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Superposition of 2 vectors

  1. Aug 23, 2006 #1
    I'm considering the superposition of 2 vectors.


    Trying to eliminate t. Its easy when the phase shift is 0 or pi/2 but I'm not sure how to go about it in this case. I can get Ey to be a function of cos(wt)-sin(wt), or cos(wt)sin(wt) just using trigonometric formulas, but I don't know where to go from there.

    The end result should be an elipse with axis that are not aligned with the x-y axis. If its quite difficult then I won't worry about it. I just get the feeling that I've done this before and I should know it, but somethings not clicking. Thanks.

    edit: For some reason the LATEX graphics don't appear for me. Just in case I put the formulas in wrong they should be:

    Ex = E1 cos(wt)
    Ey = E2 cos(wt + pi/4)
    Last edited: Aug 23, 2006
  2. jcsd
  3. Aug 23, 2006 #2

    Andrew Mason

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    How about [tex]E = \sqrt{E_x^2 + E_y^2} [/tex]?

    As far as the latex problem, you have to use lower case for the tex and /tex commands.

  4. Aug 23, 2006 #3
    Thanks for the reply, I don't think I was very clear about what I'm trying to do though.

    If I have


    then [tex]E_1y=E_2x[/tex] gives me my equation of motion for the particle.


    [tex]E_y=E_2cos(wt + \frac{\pi}{2})[/tex]


    [tex]\frac{x^2}{E_1}+\frac{y^2}{E_2}=1 [/tex]

    And the vector moves in an ellipse.

    I'm not sure how to get the same sort of equation (ie. no dependance on t), or if its possible, for





    Not sure where I can go from here
    Last edited: Aug 23, 2006
  5. Aug 23, 2006 #4


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    There's a rotation [transformation] involved.
    Complex numbers might simplify your calculation.
  6. Aug 23, 2006 #5
    I'm sorry but I'm still not sure how to get there.

    If I write it as,


    I don't know where to go from here.

    I feel like its staring me right in the face. I know what the answer should look like. I know the transform should be along the lines of [cosA,-sinA;sinA,cosA] and I could probably work out A by playing with E1 and E2, but its just not falling into place for me. Maybe I'm just having a bad day, hopefully a good night's sleep will help.
  7. Aug 23, 2006 #6
    Just for clarification, I'll explain the context in which I ask this question. For a uGrad assignment I have been asked to propose an exam question and provide a solution (The hard part about this assignment is thinking of a question that hasn't already been on previous exams).

    Part of my question goes along the lines of:
    Consider 2 EM waves, prpagating along the z axis, with angular frequency w and wave number k. Wave 1 has its E-field aligned with the x axis, wave 2 with the y axis. There is a relative phase difference of [tex]\phi[/tex] between them. Describe how the polarisation of the superposition of waves 1 & 2 varies in time.

    For the case [tex]\phi=0[/tex] we have a plane wave with an E-field of magnitude [tex]\sqrt{E_1^2+E_2^2}[/tex]. For all [tex]0 < \phi < \pi[/tex] we should have an elliptically polarized plane wave, but I only know how to demonstrate this for the case where [tex]\phi=\frac{\pi}{2}.[/tex]
  8. Aug 24, 2006 #7
    Nevermind. Worked it out. Just wasn't thinking yesterday.
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