Electric flux for closed surfaces

The idea of flux, developed in section 1.6.1 for an open (flat or curved but not fully enclosing) surface, extends naturally to a closed surface -- a surface, such as a sphere, cube, or cylinder, that completely encloses a finite volume of space, with no gaps or edges. For a closed surface, the convention is always to take the outward normal (pointing away from the enclosed interior) as the positive direction at every point on the surface. With this convention, flux through any small patch of the surface is counted as positive wherever the field has a component leaving the enclosed volume (field lines exiting), and negative wherever the field has a component entering the enclosed volume (field lines going in). The total, or net, electric flux through the entire closed surface is then obtained, exactly as before, by summing (integrating) these small signed contributions E . dA over every element of the closed surface. If a closed surface encloses no net charge, field lines that enter it must also leave it somewhere else (since field lines can only begin or end on charges), so the positive and negative contributions c …