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Gauss' Law in not simply-connected regions

Does it hold even in not simply connected regions?
I.e. is true that

SS F.n dS= Q/e_0

when the region is not simply connected (is true for ANY closed region)?

Reply 1

DFranklin

I think it's true for any closed region provided you account for the boundaries correctly.

Reply 2

coffeym

I think of it like this: around any collection of regions, regardless of how they are connected, we may draw an imaginary Gaussian Surface, say in the form of a sphere, and perform the surface integral over this surface. So I'd say it is true for any regions possible.

Reply 3

gavak

Yes, but what if I want the flux through a not simple connected surface?

Reply 4

gavak

I mean, a SPECIFIC not simple connected region.. The trick of drawing a sphere fails, since that is not the surface I want.

Reply 5

coffeym

Well according to my notes the Law is true for any surface, regardless of its complexity. As for a proof, I suppose it is to show that:

\displaystyle \iint_S{\mathbf{E}\cdot d\mathbf{A}}=\iint_\Omega {\mathbf{E}\cdot d\mathbf{A}}

where

S

is, say, the sphere of radius

R

and

\Omega

is any arbitrary closed surface enclosing the charge. There must be some way to approach this...

Reply 6

DFranklin

Gavak: So what do you want?

If you want to look up theory on this, I think it's better to concentrate on the Divergence theorem, as Gauss' Law is only a special case of that.

Reply 7

gavak

Yeah but divergence theorem holds only with simple connected regions..

Reply 8

DFranklin

From http://books.google.co.uk/books?id=9p6sUTxUoZ0C&pg=PA407&lpg=PA407&dq=divergence+theorem+%22multiply+connected%22&source=web&ots=peoPXp1x3F&sig=2RER_x-EmTWHpZrpcSyvYa0hiYU&hl=en:

We note that the divergence theorem holds for both simply and multiply connected surfaces...

But if you have a counterexample, it will almost certainly show how to give a counterexample to Gauss's law as well.

Reply 9

RichE

gavak

Yeah but divergence theorem holds only with simple connected regions..

What are commonly refered to as The Divergence Theorem and Stokes Theorem
are low dimensional versions of a general dimensional "Stokes Theorem" which is normally first met in the fourth year/graduate level. It holds for all manifolds with boundary, not just simply connected ones.

You need the simply-connected hypothesis to get from curl =0 (or div =0) to the existence of potentials.

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