# Charge distribution with charge density

1. Mar 6, 2009

### latentcorpse

The following is a worked example in my notes im having difficulty with:

A charge distribution with charge density $\rho \neq 0$ exist in the half space $V:z>0$
We can view this as a system with an earthed plate at the z=0 plane. So at z=0 we have the Dirichlet boundary condition $\varphi(x,y,0)=0$

We know that the Greens' function $G_D(\vec{r},\vec{r'}) = \frac{1}{4 \pi \epsilon_0} \frac{1}{|\vec{r}-\vec{r'}} + f_D(\vec{r},\vec{r'})$ i.e. it satisfies the Poisson equation (so essentially we're letting $\varphi = G_D(\vec{r},\vec{r'})$

Our Dirichlet boundary condition tells us that $G_D(\vec{r},(x',y',0))=0$

$\Rightarrow f_D(\vec{r},(x',y',0))=-\frac{1}{4 \pi \epsilon_0} \frac{1}{\sqrt{(x-x')^2+(y-y')^2+z^2}}$

this is generalised to give

$\Rightarrow f_D(\vec{r},(x',y',0))=-\frac{1}{4 \pi \epsilon_0} \frac{1}{\sqrt{(x-x')^2+(y-y')^2+(z+z')^2}}$

and we note that $\nabla^2 f_D(\vec{r},\vec{r'})=0$ i.e. it satisfies Laplace's equation and so G still satisfies Poisson's equation.

However it then states

$\varphi(\vec{r})=\frac{1}{4 \pi \epsilon_0} \int_V dV' \rho(\vec{r'}) \left[ \frac{1}{|\vec{r}-\vec{r'}|} - \frac{1}{|\vec{r}-\vec{r'_m}|} \right]$
where $\vec{r'}=(x',y',z'), \vec{r'_m}=(x',y',-z')$ (*)

but the formula given earlier for calculating the potential from Dirichlet boundary conditions was:

$\varphi(\vec{r})= \int_V dV' G_D(\vec{r},\vec{r'}) \rho(\vec{r'}) - \epsilon_0 \int_S dS' \frac{\partial{G_D(\vec{r},\vec{r'})}}{\partial{n}} \varphi(\vec{r'})$
and so it appears that we've neglected the second term in the formula when we've written down the potential above at (*). Is this because the formula given for phi has a phi in the second term and so we can't apply it or something???

2. Mar 7, 2009

### tiny-tim

Hi latentcorpse!

In ∫SdS'…φ, if S is the z=0 plane (and a hemisphere at infinity?), then isn't φ 0 over S?

3. Mar 7, 2009

### latentcorpse

Re: Electrostatics

sorry could u perhaps give a bit more explanation please - i can't really see what you mean?

4. Mar 7, 2009

### tiny-tim

perhaps i'm misunderstanding the question

i meant, if S is the surface with the Dirichlet boundary condition mentioned in the question,
then φ = 0 all over S, so the second integral is ∫S0 dS' = 0 ?

5. Mar 7, 2009

### latentcorpse

Re: Electrostatics

lol. good point.