24.2 The Concept of Flux
FIGURE 24.7a shows an opaque box surrounding a region of space. We can’t see what’s in the box, but there’s an electric field vector coming out of each face of the box. Can you figure out what’s in the box?
FIGURE 24.7 Although we can’t see into the boxes, the electric fields passing through the faces tell us something about what’s in them.
Of course you can. Because electric fields point away from positive charges, it seems clear that the box contains a positive charge or charges. Similarly, the box in FIGURE 24.7b certainly contains a negative charge.
What can we tell about the box in FIGURE 24.7c? The electric field points into the box on the left. An equal electric field points out on the right. An electric field passes through the box, but we see no evidence there’s any charge (or at least any net charge) inside the box. These examples suggest that the electric field as it passes into, out of, or through the box is in some way connected to the charge within the box.
To explore this idea, suppose we surround a region of space with a closed surface, a surface that divides space into distinct inside and outside regions. Within the context of electrostatics, a closed surface through which an electric field passes is called a Gaussian surface, named after the 19th-century mathematician Karl Gauss. This is an imaginary, mathematical surface, not a physical surface, although it might coincide with a physical surface. For example, FIGURE 24.8a shows a spherical Gaussian surface surrounding a charge.
FIGURE 24.8 Gaussian surface surrounding a charge. A two-dimensional cross section is usually easier to draw.
A closed surface must, of necessity, be a surface in three dimensions. But three-dimensional pictures are hard to draw, so we’ll often look at two-dimensional cross sections through a Gaussian surface, such as the one shown in FIGURE 24.8b. Now we can tell from the spherical symmetry of the electric field vectors poking through the surface that the positive charge inside must be spherically symmetric and centered at the center of the sphere.
A Gaussian surface is most useful when it matches the shape and symmetry of the field. For example, FIGURE 24.9a shows a cylindrical Gaussian surface—a closed cylinder—surrounding some kind of cylindrical charge distribution, such as a charged wire. FIGURE 24.9b simplifies the drawing by showing two-dimensional end and side views. Because the Gaussian surface matches the symmetry of the charge distribution, the electric field is everywhere perpendicular to the side wall and no field passes through the top and bottom surfaces.
FIGURE 24.9 A Gaussian surface is most useful when it matches the shape of the field.
For contrast, consider the spherical surface in FIGURE 24.10. This is also a Gaussian surface, and the protruding electric field tells us there’s a positive charge inside. It might be a point charge located on the left side, but we can’t really say. A Gaussian surface that doesn’t match the symmetry of the charge distribution isn’t terribly useful.
FIGURE 24.10 Not every surface is useful for learning about charge.
These examples lead us to two conclusions:
The electric field, in some sense, “flows” out of a closed surface surrounding a region of space containing a net positive charge and into a closed surface surrounding a net negative charge. The electric field may flow through a closed surface surrounding a region of space in which there is no net charge, but the net flow is zero.
The electric field pattern through the surface is particularly simple if the closed surface matches the symmetry of the charge distribution inside.
The electric field doesn’t really flow like a fluid, but the metaphor is a useful one. The Latin word for flow is flux, and the amount of electric field passing through a surface is called the electric flux. Our first conclusions, stated in terms of electric flux, are
There is an outward flux through a closed surface around a net positive charge.
There is an inward flux through a closed surface around a net negative charge.
There is no net flux through a closed surface around a region of space in which there is no net charge.
This chapter has been entirely qualitative thus far as we’ve established pictorially what we mean by symmetry, flux, and the fact that the electric flux through a closed surface has something to do with the charge inside. In the next two sections you’ll learn how to calculate the electric flux through a surface and how the flux is related to the enclosed charge. That relationship, Gauss’s law, will allow us to determine the electric fields of some interesting and useful charge distributions.