Notice something interesting about Equation 24.13. The electric flux depends on the amount of charge but not on the radius of the sphere. Although this may seem a bit surprising, it’s really a direct consequence of what we mean by flux. Think of the fluid analogy with which we introduced the term “flux.” If fluid flows outward from a central point, all the fluid crossing a small-radius spherical surface will, at some later time, cross a large-radius spherical surface. No fluid is lost along the way, and no new fluid is created. Similarly, the point charge in FIGURE 24.19 is the only source of electric field. Every electric field line passing through a small-radius spherical surface also passes through a large-radius spherical surface. Hence the electric flux is independent of r.

FIGURE 24.19 The electric flux is the same through every sphere centered on a point charge.

This conclusion about the flux has an extremely important generalization. FIGURE 24.20a shows a point charge and a closed Gaussian surface of arbitrary shape and dimensions. All we know is that the charge is inside the surface. What is the electric flux through this arbitrary surface?

FIGURE 24.20 An arbitrary Gaussian surface can be approximated with spherical and radial pieces.

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One way to answer the question is to approximate the surface as a patchwork of spherical and radial pieces. The spherical pieces are centered on the charge and the radial pieces lie along lines extending outward from the charge. (Figure 24.20 is a two-dimensional drawing so you need to imagine these arcs as actually being pieces of a spherical shell.) The figure, of necessity, shows fairly large pieces that don’t match the actual surface all that well. However, we can make this approximation as good as we want by letting the pieces become sufficiently small.

The electric field is everywhere tangent to the radial pieces. Hence the electric flux through the radial pieces is zero. The spherical pieces, although at varying distances from the charge, form a complete sphere. That is, any line drawn radially outward from the charge will pass through exactly one spherical piece, and no radial lines can avoid passing through a spherical piece. You could even imagine, as FIGURE 24.20b shows, sliding the spherical pieces in and out without changing the angle they subtend until they come together to form a complete sphere.

Consequently, the electric flux through these spherical pieces that, when assembled, form a complete sphere must be exactly the same as the flux $q\text{/}{ϵ}_0$ through a spherical Gaussian surface. In other words, the flux through any closed surface surrounding a point charge q is

This surprisingly simple result is a consequence of the fact that Coulomb’s law is an inverse-square force law. Even so, the reasoning that got us to Equation 24.14 is rather subtle and well worth reviewing.