Chapter 22, Problem 1Q

If the electric flux through a closed surface is zero, is the electric field necessarily zero at all points on the surface? Explain. What about the converse: If $\stackrel{\to }{E}=0$ at all points on the surface is the flux through the surface zero?

To determine

Check whether zero electric flux means zero electric filed at every point on surface and vice versa

Answer to Problem 1Q

No. Zero electric flux do not imply zero electric fled at all points on surface.

Yes. Zero electric field at all points on surface means the zero electric flux.

Explanation of Solution

Electric flux is the measure of number of electric filed lines passing through the surface. According to Gauss’s theorem, zero electric flux over a surface means that the net charge enclosed by the surface is zero. But the electric filed can be nonzero by the charges lies inside or outside the Gaussian surface.

Net electric field over a surface can be zero if electric field is pointing inwards at some points and outwards from the surface at some other points. If the electric field is zero, it is no electric flux and by Gauss’ theorem there will be not net charge.

Therefore, it is incorrect that to say zero electric flux do not imply zero electric fled at all points on surface and it is correct that zero electric field at all points on surface implies the zero electric flux.