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,
$$\vec{E} = \frac{\rho \vec{d}}{3 \epsilon_0}$$
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 [tex]\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's maybe a book problem like this, or even an article i could refer to...<br /> <br /> NW[/tex]