- #1

CharlieCW

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## Homework Statement

We have two semi-infinite coplanar planes defined by z=0, one corresponding to x<0 set at potential zero, and one corresponding to x> set to potential ##V_0##.

a) Find the Green function for the potential in this region

b) Find the potential ##\Phi(r)## for all points in space

## Homework Equations

Potential in terms of the Green function

$$\Phi(r)=\frac{1}{4\pi\epsilon_0}\int_V G\rho(r')dV'-\oint_S (\Phi_S \frac{\partial G}{\partial n}-G\frac{\partial \Phi_S}{\partial n})$$

Definition of the Green function

$$G(r,r')=\frac{1}{|r-r'|}+F(r,r') \ \ \ \ \ , \ \ \ \ \nabla^2 F(r,r')=0$$

$$\nabla^2 G(r,r')=-\delta(x-x')\delta(y-y')\delta(z-z')$$

Boundary conditions (Dirichlet)

$$\Phi(x<0,y,z=0)=0,\Phi(x>,y,z=0)=V_0,\Phi(r\rightarrow \infty)=0$$

## The Attempt at a Solution

Since we're free to choose our Green function as long as ##F(r,r')## obeys Laplace equation, we can choose it so that ##G=0## in ##S##. This allows us to cancel the last term inside the surface integral of the potential, so we get:

$$\Phi(r)=\frac{1}{4\pi\epsilon_0}\int_V G\rho(r')dV'-\oint_S \Phi_S \frac{\partial G}{\partial n}$$

Since we don't have a volumetric density for the first party, we can reduce this further to:

$$\Phi(r)=-\oint_S \Phi_S \frac{\partial G}{\partial n}$$

In other words, the problem has reduced to finding the appropiate Green function for the planes that satisfies the boundary conditions:

$$\Phi(x<0,y,z=0)=0,\Phi(x>0,y,z=0)=V_0,\Phi(r\rightarrow \infty)=0$$

From here on, I'm not entirely sure about my procedure. I started by solving the Green function as the solution to the equation:

$$\nabla^2 G(r,r')=-\delta(x-x')\delta(y-y')\delta(z-z')$$

with the boundary condition ##G=0## for ##z=0##. The Green function of this plane is equivalent to the potential of a charge placed near a grounded surface, which is a well known solution:

$$G(r,r')=\frac{1}{4\pi}(\frac{1}{\sqrt{(x-x')^2+(y-y')^2+(z-z')^2}}-\frac{1}{\sqrt{(x-x')^2+(y-y')^2+(z+z')^2}})$$

Which we can confirm indeed satisfies ##G=0## for ##z=0##.

However, this Green function doesn't satisfy the boundary conditions for my problem, so I don't know how to modify or find the proper Green function. I've tried test functions or even separating the Green function in parts (through a dirac delta), but nothing seems to work so far.

Once I find the Green function I can easily substitute into the equation for the potential and solve it without problem, so my real issue is finding the appropiate Green function.

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