TITLE: How to apply Gauss law at null points?
QUESTION [0 upvotes]: How do I use law to determine total flux at Gaussian surface at null point?
It seems , when I consider a Gaussian surface around null point, two axial electric field lines enter it but they don't exit. This leads to supposition that net flux is not zero. This creates a negative charge present in Gaussian surface when it is not.

REPLY [3 votes]: Imagine that the electric field in the region of the neutral (null) point looks something like this with a Gaussian surface which is a cylinder shown in red.  The exact shape does not matter.

As drawn all looks well in that grey electric field lines enter at faces $A$ and $B$ and leave out of curved area $C$.  One can imagine that all the sums work out as the total electric flux entering the Gaussian surface is equal to the total electric flux leaving the Gaussian surface resulting in a net flux through the Gaussian surface of zero ie no net electric charge within the cylinder.
Your contention is that there are two electric field lines which enter the Gaussian surface but do not leave the cylinder and this indicates a net negative charge within the cylinder.
The first thing to note is that those two field lines shown in green are unique in that all other field lines enter and leave the Gaussian surface no matter how close those field lines are to the green field lines.
Now a line is a one dimensional entity as it has no width (cross sectional area) and so when one computes the contribution to the electric flux $\vec E \cdot d\vec A$ of those two electric field lines shown in green the result is zero as $d\vec A$ (the cross-sectional area of the lines) is zero.
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