Demystifying Electromagnetic Equations

Excerpt

In classical Newtonian mechanics, equations and formulas never change form. However, the same thing cannot be said about the equations and formulas of electromagnetic theory, which often change form when converted from one system of units to another. For this reason electromagnetic textbooks are almost always written using a single system of units, and the technical professionals who read them end up being comfortable in only that system. When they encounter a new and important formula in unfamiliar units later on, they must either use a conversion table to change the formula to their preferred system of units or try to become familiar with the formula's units. Although conversion tables usually give the correct answer, they turn their users into computers who must push around numbers and variables without any true understanding of what is being done. It is probably unwise to rely blindly on conversion tables if one must be absolutely sure the transformed formula is correct. That leaves the second option: becoming familiar with the formula's units. The drawback here is that even if a textbook can be found that uses the formula's units, it has been written to teach the basic principles of electromagnetism rather than what the technical professional is looking for, i.e., a detailed explanation of how to convert equations from one system of units to another. This book provides exactly that, while at the same time assuming a good—but not necessarily advanced—understanding of electricity and magnetism.

There are five widely recognized systems of electromagnetic units; four are connected to the centimeter-gram-second (cgs) system of mechanical units and one is connected to the meter-kilogram-second (mks) system of mechanical units. The four connected to the cgs mechanical units are the cgs Gaussian system, the Heaviside-Lorentz system, the cgs electrostatic system, and the cgs electromagnetic system. The system connected to the mks mechanical units is the Système International or rationalized mks system. The units of the Système International or rationalized mks system are often called SI units. The cgs electrostatic and cgs electromagnetic systems of units were developed first. These are the units in which Maxwell's equations—the foundation of classical electromagnetic theory—were first proposed during the middle of the nineteenth century. The Heaviside-Lorentz and cgs Gaussian systems were introduced at the end of the nineteenth century, followed almost immediately at the beginning of the twentieth century by the rationalized mks system (SI units). The rationalized mks system is the most popular electromagnetic system in use today; almost all introductory textbooks use SI units to explain the principles of electricity and magnetism. This book explains all five systems in depth, along with two systems of mostly historical interest; the nineteenth century system of “practical” units and the unrationalized mks system.

One chronic problem found in many articles and books about systems of units is that the customary language of physics and engineering can permit ambiguity while sounding exact. Suppose, for example, we say “The electric-current unit in the cgs electrostatic system is the statamp and the electric-current unit in the cgs electromagnetic system is the abamp, with

To avoid this sort of ambiguity, we introduce here the idea of U and N operators, with a U operator returning just the units associated with a physical quantity and an N operator returning just the pure number, or numeric component, associated with a physical quantity. In the cgs system, for example, we have

(c)

(I)

(I)

The abbreviations of the SI units are, unfortunately, another possible source of confusion when separating equations into numeric components and units. For example, the standard abbreviation for the SI unit of charge, the coulomb, is C. The capacitance of a circuit element is also traditionally represented as C, and we have already seen that c is used to represent the speed of light. If all three quantities—the coulomb, the speed of light, and the capacitance—have to be included in the same equation, there will be problems. To avoid this source of confusion, we have lengthened the standard abbreviations for the electromagnetic units, representing coulomb by coul, ampere by amp, and so on. This makes the notation less confusing, but the reader should note that the abbreviations used here, although easily understandable, are not the official, internationally approved symbols for the SI units. These international symbols are, in any case, of fairly recent vintage and can be found in virtually all modern textbooks on electromagnetic theory.

One final point worth mentioning is how we treat rationalization of electromagnetic equations. During the middle of the twentieth century it became clear that there were two different schools of thought concerning the rationalization of electromagnetic equations: one that it was a rescaling of the electromagnetic units, and the other that it was a rescaling of the electromagnetic quantities themselves. Both views can be used to deduce the same systems of electromagnetic equations, and both views allow engineers and scientists to transform electromagnetic measurements from one system to another correctly. In the end, neither side convinced the other of the correctness of its views and the controversy faded away. For the purposes of this book, we take the position that rationalization is a rescaling of electromagnetic physical quantities rather than a change of units, not only because it is then easier to describe the units of the rationalized and unrationalized electromagnetic systems but also because it makes the transformation of equations to and from rationalized electromagnetic systems a straightforward process. The opposite position, that rationalization just involves rescaled units, is not necessarily incorrect—that is, after all, how the idea of rationalization was first proposed in the nineteenth century—but it can easily become confusing in a book of this sort.