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Electric Field between two opposite, gaussian distributed charged spheres

  1. May 16, 2010 #1

    I understand how to get the electric field between two spheres of uniform charge,
    \vec{E} = \frac{\rho \vec{d}}{3 \epsilon_0}
    which is simplified because at a point [tex]\vec{r}[/tex], the vectors from each charge center combine to give the distance, [tex]\vec{d}[/tex], between centers (since [tex]\rho[/tex]'s can be factored).

    So far, calculating this for two overlapping spheres of gaussian charge distribution seems non-trivial since you can't make this factorization and simply obtain an expression proportional to [tex]\vec{d}[/tex]...

    Am I correct with the complexity of this problem, or is there some way more efficient to approach this problem?

    I hope to use this electric field to model a harmonic oscillator (electron sphere oscillating around stationary ion sphere) and find a frequency...

  2. jcsd
  3. May 16, 2010 #2
    I think you have first to calculate which is the total rho of you 3D distribution.

    This seems to be the sum of the two rho's you presented. After that I guess you will have to use integration over the whole space to get the field at point P = (x,y,z).

    Best Regards,

  4. May 16, 2010 #3
    Thanks for the response DaTario.

    I've started with that method - finding the field at a point P = (x,y,z) (point [tex]\vex{r}[/tex] in spherical) by integrating the charge from the positive distribution [tex]\rho_{+}=\rho_{0}exp(-(r^{2})/(2\sigma^{2}))[/tex] and then the negative distribution [tex]\rho_{-}=\rho_{0}exp(-(r-d)^{2}/(2\sigma^{2})), assuming [tex]d[/tex] is the displacement of the negative charge center along the z-axis for convenience, but i think it get hairy because these end up being factors for the vectors [tex]\vec{r}_{+}[/tex] and [tex]\vec{r}_{-}[/tex] when you sum the electric fields to get the total, in the overlap.

    I was wondering if there's maybe a book problem like this, or even an article i could refer to...

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