Statics
We begin now our detailed study of the theory of electromagnetism. All of electromagnetism is contained in the Maxwell equations. 
  Maxwell’s equations: \begin{align} \label{Eq:II:4:1} \FLPdiv{\FLPE}&=\frac{\rho}{\epsO},\\[1ex] \label{Eq:II:4:2} \FLPcurl{\FLPE}&=-\ddp{\FLPB}{t},\\[1ex] \label{Eq:II:4:3} c^2\,\FLPcurl{\FLPB}&=\ddp{\FLPE}{t}+\frac{\FLPj}{\epsO},\\[1ex] \label{Eq:II:4:4} \FLPdiv{\FLPB}&=0. \end{align}  The situations that are described by these equations can be very complicated.  We will consider first relatively simple situations, and learn how to handle them before we take up more complicated ones. The easiest circumstance to treat is one in which nothing depends on the time—called the static case. All charges are permanently fixed in space, or if they do move, they move as a steady flow in a circuit (so $\rho$ and $\FLPj$ are constant in time). In these circumstances, all of the terms in the Maxwell equations which are time derivatives of the field are zero. In this case, the Maxwell equations become: 
  Electrostatics: \begin{align} \label{Eq:II:4:5} \FLPdiv{\FLPE}&=\frac{\rho}{\epsO},\\[1ex] \label{Eq:II:4:6} \FLPcurl{\FLPE}&=\FLPzero. \end{align}   Magnetostatics: \begin{align} \label{Eq:II:4:7} \FLPcurl{\FLPB}&=\frac{\FLPj}{\epsO c^2},\\[1ex] \label{Eq:II:4:8} \FLPdiv{\FLPB}&=0. \end{align}  You will notice an interesting thing about this set of four equations. It can be separated into two pairs. The electric field $\FLPE$ appears only in the first two, and the magnetic field $\FLPB$ appears only in the second two. The two fields are not interconnected. This means that electricity and magnetism are distinct phenomena so long as charges and currents are static. The interdependence of $\FLPE$ and $\FLPB$ does not appear until there are changes in charges or currents, as when a condensor is charged, or a magnet moved. Only when there are sufficiently rapid changes, so that the time derivatives in Maxwell’s equations become significant, will $\FLPE$ and $\FLPB$ depend on each other.  Now if you look at the equations of statics you will see that the study of the two subjects we call electrostatics and magnetostatics is ideal from the point of view of learning about the mathematical properties of vector fields. Electrostatics is a neat example of a vector field with zero curl and a given divergence. Magnetostatics is a neat example of a field with zero divergence and a given curl. The more conventional—and you may be thinking, more satisfactory—way of presenting the theory of electromagnetism is to start first with electrostatics and thus to learn about the divergence. Magnetostatics and the curl are taken up later. Finally, electricity and magnetism are put together. We have chosen to start with the complete theory of vector calculus. Now we shall apply it to the special case of electrostatics, the field of $\FLPE$ given by the first pair of equations.  We will begin with the simplest situations—ones in which the positions of all charges are specified. If we had only to study electrostatics at this level (as we shall do in the next two chapters), life would be very simple—in fact, almost trivial. Everything can be obtained from Coulomb’s law and some integration, as you will see. In many real electrostatic problems, however, we do not know, initially, where the charges are. We know only that they have distributed themselves in ways that depend on the properties of matter. The positions that the charges take up depend on the $\FLPE$ field, which in turn depends on the positions of the charges. Then things can get quite complicated. If, for instance, a charged body is brought near a conductor or insulator, the electrons and protons in the conductor or insulator will move around. The charge density $\rho$ in Eq. (4.5) may have one part that we know about, from the charge that we brought up; but there will be other parts from charges that have moved around in the conductor. And all of the charges must be taken into account. One can get into some rather subtle and interesting problems. So although this chapter is to be on electrostatics, it will not cover the more beautiful and subtle parts of the subject. It will treat only the situation where we can assume that the positions of all the charges are known. Naturally, you should be able to do that case before you try to handle the other ones. 