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Force on a point charge at the tip of a solid uniformly charged insulating cone

  1. Apr 11, 2012 #1
    1. The problem statement, all variables and given/known data
    A solid insulating cone has a uniform charge density of rho and a total charge of Q. The base of teh cone had a radius of R and a height of h. We wish to find the electric force on a point charge of q' at point A, located at the tip of the cone. (Hint: You may use the result of the electric force along the axis of a disk when solving this problem.)


    2. Relevant equations
    [
    [itex]E= \frac{Q}{2\pi \epsilon_{0} R^{2} }(1 -\frac{z}{ (z^{2}+R^{2}) ^{1/2} })[/itex]


    3. The attempt at a solution
    I decide to lay the cone flat along the z axis. My calculations are independent of the coordinate system though ( which I think may be wrong). I choose a flat disc(area of a circle) for my charge element dQ.
    [itex] \rho = \frac{Q}{V}[/itex]
    [itex]dq = \rho \pi r^{2}[/itex]
    [itex]dF = \frac{\rho \pi r^{2}dr}{4\pi \epsilon_{0}h^{2}}[/itex]
    [itex]F = \frac{\rho \pi}{4\pi \epsilon_{0}h^{2}}\int^{0}_{R}r^{2}dr[/itex]
    [itex]F = \frac{-\rho\pi R^{3}}{12\pi \epsilon_{0}H^{2}}[/itex]
    [itex]F = \frac{-Q R^{3}}{12\pi \epsilon_{0}H^{4}}[/itex]
     
    Last edited: Apr 12, 2012
  2. jcsd
  3. Apr 11, 2012 #2
    first, the formula you have given for the electric field on the axis of the disk seems to be wrong. It should be

    [tex]E= \frac{Q}{2\pi\epsilon_o R^2} \left[1 - \frac{z}{(z^2+R^2)^{1/2}}\right ][/tex]

    where z is the distance of point A on the axis of the disk. Now you can think of a cone as
    pile of disks. So when you consider the charge element [itex]dQ[/itex], it should be
    [itex]dQ=\rho\pi R^2 t[/itex] where t is the thickness of the disk. You can take t as
    another small element [itex]dx[/itex] where x is the distance of the disk from the base.

    Edit: For writing quotients in latex use \frac and not \stackrel. The later is used for
    writing some content above another. look here
    http://www.ee.iitb.ac.in/~trivedi/LatexHelp/latex/stackrel.html
     
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