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General Solutions for a particle in a magnetic field

  1. Dec 5, 2011 #1
    Hello everyone!
    I encountered a curious problem while trying to solve the case of a particle with v(v_x0, v_y0, 0), and B(0, 0, Bz); The elucidation of the differential equations obtained through the Lorentz force in this case, should coincide with those obtained through a simplification granted by using the Centripetal force, but here, instead of circular motion I get an unattractive ellipse :(.
    Assuming the particle moves only on the x-y plane, and starts at (x0, y0, 0), the Lorentz force yields the following:
    [itex]
    \large
    \vec{F} = q(\vec{v}\times\vec{B})
    [/itex]
    [itex]
    ma_x(t) = -qv_yB
    [/itex]
    [itex]
    ma_y(t) = qv_xB
    [/itex]
    Which in turn, with these initial conditions leads to:
    With v0 = Sqrt(vx0^2+vy0^2);
    Alpha derived from: Arctan[vy/vx] = alpha;
    [itex]
    x(t) = x_0+ \frac{v_0(-\sin(\alpha)+\sin(\omega t + \alpha))}{\omega}
    [/itex]
    [itex]
    y(t) = y_0+ \frac{v_0(\cos(\alpha)-\cos(\omega t + \alpha))}{\omega}
    [/itex]
    [itex]
    \omega = \frac{qB}{m}
    [/itex]
    Plugging in some random values, leads to the attached image, while we all know that motion in a magnetic field should be accompanied by uniform circular motion;
    Where have I gone wrong?
    Thanks,
    Daniel
    P.S
    This is not related in anyway, to homework.
     

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  3. Dec 5, 2011 #2

    Born2bwire

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    Science Advisor
    Gold Member

    You don't think it might not be caused by the fact that your axes have different scales?
     
  4. Dec 5, 2011 #3
    Firstly, let me thank you for a very prompt reply!
    Yes, I have thought of that, as a matter of fact, but the problem is, that attempting to prove that curve is a circle, i.e, by placing it in (x-a)^2+(y-b)^2=R^2, doesn't produce the desired result. In other words, after expanding the left-hand-side, the time dependent component does not vanish, further suggesting that this is somehow, not a circle.
    Anyhow, adjusting the scales does little good :(.
    What else could there be?
    Thankful as always,
    Daniel
     
  5. Dec 5, 2011 #4

    Born2bwire

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    Science Advisor
    Gold Member

    When I make a plot it looks like a circle to me. And from what I can tell x^2+y^2 = R^2, where R is a constant. You forget that x and y have constant offsets dependent upon x_0, y_0 and \alpha. If you retain the time dependent parts you can easily see that they create a circle.
     
  6. Dec 5, 2011 #5
    Hi,
    Thanks again for your response;
    I guess I did overreact, and that it should turn out alright; as for x^2+y^2..., I think the problem lied in my not taking the explicit form of y as a function of x. Thus, the problem resides, in this case, in the function being a parametric one.
    Thanks again for all your help!
    Couldn't have done it with you!
    Beholden,
    Bound,
    Daniel
     
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