Let’s remind ourselves where we are in terms of discovering laws about the electromagnetic field. Gauss’s law, which you studied in Chapter 24, states a very general property of the electric field. It says that charges create electric fields in such a way that the electric flux of the field is the same through any closed surface surrounding the charges. FIGURE 31.11 illustrates this idea by showing the field lines passing through a Gaussian surface enclosing a charge.

FIGURE 31.11 A Gaussian surface enclosing a charge.

The mathematical statement of Gauss’s law for the electric field says that for any closed surface enclosing total charge $Q_{\text{in}}$, the net electric flux through the surface is

The circle on the integral sign indicates that the integration is over a closed surface. The units of electric flux are those of electric field times area, namely V m.

There’s an analogous equation for magnetic fields, an equation we implied in Chapter 29—where we noted that isolated north or south poles do not exist—but didn’t explicitly write it down. FIGURE 31.12 shows a Gaussian surface around a magnetic dipole. Magnetic field lines form continuous curves, without starting or stopping, so every field line leaving the surface at some point must reenter it at another. Consequently, the net magnetic flux over a closed surface is zero.

FIGURE 31.12 There is no net flux through a Gaussian surface around a magnetic dipole.

We’ve shown only one surface and one magnetic field, but this conclusion turns out to be a general property of magnetic fields. Because every north pole is accompanied by a south pole, we can’t enclose a “net pole” within a surface. Thus Gauss’s law for magnetic fields is

Equation 31.14 is the mathematical statement that Coulomb electric field lines start and stop on charges. Equation 31.15 is the mathematical statement that magnetic field lines form closed loops; they don’t start or stop (i.e., there are no isolated magnetic poles). These two versions of Gauss’s law are important statements about what types of fields can and cannot exist. They will become two of Maxwell’s equations.

The third field law we’ve established is Faraday’s law:

where the line integral of $\vec{E}$ is around the closed curve that bounds the surface through which the magnetic flux ${\text{Φ}}_{\text{m}}$ is calculated. Equation 31.16 is the mathematical statement that an electric field can also be created by a changing magnetic field. The correct use of Faraday’s law requires a convention for determining when fluxes are positive and negative. The sign convention will be given in the next section, where we discuss the fourth and last field equation—an analogous equation for magnetic fields.