The closed surface shown in FIGURE 24.21a has a point charge q outside the surface but no charges inside. Now what can we say about the flux? By approximating this surface with spherical and radial pieces centered on the charge, as we did in Figure 24.20, we can reassemble the surface into the equivalent surface of FIGURE 24.21b. This closed surface consists of sections of two spherical shells, and it is equivalent in the sense that the electric flux through this surface is the same as the electric flux through the original surface of Figure 24.21a.

FIGURE 24.21 A point charge outside a Gaussian surface.

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If the electric field were a fluid flowing outward from the charge, all the fluid entering the closed region through the first spherical surface would later exit the region through the second spherical surface. There is no net flow into or out of the closed region. Similarly, every electric field line entering this closed volume through one spherical surface exits through the other spherical surface.

Mathematically, the electric fluxes through the two spherical surfaces have the same magnitude because ${\text{Φ}}_{\text{e}}$ is independent of r. But they have opposite signs because the outward-pointing area vector $\vec{A}$ is parallel to $\vec{E}$ on one surface but opposite to $\vec{E}$ on the other. The sum of the fluxes through the two surfaces is zero, and we are led to the conclusion that the net electric flux is zero through a closed surface that does not contain any net charge. Charges outside the surface do not produce a net flux through the surface.

This isn’t to say that the flux through a small piece of the surface is zero. In fact, as Figure 24.21a shows, nearly every piece of the surface has an electric field either entering or leaving and thus has a nonzero flux. But some of these are positive and some are negative. When summed over the entire surface, the positive and negative contributions exactly cancel to give no net flux.