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Homework Help: E&M Griffiths 3.34

  1. Nov 9, 2005 #1
    A point charge (q, mass m) is released from rest at a distance d from a grouned infinite conducting plane. How long does it take to hit the plane?
    Answer pi*(d/q)*sqrt(2pi*eps m d)
    This problem seemed easy to me at very, but it leads to a second order nonlinear equation
    [tex]m\frac{d^2 z}{dt^2} = \frac{q^2}{16 \pi \epsilon_0 z^2}[/tex].

    I tried using energy considerations to write v as v(z), I then solved for z as z(v), and integrated the above equation for v(t), putting in the limits 0 and infinity. This did not give the correct answer, although it appeared to be close. Any suggestions?
  2. jcsd
  3. Nov 10, 2005 #2
    I haven't actually worked out the problem, so I'm not sure if the right side of your equation is correct. Just as a note incase you didn't do this - since the infinite conducting plane is grounded, you need to use the method of images to get your potential function.
  4. Nov 10, 2005 #3
    Maple choked on the d.e. but, you should be able to show
    v2 = 2c(1/z - 1/d) with c = q2/(16 Pi Eps0 m)
    with conservation of energy or integrating the force equation once.
    now to solve for time put solve for v and use the positive root,
    the negative one leads to t<0.
    thus [tex] t = \frac{1}{\sqrt{2c}} \int_d^0 \sqrt{ \frac{ dz}{ d - z}} \. d z [/tex]
    The substitution z = d cos2(theta) makes this doable.
  5. Nov 10, 2005 #4
    Yeah, that's what I indicated I tried above. It appeared to fail the first time I did it, but the second time it worked out.
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