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Using a double angle formula with Snell's Law

  1. Nov 27, 2011 #1
    1. The problem statement, all variables and given/known data
    A light ray is incident from air onto a glass surface with an index of refraction n = 1.56. Find the angle of incidence for which the corresponding angle of refraction is one-half the angle of incidence. Both angles are defined with the normal to the surface.


    2. Relevant equations
    n = sin a1/sin a2 (Snell's Law)

    sin(a + b) = (sin a x cos b) + (cos a x sinb)

    sin(2a) = 2 sin a cos a

    3. The attempt at a solution

    Somehow you plug in the double angle formula into snells law. but I don't understand how this is a down. a breakdown of the equations used to solve problem would be extremely helpful :)
     
  2. jcsd
  3. Nov 27, 2011 #2

    ehild

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    The angle of incidence is 2a, the angle of refraction is a. Plug in to Snell's law.

    ehild
     
  4. Nov 27, 2011 #3
    but then how do you solve for a?
     
  5. Nov 27, 2011 #4

    ehild

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    Show your equation.

    ehild
     
  6. Nov 28, 2011 #5
    sin(a)/sin(2a) = 1.56
     
  7. Nov 28, 2011 #6

    gneill

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    What happens if you apply the sine double angle formula here?
     
  8. Nov 28, 2011 #7
    Your equation is actually incorrect. by Snell's formula,
    sin a = 1.56Apply the duble angle formula to sin (2a) to get 2sin(a)cos(a). since a does not equal an integral multiple of pi, you can divide out sin(a). From there, you can find that cos(a) = 0.321
    Use the inverse of cosine function to solve for a = 71.306 degrees.
     
    Last edited: Nov 28, 2011
  9. Nov 28, 2011 #8
    if sin a = 1.56, how do you apply the double angle formula?
     
  10. Nov 28, 2011 #9

    gneill

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    Snell's Law:
    [tex] \frac{sin(\theta 1)}{sin(\theta 2)} = \frac{v1}{v2} = \frac{n2}{n1} [/tex]
     
  11. Nov 28, 2011 #10
    im so confused.. could someone just show the steps on how to solve it? i think much better in equations than words.
     
  12. Nov 28, 2011 #11

    gneill

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    Take Snell's law as I wrote it above (You can ignore the velocity ratio, I only included it for completeness. It's not needed for this problem). Substitute the given values for the angles and indexes of refraction. What do you get?
     
  13. Nov 28, 2011 #12
    isnt that what i wrote before? sin a/sin 2a = 1.56/1.
     
  14. Nov 28, 2011 #13
    sorry, i see. sin2a/sina = 1.56/1
     
  15. Nov 28, 2011 #14
    ah ha! so sin 2a = 2sinacosa and the sins cancel out, giving you a final equation of cos a = 0.78. thanks very much everyone!
     
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