Question

If the total charge enclosed by a surface is zero, does it imply that
the elecric 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

NCERT Class XII
Physics - Exemplar Problems

Chapter 1. Electric Charges and Fields

Answer 1

first statement  need not be true          and         
   2nd statement is True.

This is as per the Gauss's law of electrostatics, the flux over a closed surface calculated by surface integral of Electric field over the closed surface , which encloses a net total charge Q divided by permittivity of the medium.

   If the electric field every where on the closed surface is 0, then flux is 0 and hence the net charge enclosed is 0.
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  If the net charge enclosed is 0, then the surface integral is 0.  But that does not mean that the electric field is 0 every where along the surface.   It is possible that dot product of electric field with surface area on one part of the surface is equal and opposite to the minus of dot product of electric field with surface area on another part of the surface.

   thus only net flux is 0, but flux every where need not be 0.