Charge Density Function to Solve Poisson Eq.

[sarperb]

Homework Statement

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.

Homework Equations

$-\nabla^{2}V(\vec{r}) = \frac{\rho(\vec{r})}{\epsilon_{0}}$

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;

$\rho(\vec{r}) = A(r) U(R-r) (δ(w-1)+δ(w+1))$

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

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

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 $1/r^{2}$ all of which will give me an answer which is dimensionally correct.

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

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

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.

$\rho(\vec{r}) = A(r) U(b-r) U(r-a) δ(w)$

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

$\frac{Q}{2\pi\;} = \int^{b}_{a}r^{2}A(r)dr$

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

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 $1/r^{2}$.So in the end my question is, how do I know what A(r) should be in terms of r dependance?

 

[Nate810]

Because in the Rod case it is 1/r^{2} and in the Annulus case it is 1/r. I know we can assume A(r) to be a constant, or a function of 1/r, or 1/r^2. But how do I know which one to choose? Thank you for any help!