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    I confused you by incorrectly describing the mismatch.
    The diagram shows an example of c1>c2, as you say. But in that case there is no limit (before π/2) on θ1. n12>0, so the arcsin of it is not defined.
    Your second diagram, with c2>c1, makes more sense. E.g. total internal reflection can occur from water (low c) to air (high c), but not the other way around.
     
  6. Nov 23, 2016 #5
    True, that is also why i thought my answer wasn't right, since there is no limit for ##\theta_1## if ##n_{12}>1##
     
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