Gauss law also implies that when the surface is so chosen that there are some charges inside and some outside.

The flux in such a situation is given by $\oint \mathrm{E}.\mathrm{dS}=\frac{q}{{\epsilon }_0}.$

Step 2: Identify the terms involved in Gauss' law.

In such situations, the electric field in the LHS is due to all the charges both inside and outside the surface. The term q on the right side of the equation given by Gauss' law represents only the total charge inside the surface.

Thus, despite being total charge enclosed by a surface zero, it doesn't imply that the electric field everywhere on the surface is zero, the field may be normal to the surface.

Also, conversely, if the electric field everywhere on a surface is zero, it doesn't imply that the net charge inside it is zero.

i.e., putting E = 0 in $\oint \mathrm{E}.\mathrm{dS}=\frac{q}{{\epsilon }_0},$
we get, q = 0.