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Charge Density Function to Solve Poisson Eq.

  1. Mar 15, 2012 #1
    1. The problem statement, all variables and given/known data
    This is not really a homework just studying but I'm kinda stuck.

    So I am trying to find out how to formally write down the Charge Density for any distribution.

    Although I will not get into Green's Function or how to find V, I got that fine.

    My example will be a Rod of uniformly distributed charge (total Q) inside a grounded sphere.

    2. Relevant equations

    [itex]-\nabla^{2}V(\vec{r}) = \frac{\rho(\vec{r})}{\epsilon_{0}}[/itex]


    3. The attempt at a solution

    Note: In the below equations w = cosθ for simplicity.
    So trying to write down the charge density for a Rod of Charge;

    [itex]\rho(\vec{r}) = A(r) U(R-r) (δ(w-1)+δ(w+1))[/itex]

    [itex]Q = \int^{\infty}_{0}r^{2}A(r)dr\;U(R-r)\;(1+1)\;2\pi[/itex]

    [itex]\frac{Q}{4\pi\;} = \int^{R}_{0}r^{2}A(r)dr[/itex]

    So now I can take the integral if I assume A(r) to be a constant in r, or I can say it is proportial to 1/r, or I can say it is proportional to [itex]1/r^{2}[/itex] all of which will give me an answer which is dimensionally correct.

    Although only when I say A(r) is a function of [itex]1/r^{2}[/itex] I get the right answer for V after I go thru the Green's Function process.

    Which is;
    [itex]V_{in}(r,w) = \frac{Q}{4\pi\epsilon_{0}R}[ln(\frac{r}{R})+\sum^{\infty}_{\ell=2 (Even)}\frac{(2\ell+1)}{\ell(\ell+1)}(1-(\frac{r}{R})^{\ell})P_{\ell}(w)][/itex]

    Note that this confusion does not arise when one has a dirac delta in r because it doesn't really matter, the delta will kill all r when we are taking the Green's Function integral anyway - so the answer to the question does not change. It only arises when one has a Step Function.

    As another example for comparison, a very similar situation arises for an Annulus of Charge with radii a and b.

    [itex]\rho(\vec{r}) = A(r) U(b-r) U(r-a) δ(w)[/itex]

    [itex]Q = \int^{\infty}_{0}r^{2}A(r)dr\;U(b-r) U(r-a)\;2\pi[/itex]

    [itex]\frac{Q}{2\pi\;} = \int^{b}_{a}r^{2}A(r)dr[/itex]

    Now in the notes of our instructor this charge density is given as:
    [itex]\rho(\vec{r}) = \frac{Q}{\pi(b^{2}-a^{2})r}\;U(b-r) U(r-a) δ(w)[/itex]

    So although this looked almost exactly the same while trying to find A(r), here I can see that A(r) is a function of 1/r, but in the case of the rod it was [itex]1/r^{2}[/itex].


    So in the end my question is, how do I know what A(r) should be in terms of r dependance?
     
    Last edited: Mar 15, 2012
  2. jcsd
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