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The question I have is a succession from the ones below.

Why do outside charges do not contribute to net flux of a Gaussian Surface?

Gauss' law and an external charge

The responses to these left me dissatisfied - there is accounting for the difference in the magnitude of the fields passing through a closed surface, however there is no expression of distance from the closed surface as being important.

I'm sympathetic to the latter threads' correspondents in some comments, pointing out that, while for an enclosing surface which passes through exponentially more area as it is more distant from a charge, that explanation of outside charges not affecting flux through a closed shape doesn't make any sense when thinking about a cylinder or an asymmetrical rectangular prism or most possible shapes for the flux to go through and a corresponding cross sectional charge source.

I think the question I have is essentially, could somebody answer the initial question (Why do outside charges not contribute to net flux through a surface?) pertaining specifically to a very long rod with a disk of charge very near at one end, with the disk having identical area to a cross-section of the rod.

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Think of the electric field lines produced by charges outside the closed surface. They pass through some portion of the surface only to exit at some other portion of the surface. Result: No net flux across the surface (lines in = lines out).

Hope this helps.

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Gauss' law implies that if there is no enclosed charge in a given boundary, then the net electric flux will be zero.

Your confusion is regarding field lines. Think of $\phi$ as being analogous to the number of field lines. Then, the electric field would turn out to be the electric flux density.

$$\phi=EA$$ Taking the derivative of $\phi$ with respect to $A$, $$d\phi=EdA$$ and then $$E=d\phi/dA$$

Basically, you can think of this whole problem as related to the number of field lines that passes through the sphere. The electric field is not so important regarding visualisig it, as the flux that we are bothered about is the field line count themselves. Whatever that enters the sphere exits out. And so, Gauss' Law holds.

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