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16.4: The Gaussian Mixed System

A problem arises if we are dealing with a situation in which there are both “electrostatic” and “electromagnetic” quantities. The “mixed system”, which is used very frequently, in CGS literature, uses esu for quantities that are held to be “electrostatic” and emu for quantities that are held to be “electromagnetic”, and it seems to be up to each author to decide which quantities are to be regarded as “electrostatic” and which are “electromagnetic. Because different quantities are to be expressed in different sets of units within a single equation, the equation must include the conversion factor \(c = 2.997 \ 924 \ 58\) \(\times 10^{10}\)      in strategic positions within the equation.

The most familiar example of this is the equation for the force \(\textbf{F}\) experienced by a charge \(Q\) when it is moving with velocity \(\textbf{v}\) in an electric field \(\textbf{E}\) and a magnetic field \(\textbf{B}\). This equation is liable to appear either as

\[F = Q \left( \textbf{E} + \frac{\textbf{v} \times \textbf{H}}{c} \right) \]

or as        \[F= Q \left( \textbf{E} + \frac{\textbf{v} \times \textbf{B}}{c}\right) .\]

It can appear in either of these forms because, if CGS emu are used, \(B\) and \(H\) are numerically equal in vacuo. The conversion factor \(c\) appears in these equations, because it is understood (by those who understand CGS units) that \(Q\) and \(E\) are to be expressed in esu, while \(B\) or \(H\) is to be expressed in emu, and the conversion factor \(c\) is necessary to convert it to esu.

It should be noted that in all previous chapters in these notes, equations balance dimensionally, and the equations are valid in any coherent system of units, not merely SI. Difficulties arise, of course, if you write an equation that is valid only so long as a particular set of units is used, and even more difficulties arise if some quantities are to be expressed in one system of units, and other quantities are to be expressed in another system of units.

An analogous situation is to be found in some of the older books on thermodynamics, where it is possible to find the following equation:

\[C_P - C_V = R \ / \ J.\]

This equation expresses the difference in the specific heat capacities of an ideal gas, measured at constant pressure and at constant volume. In equation 16.4.3, it is understood that \(C_P\) and \(C_V\) are to be expressed in calories per gram per degree, while the universal gas constant is to be expressed in ergs per gram per degree. The factor \(J\) is a conversion factor between erg and calories. Of course the sensible way to write the equation is merely

\[C_P - C_V = R.\]

This is valid whatever units are used, be they calories, ergs, joules, British Thermal Units or kWh, as long as all quantities are expressed in the same units.  Yet it is truly extraordinary how many electrical equations are to be found in the literature, in which different units are to be used for dimensionally similar quantities.

Maxwell’s equations may appear in several forms. I take one at random from a text written in CGS:

\[div \textbf{B} = 0,\]

\[div \textbf{D} = 4 \pi \rho,\]

\[c \textbf{curlH} = \dot{\textbf{D}} + 4 \pi \textbf{J},\]

\[c \textbf{curlE} = - \dot{\textbf{B}}.\]

The factor \(c\) occurs as a conversion factor, since some quantities are to be expressed in esu and some in emu. The \(4\pi\) arises because of a different definition (unrationalized) of permeability. In some versions there may be no distinction between \(\textbf{B}\) and \(\textbf{H}\), or between \(\textbf{E}\) and \(\textbf{D}\), and the \(4 \pi\) and the \(c\) may appear in various places in the equations.

(It may also be remarked that, in the earlier papers, and in Maxwell’s original writings, vector notation is not used, and the equations appear extremely cumbersome and all but incomprehensible to modern eyes.)

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