Green Function Method, and reciporcity

We know that the core of electrostatic problems is Possion’s equation:

Solving the special case where $\rho=0$ ,that is, the Laplace equation, is easier(not easy, just a tiny bit easier). There’s a variety of method avaiable, for example separation of variables. Also, due to numerous properties of the Laplace equation, solving it numerical is also easier. However, for most practical problems, there’s bound to be some charges. So what can we do?

Well, not all boundary problems are that diffcult to solve! Boundary problems that only include point charges are (a tiny bit) easier. For example, there’s the method of images that we could make use of. So, could we utilize the results of point charge problem and apply them to general situations? The answer is an absolute yes!

Dirac’s delta function

To describe the charge density of point charges, we introduce the Dirac delta function: $\delta(x) = 0$ when $x \ne 0$, and $\delta(x) = \infty$ when $x = 0$.

Note that this is not a literal function. It’s actually a distribution, a limit of functions. For example, the standard normal distribution $\frac{1}{2\pi\sqrt{\sigma}} e^{-\frac{x^2}{2\sigma^2}}$ will become the delta function when $\sigma$ approches zero. The rigorous defination of the delta function is done in this way.

A point charge occupies infitismal space, yet carries a finite amount of charge. So while the delta function is infinite at zero, it’s integral is not. That is, $\int{\delta(x)dx} = 1$ for any interval which contains zero, and $\int{\delta(x)dx} = 0$ for all the intervals without. The “one” there is chosen to keep things simple. Any finite number will do. In fact, one should think of the delta function as a symbol that can only be used under the integral. For example, one of delta function’s most important property: $\int{f(x)\delta(x)dx} = f(0)$ where $f$ is some ordinary function. Only the value of $f$ at zero matters. Everything else is swept “under the rug”. It is best to think of the delta function as something waiting to be integrated.

Of course, is easy to generalize the function to more dimensions. Simply have the integral integrate over volume, instead of length. Also, its easy to change the point where the value is infinte. For example, $\delta(x-3)$ is infinite at $x=3$, and critical point here is three.

So now we can describe a point charges’ density using the delta function. A point charge $q$ at $\vec{x}'$ could be represented as $\rho(\vec{x}) = q\delta(\vec{x} - \vec{x}')$ note the delta function here is its three dimensional generalization.

Green Function

Now we are ready to tackle the real challenge. The potential a point charge at $x'$ causes satisfies Possions equation: $\nabla^2V(\vec{x}) = -\frac{q}{\epsilon_0}\delta(\vec{x} - \vec{x}')$