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Electromag Question?

  1. Nov 18, 2008 #1
    1. The problem statement, all variables and given/known data

    An electron beam with negligible velocity in the x direction is emitted from a heated plane cathode through a vacuum towards a plane anode at x = s, there the potential is V = Vo. At the cathode V(x) = 0 and x = 0. The charge density of the beam is;

    p = - (4eoVo)/(9s^(4/3)x^2/3

    Show that V(x) takes the form V(x) = Vos^p.x^q

    2. Relevant equations

    3. The attempt at a solution

    I have worked out that I have to use: divE = p/e0

    So when I substitute the expression I have for p into this I ger:

    divE = (4Vo)/(9s^(4/3)x^2/3

    However I do not know how to get this into a form to find V(x), im stuck. Would appretiate some helpful hints or even something a bit more forceful. Thanks
  2. jcsd
  3. Nov 18, 2008 #2


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    Homework Helper
    Gold Member

    Using Poisson's equation is a much easier way to go here....since V is a function of x only, [tex]\nabla ^2V= \frac{d^2V(x)}{dx^2}[/tex] and so applying Possion's equation gives you an ordinary differential equation which you can solve....try it and see what you get.
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