Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Potential due to anisotropic gaussian distrubution

  1. Sep 15, 2011 #1
    Hello All

    If there is spherically symmetric gaussian charge density (http://en.wikipedia.org/wiki/Poisson's_equation)
    [tex] \rho(\mathbf{r})=\frac{q_{i}}{(2 \pi)^{3/2} \sigma^3} e^{- \frac{\lvert r \rvert^2}{2 \sigma^2}}[/tex]

    then it will have have potential [itex]\phi(r)[/itex] by solving Poisson equation [itex]\bigtriangledown^{2} \phi(r)=-4\pi\rho(\mathbf{r})[/itex]
    [tex] \phi(r) = \frac{\mbox{erf} \bigg(\frac{r}{2\sigma}\bigg)}{r}[/tex]

    But this is the case if gaussian is spherical symmetric which is diagonal element of convariance matrix are same [itex]\sigma_{x}^{2} = \sigma_{y}^{2} = \sigma_{z}^{2} = \sigma^2[/itex]

    Since in general covariance matrix for tri-variate case give by
    [tex]\varSigma=
    \begin{bmatrix}
    \sigma_{x}^{2} & \sigma_{x}\sigma_{y} & \sigma_{x}\sigma_{z} \\
    \sigma_{y}\sigma_{x} & \sigma_{y}^{2} & \sigma_{y}\sigma_{z} \\
    \sigma_{z}\sigma_{x} & \sigma_{z}\sigma_{y} & \sigma_{z}^{2} \\
    \end{bmatrix}[/tex]

    Now i want potential due to anisotropic gaussian distribution (i.e. if i have full covariance matrix). Will it have same analytic form of potential as give by spherically symmetric gaussian?

    Is there article/ reference in literature where such problem solved. As i didn't found till now.

    Any help will be highly thank full :cool:
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: Potential due to anisotropic gaussian distrubution
Loading...