If the total charge enclosed by a surface is zero,
Question:

If the total charge enclosed by a surface is zero, does it imply that the electric field everywhere on the surface is zero? Conversely, if the electric field everywhere on a surface is zero, does it imply that net charge inside is zero.

Solution:

According to the Gauss’s law,

$\int_{S} E . d s=\frac{q}{\varepsilon_{o}}$

Here, the term q on the right side of the equation includes the sum of all charges enclosed by the surface (irrespective of the position of the charges inside the surface).

If the surface is so chosen that there are some charges inside and some outside, the electric field on the left side of equation will be due to all charges (both inside and outside)

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

However, if total electric field everywhere on a surface is zero then it implies that net charge inside is zero.