Rationalizing Maxwell's Equations

Maxwell's equations in Gaussian units are shown at the right. Gaussian units are based on the cgs system of mechanical units, and were long dominant for theoretical investigations. Absolute electrostatic units (esu) are used for electrical quantities, absolute electromagnetic units (emu) for magnetic quantities. The electric field **E** and the magnetic flux density **B** are the basic vectors; **D** and **H** are obtained from them by adding the polarization **P** or subtracting the magnetization **M**, respectively. If **P** and **M** are proportional to the applied field, then **D** and **B** are related to **E** and **H** through the dielectric constant κ and the permeability μ as shown. The sources of the fields are charges and currents. The equivalent charge density and current due to polarization and magnetization are ρ_{b} and **J**_{b}, where the "b" stands for "bound." The divergence of **D** and the curl of **H** give the "free" charge and current densities, respectively, while the divergence of **E** and the curl of **B** give the total charge and current densities. The Lorentz force on a point charge q is also given, where **E** and **B** are the effective fields on the charge q, where polarization and magnetization are zero. The energy definitions of resistance R, capacitance C and inductance L are shown. Charges are in esu, currents in esu/s in these equations. The universal constant c = 2.9979 x 10^{10} cm/s, the speed of light.

The factors of 4π in the equations come from the 4π steradians of solid angle surrounding a point. If the radial electric field from a point charge q is q/r^{2}, then the flux of the electric field across a sphere of radius r with center at the charge is 4πr^{2}(q/r^{2}) = 4πq. This implies that div **E** = 4πρ when the divergence theorem is used to turn this into a differential relation.

The factors of 4π in Maxwell's equations can be eliminated by a scale change on the fields and their sources, otherwise retaining the form of each equation. This is called "rationalization" of the units, which is a convenience in theoretical studies, especially of electromagnetic waves and radiation. The greatest convenience is the elimination of the 4π in the definitions of **D** and **B**. The name was probably chosen to encourage the step, by disparaging "irrational" units, but has no other significance. Rationalization was encouraged by Oliver Heaviside and H. A. Lorentz, after which the resulting system of units is named. Heaviside-Lorentz units (hlu) are rationalized Gaussian units. The engineer's Giorgi or MKSA units have also been rationalized.

What we need to do can be found from the equation div **D** = 4πρ, or from its equivalent in the absence of polarization, div **E** = 4πρ. If we multiply ρ by a constant α, we must also divide **E** by the same constant, so that the equation **F** = q**E** is preserved. Therefore, we write div (**E**/√4π) = √4π ρ. In terms of the new field **E**' = **E**/√4π and new charge density √4π ρ, we have div **E**' = ρ', which is what we desire. Therefore, to rationalize the Gaussian units, we multiply the measures of the sources by √4π, and divide the measures of the fields by the same factor. As you can easily check, this removes the factors of 4π from Maxwell's equations while preserving the form of each equation. The primes refer to hlu, the unprimed quantities to Gaussian.

To avoid confusion, we must state that we are working with the *measures* of the quantities in terms of their units. This is the number in expressions like 2.5 ft. To convert to inches, we multiply the *measure* by 12: 12 x 2.5 ft = 30 in. The *unit* has changed from 1 ft to 1 ft/12 = 1 in in the process. Of course, to convert units, we use factors of unity, such as 12 in / 1 ft, and this factor, multiplying the combination of measure with the unit, takes care of both.

The equations that result are shown at the right. The factor √4π = 3.544907. In free space, where ρ = 0, **J** = 0, **P** = 0 and **M** = 0, they are exactly the same as in Gaussian units, so not a lot has been gained here. From what we have said, the hlu of charge is (1/3.545) esu (statcoulomb), and the hlu of potential is 3.545 esu (statvolt). The hlu of magnetic flux density is 3.545 gauss. Therefore, the charge on the electron is -1.703 x 10^{-9} hlu, and the hlu of potential is 1063.5 V. The practical current corresponding to 1/c hlu is 10/3.545 = 2.821 A.

The effect on resistance, capacitance and inductance can easily be found by the equations defining them. It is convenient to define a resistance that is c times the actual esu resistance, v = (cR) x (i/c), relating v and i/c. Then, W = cR(i/c)^{2} becomes W = (cR/4π)(√4πi/c)^{2}, or (cR)' = (cR)/4π. 1 hlu of resistance cR' is 377.1 Ω in practical units. Similarly, U = Cv^{2}/2 gives U = 4πC(v/√4π)^{2}/2, or C' = 4πC. Finally, from U = Li^{2}/2 we find U = L/4π(√4π i)^{2}/2, or L' = L/4π. Just as cR is sometimes used instead of R, so c^{2}L may be used instead of L, but the conversion is the same. The 4π's, banished from Maxwell's equations, pop up elsewhere!

Coulomb's Law in hlu is F = qq'/4πr^{2} dyne. The capacitance of a parallel-plate capacitor of area A and spacing d is C = A/4πd cm in Gaussian units, so it is C' = A/d in hlu. Similarly, a sphere of radius a has C = a in Gaussian, C' = 4πa in hlu. These, and many other similar results, are easily obtained from what we have said. Note that the hlu cm of capacity is not equal to the esu cm of capacity! It is very easy to convert between hlu and Gaussian units, since only factors of √4π are required. A theoretical argument can use hlu, then convert ot Gaussian at the end. Conversions from Gaussian to practical units are, of course, well known.

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Composed by J. B. Calvert

Created 9 October 2002

Last revised 10 October 2002