The magnetic field of steady currents; Ampère’s law
We have seen that there is a force on a wire in the presence of a magnetic field, produced, say, by a magnet. From the principle that action equals reaction we might expect that there should be a force on the source of the magnetic field, i.e., on the magnet, when there is a current through the wire.1 There are indeed such forces, as is seen by the deflection of a compass needle near a current-carrying wire. Now we know that magnets feel forces from other magnets, so that means that when there is a current in a wire, the wire itself generates a magnetic field. Moving charges, then, produce a magnetic field. We would like now to try to discover the laws that determine how such magnetic fields are created. The question is: Given a current, what magnetic field does it make? The answer to this question was determined experimentally by three critical experiments and a brilliant theoretical argument given by Ampère. We will pass over this interesting historical development and simply say that a large number of experiments have demonstrated the validity of Maxwell’s equations. We take them as our starting point. If we drop the terms involving time derivatives in these equations we get the equations of magnetostatics: \begin{equation} \label{Eq:II:13:12} \FLPdiv{\FLPB}=0 \end{equation}  and \begin{equation} \label{Eq:II:13:13} c^2\FLPcurl{\FLPB}=\frac{\FLPj}{\epsO}. \end{equation}  These equations are valid only if all electric charge densities are constant and all currents are steady, so that the electric and magnetic fields are not changing with time—all of the fields are “static.” We may remark that it is rather dangerous to think that there is such a thing as a static magnetic situation, because there must be currents in order to get a magnetic field at all—and currents can come only from moving charges. “Magnetostatics” is, therefore, an approximation. It refers to a special kind of dynamic situation with large numbers of charges in motion, which we can approximate by a steady flow of charge. Only then can we speak of a current density $\FLPj$ which does not change with time. The subject should more accurately be called the study of steady currents. Assuming that all fields are steady, we drop all terms in $\ddpl{\FLPE}{t}$ and $\ddpl{\FLPB}{t}$ from the complete Maxwell equations, Eqs. (2.41), and obtain the two equations (13.12) and (13.13) above. Also notice that since the divergence of the curl of any vector is necessarily zero, Eq. (13.13) requires that $\FLPdiv{\FLPj}=0$. This is true, by Eq. (13.8), only if $\ddpl{\rho}{t}$ is zero. But that must be so if $\FLPE$ is not changing with time, so our assumptions are consistent.  The requirement that $\FLPdiv{\FLPj}=0$ means that we may only have charges which flow in paths that close back on themselves. They may, for instance, flow in wires that form complete loops—called circuits. The circuits may, of course, contain generators or batteries that keep the charges flowing. But they may not include condensers which are charging or discharging. (We will, of course, extend the theory later to include dynamic fields, but we want first to take the simpler case of steady currents.) Now let us look at Eqs. (13.12) and (13.13) to see what they mean. The first one says that the divergence of $\FLPB$ is zero. Comparing it to the analogous equation in electrostatics, which says that $\FLPdiv{\FLPE}=\rho/\epsO$, we can conclude that there is no magnetic analog of an electric charge. There are no magnetic charges from which lines of $\FLPB$ can emerge. If we think in terms of “lines” of the vector field $\FLPB$, they can never start and they never stop. Then where do they come from?  Magnetic fields “appear” in the presence of currents; they have a curl proportional to the current density. Wherever there are currents, there are lines of magnetic field making loops around the currents. Since lines of $\FLPB$ do not begin or end, they will often close back on themselves, making closed loops. But there can also be complicated situations in which the lines are not simple closed loops. But whatever they do, they never diverge from points. No magnetic charges have ever been discovered, so $\FLPdiv{\FLPB}=0$. This much is true not only for magnetostatics, it is always true—even for dynamic fields. The connection between the $\FLPB$ field and currents is contained in Eq. (13.13). Here we have a new kind of situation which is quite different from electrostatics, where we had $\FLPcurl{\FLPE}=\FLPzero$. That equation meant that the line integral of $\FLPE$ around any closed path is zero: \begin{equation*} \underset{\text{loop}}{\oint}\FLPE\cdot d\FLPs=0. \end{equation*}  We got that result from Stokes’ theorem, which says that the integral around any closed path of any vector field is equal to the surface integral of the normal component of the curl of the vector (taken over any surface which has the closed loop as its periphery). Applying the same theorem to the magnetic field vector and using the symbols shown in Fig. 13–6, we get \begin{equation} \label{Eq:II:13:14} \oint_\Gamma\FLPB\cdot d\FLPs= \int_S(\FLPcurl{\FLPB})\cdot\FLPn\,dS. \end{equation}  Taking the curl of $\FLPB$ from Eq. (13.13), we have \begin{equation} \label{Eq:II:13:15} \oint_\Gamma\FLPB\cdot d\FLPs= \frac{1}{\epsO c^2} \int_S\FLPj\cdot\FLPn\,dS. \end{equation}  The integral over $S$, according to (13.5), is the total current $I$ through the surface $S$. Since for steady currents the current through $S$ is independent of the shape of $S$, so long as it is bounded by the curve $\Gamma$, one usually speaks of “the current through the loop $\Gamma$.” We have, then, a general law: the circulation of $\FLPB$ around any closed curve is equal to the current $I$ through the loop, divided by $\epsO c^2$: \begin{equation} \label{Eq:II:13:16} \oint_\Gamma\FLPB\cdot d\FLPs= \frac{I_{\text{through $\Gamma$}}}{\epsO c^2}. \end{equation}  This law—called Ampère’s law—plays the same role in magnetostatics that Gauss’ law played in electrostatics. Ampère’s law alone does not determine $\FLPB$ from currents; we must, in general, also use $\FLPdiv{\FLPB}=0$. But, as we will see in the next section, it can be used to find the field in special circumstances which have certain simple symmetries.