The Poisson Equation

This is not a mandatory required section, but one that physics majors might well read as you're going to be learning it soon anyway (and it is very cool).

Let us start with our old friend, Gauss's Law:

$$\oint_S \vec{E} \cdot \hat{n} dA = \frac{1}{\epsilon_0} \int_{V/S} \rho(\vec{r}_0) d^3r_0.$$ (12)

In your homework, you consider the application of this to a very small differential volume $dV = dx dy dz$ with surface areas $dA_x = \pm dydz$, $dA_y = \pm dxdz$, $dA_z = \pm dxdy$. Using this and dividing through by $dV$ it is easy to show that

$$\frac{\partial E_x}{\partial x} + \frac{\partial E_y}{\part... ...+ \frac{\partial E_z}{\partial z} = \frac{\rho}{\epsilon_0}$$ (13)

or

$$\vec{\nabla} \cdot \vec{E} = \frac{\rho}{\epsilon_0}$$ (14)

which is the vector differential way to write Gauss's Law. The quantity on the left is called the divergence of the electric (vector) field, and we see that the divergence of the field at each point in space is directly proportional to the charge (density) at that point.

Recalling that $E_x = -\frac{\partial V}{\partial x}$, etc, we write this in terms of the scalar potential as:

$$\vec{\nabla} \cdot \vec{\nabla} V = - \frac{\rho}{\epsilon_0}$$

$$\nabla^2 V = - \frac{\rho}{\epsilon_0}$$ (15)

The $\nabla^2$ is called the Laplacian operator and this scalar second order partial differential equation is called the Poisson equation. It's solution is simple and given (more or less) by equation (11) avove. This equation is remarkably deep. Think of how you might represent $\rho(\vec{r})$ for a point charge, for example.

The antidifferential operator

$$G(\vec{r},\vec{r}_0) = - \frac{1}{4\pi\vert \vec{r} - \vec{r}_0 \vert}$$ (16)

plays a special role for the point charge or extended charge distributions. It is called a Green's Function (where there are periodically aguments over whether it ought to be called a a ``Green Function'' or just plain ``Green's Function'' with no article; I really don't care as long as it is Green.

Green's Functions (there are more than one, for different differential equations) are Your Friend, as an old professor of mine once told me². However, we will regretfully bid them adieu at this point, hopefully having laid the ground for a much deeper exploration of their properties (and the associated properties of general solutions to the related partial differential equations).