Electric Field between two opposite, gaussian distributed charged spheres

[Nick White]

Hi,

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

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...

Thanks

 

[DaTario]

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,

DaTario

 

[Nick White]

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,

DaTario

Thanks for the response DaTario.

I've started with that method - finding the field at a point P = (x,y,z) (point \vex{r} in spherical) by integrating the charge from the positive distribution \rho_{+}=\rho_{0}exp(-(r^{2})/(2\sigma^{2})) and then the negative distribution \rho_{-}=\rho_{0}exp(-(r-d)^{2}/(2\sigma^{2})), assuming d 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 \vec{r}_{+} and \vec{r}_{-} when you sum the electric fields to get the total, in the overlap.<br /> <br /> I was wondering if there&#039;s maybe a book problem like this, or even an article i could refer to...<br /> <br /> NW