The same equations have the same solutions
The total amount of information which has been acquired about the physical world since the beginning of scientific progress is enormous, and it seems almost impossible that any one person could know a reasonable fraction of it. But it is actually quite possible for a physicist to retain a broad knowledge of the physical world rather than to become a specialist in some narrow area. The reasons for this are threefold: First, there are great principles which apply to all the different kinds of phenomena—such as the principles of the conservation of energy and of angular momentum. A thorough understanding of such principles gives an understanding of a great deal all at once. Second, there is the fact that many complicated phenomena, such as the behavior of solids under compression, really basically depend on electrical and quantum-mechanical forces, so that if one understands the fundamental laws of electricity and quantum mechanics, there is at least some possibility of understanding many of the phenomena that occur in complex situations. Finally, there is a most remarkable coincidence: The equations for many different physical situations have exactly the same appearance. Of course, the symbols may be different—one letter is substituted for another—but the mathematical form of the equations is the same. This means that having studied one subject, we immediately have a great deal of direct and precise knowledge about the solutions of the equations of another. We are now finished with the subject of electrostatics, and will soon go on to study magnetism and electrodynamics. But before doing so, we would like to show that while learning electrostatics we have simultaneously learned about a large number of other subjects. We will find that the equations of electrostatics appear in several other places in physics. By a direct translation of the solutions (of course the same mathematical equations must have the same solutions) it is possible to solve problems in other fields with the same ease—or with the same difficulty—as in electrostatics. The equations of electrostatics, we know, are \begin{align} \label{Eq:II:12:1} \FLPdiv{(\kappa\FLPE)}&=\frac{\rho_{\text{free}}}{\epsO},\\[1ex] \label{Eq:II:12:2} \FLPcurl{\FLPE}&=\FLPzero. \end{align}  (We take the equations of electrostatics with dielectrics so as to have the most general situation.) The same physics can be expressed in another mathematical form: \begin{align} \label{Eq:II:12:3} \FLPE=&-\FLPgrad{\phi},\\[1ex] \label{Eq:II:12:4} \FLPdiv{(\kappa\,\FLPgrad{\phi})}&=-\frac{\rho_{\text{free}}}{\epsO}. \end{align} Now the point is that there are many physics problems whose mathematical equations have the same form. There is a potential ($\phi$) whose gradient multiplied by a scalar function ($\kappa$) has a divergence equal to another scalar function ($-\rho_{\text{free}}/\epsO$). Whatever we know about electrostatics can immediately be carried over into that other subject, and vice versa. (It works both ways, of course—if the other subject has some particular characteristics that are known, then we can apply that knowledge to the corresponding electrostatic problem.) We want to consider a series of examples from different subjects that produce equations of this form.