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If the electric field at all points on a closed Gaussian surface is zero, does this mean that the net charge enclosed by the surface is zero?
Yes; zero field everywhere means zero flux everywhere.
Yes; zero electric field means that there are no charges of any kind in the region.
No; there may be some positive and some negative charges enclosed within the surface.
No; zero field could be caused by charges outside of the surface as well as inside the surface.
No; it is impossible for charges to exist and yet electric field to be zero everywhere.