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Finding the angle of refraction

  1. Sep 18, 2016 #1
    1. The problem statement, all variables and given/known data
    The refractive index of diamond is 2.42. A ray of light passing from the inside of a diamond into air makes an angle of incidence of 20 degrees. Find the angle of refraction.

    2. Relevant equations
    n = refractive index
    aNd = 2.42 (my book said the speed of light in a vacuum and in air are close enough to treat them as the same for our purposes)
    dNa = 1/aNd
    Sin i / Sin r = dNa
    i = 20 degrees
    3. The attempt at a solution

    0.413 = dNa
    Sin 20 degrees = 0.3420
    0.3420 / sin r = 0.413
    therefore 0.3420 x 0.413 = sin r
    sin r = 0.142
    arc sin of 0.142 = 8.159 degrees

    This is wrong, the answer should be 55.9 degrees..

    Where did I go wrong?
    Did I go wrong in my maths or in my use of the formula?

    If someone could point me in the right direction that would be wonderful.
     
  2. jcsd
  3. Sep 18, 2016 #2

    Simon Bridge

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    I think you los track of the refractive index ration and which to use where.
    Instead use Snell's law as she is writ:
    ##\qquad n_1\sin\theta_1 = n_2\sin\theta_2##
    ... that way you can keep track.

    Since the ray passes from the diamond into the air, which way will it bend (towards the normal or away from the normal)?
     
  4. Sep 18, 2016 #3

    kuruman

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    Why is this correct?

    Start with ## n \sin \theta_i = \sin \theta_r ## and solve for ## \sin \theta_r ##.
     
  5. Sep 18, 2016 #4
    That formula isn't taught in my book. I saw them explain it on Kahn Academy but I don't really understand it. I think it is just another way to express the same thing. I have to stick with the formula given for my exam. If I use any formula which is not on the syllabus there is a good chance I will be marked wrong, depending on who corrects my test. These are the formula:

    Snell's law: n = sin i / sin r
    xny = 1 / ynx

    It will be bent away from the normal.


    Why isn't it correct? The formula says sin i over sin r = dNa
    So I have to multiply sin i by dNa to get sin r, right? Is my algebra wrong again?
     
  6. Sep 18, 2016 #5

    kuruman

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    OK, we will go by the formula that you are familiar with. I understand sin i is the sine of the angle of incidence and that sin r is the sine of the angle of refraction. Please explain what dNa stands for because I am unfamiliar with the notation.
     
  7. Sep 18, 2016 #6
    Sorry about that!
    I should have realized it wouldn't be clear.

    we are using n for refractive index as well, but
    aNd (and)
    d stands for diamond and a stands for air, so aNd is the refractive index of light travelling from air into a diamond and dNa is the refractive index of light travelling from a diamond into air.
     
  8. Sep 18, 2016 #7

    kuruman

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    And, as the problem states, the refractive index of diamond is 2.42, which one do you think is given, dna or and?
     
  9. Sep 18, 2016 #8
    it has to be and, doesn't it? Because all it says is "the refractive index", and the normal one to give is light travelling from air (or a vacuum) into the medium.
     
  10. Sep 18, 2016 #9

    kuruman

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    OK. Then, if light goes from air to diamond, Snell's law is
    $$ \frac{sin \theta_{air}}{sin \theta_{diamond}} = {_an_d} $$
    Right?
    And when light goes from diamond into air,
    $$ \frac{sin \theta_{diamond}}{sin \theta_{air}} = {_dn_a} $$
    Right?
    Can you solve for what you need? Note that these two equations say that it doesn't matter what is "incident" and what is "refracted". When you send light back where it came from, it will retrace its path. Also note that I used subscripts "diamond" and "air" instead of "i" and "r" to avoid confusion of what's what. I recommend you do the same if you are to use the form of Snell's law that was given to you. The rule is this "The subscript of the sine in the numerator matches the first subscript of n and the subscript of the sine in the denominator matches the second subscript of n."
     
  11. Sep 18, 2016 #10
    Ah! Thank you, that was a silly mistake. Let me see if I can do it now

    0.413 = dNa
    Sin 20 degrees = 0.3420
    sin r [sin theta diamond] / 0.3420 = 0.413

    I think I am still going wrong somewhere because I can't see how to get 55.9 degrees out of what I have o_O Sorry about this
     
  12. Sep 18, 2016 #11

    kuruman

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    OK
    OK
    Use the two OK'ed numbers above and forget the others. Start with ## \frac{\sin \theta_{diamond}}{ \sin \theta_{air}}={_dn_a} ##
    Which one of the quantities in the equation are you looking for and how can you find it?
     
  13. Sep 18, 2016 #12

    Simon Bridge

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    Wow - no wonder you are confused. So besides what kuruman is telling you ...
    If the speed of light in the medium is v, and the speed of light in vaccuum is c, then, by definition, the refractive index for the material is n=c/v
    Snells law is for light passing through an interface between two arbitrary transparent materials.
    I wanted to point this out because it will help you later when you talk to people outside your particular course.

    Refractive index for air is usually treated as 1 ... so for light passing from air to a material, ##\sin\theta_i = n\sin\theta r## and for light passing from the material to the air, ##n\sin\theta_i = \sin\theta_r## ... these are mathematically the same equation.
    So the ratio ##\sin\theta_i/\sin\theta_r## is 1/n, if light goes from air to medium, and n if the light goes from medium to air.

    The genaral Snells law is better because you only have one equation to get straight.
    The maths does not care which direction the light travels, so there's one thing you don't need to worry about, and those mNa and aNm things take care of themselves so there's another two things not to worry about. The downside is that you need to be able to do algebra.
    I'm pretty bad at memorizing stuff, so I always feel it is better to reduce the amount of memorization needed to do anything.

    I'm gonna let kuruman help you out here though... you don't need two people talking at you.
     
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