So I thought I was understanding Green's functions, but now I am unsure. I'll start by explaining (briefly) what I think I know then ask the question.
Background
Greens are a way of solving inhomogeneous differential equations by first solving the response to a unit impulse. I am aware they are a tool that can sometimes be given a physical meaning. I fear those two realms may be crossing for me where they shouldn't.
$\hat{L} f(x) =s (x)$
$\hat{L} G(x) =\delta (x)$
Using these we can obtain the following general expression for the solution:
$f(x)=\int G(x-x')s(x')dx'$
Physically, in EM, the Greens function appears to be in the form of a potential due to a single point charge. I.e. we can construct our solution for an arbitrary charge distribution by 'summing' over everything to create our known charge distribution.
$\nabla^2\Phi=-\frac{\rho}{\epsilon}$
$\nabla^2G(x-x')=-\frac{\delta(x-x')}{\epsilon}$
$\Phi(x)=\int G(x-x')\rho(x')dx'$
Where
$G(x-x')=\frac{1}{|x-x'|}$
When we start to talk about boundary conditions there is an ambiguity in the definition of our Greens function such that we can have a form as follows:
$G(x-x')=\frac{1}{|x-x'|}+F(x-x')$ so that $\nabla^2F(x-x')=0$
I guess this is where it starts to get blurred for me.
Question:
Jackson 2.7 a)
Consider the half plane z>0 with drichlet boundary conditions on the z=0 plane. We wish to write down the greens function for this situation.
Many solutions I have found jump right to the greens function of:
$G(x,x')=\frac{1}{|x-x'|}-\frac{1}{|x-x''|}$
But I do not understand how they got here – they say it is obvious- but given what I understand I must be missing something.
How is this the greens function if there is no charge? They appear to allude to the existence of a charge and an image charge. It seems like we are trying to solve the laplace equation which is homogeneous (so I don't understand the use of greens here). I am guessing the boundary conditions are the critical thing I am not understanding fully.
Feel free to correct any of my misinterpretations and mistakes in general.
Best Answer
Good questions; I'm sure a lot of people are confused on this stuff (as I was the first time I used Jackson).
Essentially your confusion boils down to being careful to consider the following fact:
The Green's function for a particular boundary value problem depends on the boundary conditions.
In particular, let's say you have a Dirichlet boundary value problem. Then, as Jackson shows on page 39, the appropriate Green's function for such a boundary value problem must (a) satisfy Poisson's equation with a delta function source in that region and (b) vanish on the boundary (see eq. 1.43) of that region. If you can find a function that has these two properties in the region you are considering, then you have found the Green's function for the Dirichlet problem.
So if we consider the half space ($z>0$) with dirichlet boundary conditions at $z=0$, then we are looking for a function that satisfies Poisson's equation in the upper half space with unit source and which vanishes on the boundary, which in this case is $z=0$ plus the "boundary at infinity."
You can check yourself that the function $$ G(\mathbf x, \mathbf x') = \frac{1}{|\mathbf x - \mathbf x'|} - \frac{1}{|\mathbf x + \mathbf x'|} $$ has these properties. The intuition for this, and the reason why most people say you can just immediately write this down, is that the first term clearly satisfies Poisson's equation with unit source in the upper half space where $\mathbf x'$ is being taken to have $z>0$, and the second term corresponds to having a unit source in the lower half space with opposite sign. Our intuition about potentials of points charges indicates that this will cause that combination to vanish at $z=0$. Also, the Laplacian of the second term vanishes in the upper half space, so it doesn't affect the fact that in the upper half space, this function satisfies Poisson's equation with unit source at $\mathbf x'$.
I hope that was clear? I can definitely try to clean it up or expand on this. I know from personal experience that it's a confusing topic!
Addendum in response to comments.
Green's functions are associated with a set of two data (1) A region (2) boundary conditions on that region. The function $1/|\mathbf x-\mathbf x'|$ is the Green's function for (1) All of space with (2) Dirichlet boundary conditions. This is because it (a) satisfies Poisson's equation with unit source in that region and (b) vanishes at the boundary of that region (which in this case is at infinity). In general, for any region $R$, for Dirichlet boundary conditions, as long as we simply find a function $G(\mathbf x,\mathbf x')$ that, (a) satisfies Poisson's equation with a unit source placed in that region, and that (b) vanishes on the boundary of the region, then we have found the Green's function for that Dirichlet problem (by the definition of a Green's function).
When we are trying to find the Green's function for the Dirichlet problem on the upper half space, we first imagine putting a point charge in the upper half space so that condition (a) is satisfied, this leaves us with the function $1/|\mathbf x-\mathbf x'|$ where $\mathbf x'$ is a point in the upper half space. Then, we notice that although this function is an appropriate solution to Poisson's equation, it does not vanish for $x$ on the boundary, so this can't be the Green's function for this Dirichlet problem. We need to do something to this function which does not spoil the fact that it satisfies the unit source Poisson equation in the upper half space but such that the resulting function additionally satisfies the appropriate boundary condition.
So we ask ourselves "what can we do to this function so that (a) remains satisfied, but so that (b) is also satisfied in the upper half space. Well we notice that if we add any function that satisfies Laplace's equation (Laplacian equaling zero) in the upper half space to $1/|\mathbf x-\mathbf x'|$, then the resulting function will still satisfy (a).
Now what sorts of functions satisfy Laplace's equation in the upper half space?
The answer is any charge distribution whose charge density is only nonzero outside of the upper half space will create a potential that satisfies Laplace's equation in the upper half space.
So if we can find a charge distribution that when placed in the lower half space produces a potential that when added to $1/|\mathbf x-\mathbf x'|$ causes their sum to vanish on the boundary, then their sum will satisfy the properties required of a Green's function. This is where we notice that an "image" point charge will do exactly this!
All we are doing with this point charges is an intuitive way of finding a function that satisfies the appropriate mathematical properties (a) and (b) that a Green's function for a Dirichlet problem must satisfy.