I am reading Jackson Electrodynamics (section 1.10 in 3rd edition) and he is discussing the Poisson eqn $$\nabla^2 \Phi = -\rho / \epsilon_0$$ defined on some finite volume V, the solution using Greens theorem is(adsbygoogle = window.adsbygoogle || []).push({});

$$\Phi (x) = \frac{1}{4 \pi \epsilon_0} \int_V G(x,x') \rho(x')d^3x' +\frac{1}{4 \pi } \int_{\partial V} G(x,x') \frac{\partial \Phi (x') }{\partial n'} - \Phi(x') \frac{\partial G (x, x') }{\partial n'} da'$$

where G(x,x') is any Greens function

He says that for Neumann boundary conditions i.e. $$\partial \Phi / \partial n$$ is given as an explicit function on the boundary, it might seem that the obvious solution would be to choose G so that on the boundary

$$\frac{\partial G (x') }{\partial n'} = 0 $$

However he says that this isnt so simple since by Gauss theorem

$$\int_{\partial V} \frac{\partial G (x,x') }{\partial n'} da' = -4 \pi $$

for any Green's function. This leads me to my first question. So what? The integral we are interested in is not

$$\int_{\partial V} \frac{\partial G (x,x') }{\partial n'} da'$$

but rather

$$\int_{\partial V} \Phi(x') \frac{\partial G (x,x') }{\partial n'} da'$$

which doesnt do anything funny through Gauss theorem, so

$$\frac{\partial G (x') }{\partial n'} = 0 => \Phi(x') \frac{\partial G (x') }{\partial n'} = 0$$

and so that term in the solution should just equal 0.

He then goes on to say that what we should do is set

$$\frac{\partial G (x') }{\partial n'} = - 4\pi/S $$

where

$$S =\int_{\partial V} da' $$

so that

$$ -\frac{1}{4 \pi } \int_{\partial V}\Phi(x') \frac{\partial G (x') }{\partial n'} da' = \int_{\partial V} \frac{\Phi(x')}{S} da' := <\Phi>_{\partial V}$$

in other words it is the average value of phi on the boundary. He then says that in the limit that the surface goes to infinity this term goes to zero. I dont see why that should be the case, I can actually show explicitly functions that its average value on some surface does not depend on the "size" of the surface at all, and others which will blow up to infinity as the surface grows. What am I missing here?

Finally, his opening line was about finite volumes how is the surface at infinity (unless he is talking about the weird mathematical cases where an infinite surface bounds a finite volume, but somehow I doubt it)

TIA

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# Q about Poisson eqn w/ Neumann boundary conditions as in Jackson

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