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Homework Help: Simple Harmonic Motion of Charges

  1. Jul 18, 2010 #1
    1. The problem statement, all variables and given/known data
    A negatively charged particle -q is placed at the center of a uniformly charged ring of radius a having positive charge Q. The particle, confined to move along the x-axis, is moved a small distance x along the axis (x << a) and released. Show that th eparticle oscillates in simple harmonic motion with a frequency given by
    f = (1/2pi)(kqQ/ma^3)^(1/2).


    2. Relevant equations
    F = kq1q2/r^2
    torque = r x F
    torque = (I)(alpha)
    I (moment of inertia) = mL^2 (L = distance from point about which rotation occurs, in this case, approx. L = a)
    alpha = d^2(theta)/dt^2
    *Where theta is angle between a and the hypotenuse in the triangle with base and height a and x.


    3. The attempt at a solution
    I tried finding net force on -q as exerted by the ring (relevant force is only in x direction)
    F = -kQq/a(a-x) + kQq/a(a+x)
    F = -2kQqx/a^3

    then I plugged this into torque = r x F
    where r is approx. a
    and equated with torque = I(alpha)
    which gave me the motion for simple harmonic motion d^2(theta)/dt^2 + 2kQqx(theta)/ma^4
    (I made use of the limit lim(∆theta-->0) sin(theta)/theta = 1)

    and from this, the angular frequency should be (2kQqx/ma^4)^(1/2)
    which gives f = (1/2(pi))(kQqx/ma^4)^(1/2)
    which obviously isn't the answer.

    What am I doing wrong?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jul 18, 2010 #2

    vela

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    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    It appears what you're doing wrong is blindly using formulas without understanding what they mean.

    You didn't explain how the ring is oriented relative to the x-axis. Does it lie in the yz-plane with its center at the origin?
     
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