## 2012-02-22

### Requiem for a Gauss

During the spring semester of last year, I took 8.022 which is the slightly more difficult version of freshman E&M. Contrary to most classes which use SI units (meter, kilogram, second), 8.022 used Gaussian CGS units (centimeter, gram, second — hereafter referred to simply as "CGS units") partly because of tradition and partly because the professor is personally more comfortable using CGS units as he uses it in his own research too.

I have a few friends who are freshmen taking 8.022 during this spring semester, and I was rather surprised and disappointed to see that 8.022 has switched to SI units. I bumped into the professor recently and asked him about it, and he said that while he is not in favor of the change, he has to do it because the textbook ("Electricity and Magnetism" by Edward Purcell) now has a 3rd edition which has switched from CGS units to SI units, as specified by archived letters from Purcell himself when he was still alive. The professor also said that as a result, except for possibly some very specialized fields, CGS units are on the way out. Given that, I think it is appropriate to take a moment to eulogize the values of CGS units.

First of all, in electricity and magnetism, CGS units are not simply corresponding SI units divided by powers of 10. Constants in fundamental laws are redefined, meaning quantities in E&M have different dimensions altogether in CGS versus SI, so they cannot be compared directly. I'm going to give just a few examples (though eventually I hope to write up something a little more comprehensive) of where CGS shines.

Coulomb's law: In CGS, Coulomb's law is very sensibly defined. Although the consequence is that charge can be expressed dimensionally in terms of mass, length, and time, there are no annoying prefactors in Coulomb's law compared to those present in SI.

Relativity: This is where CGS really shines. The point of E&M in relativity is to show that the electric and magnetic fields are components of the same geometric object viewed in different frames of reference. In CGS, those fields have the same dimensions, so the comparison becomes even easier. Plus, there is a greater amount of symmetry in the rules for transforming the fields between frames. Finally, invariant quantities involving the fields can be constructed by adding and subtracting the fields freely in CGS, which cannot be done in SI.

Biot-Savart law: In either system, the magnetic field created by a moving charged particle is very weak. In SI, this is explained on account of the prefactors in front of that law, but that is quite misleading. The truth is that strong magnetic fields require very fast moving charged particles, and CGS shows exactly that often the speed of a moving charged particle is very low compared to the speed of light, thus explaining the weakness of the magnetic field relative to the electric field. In SI, comparing the magnetic and electric fields is like comparing apples to oranges.

Ampère's law: In SI, the prefactor in front of the current term is the permeability of free space, but there is no real reason for this, and the equation just looks more messy. In CGS, the prefactor is exactly equal to the impedance of free space; although there may not be much meaning, the symmetry is quite nice, because in general the opposite of the path integral of the electric field (i.e. the voltage) along a wire that may have some impedance is equal to the current through that impedance multiplied by the value of the impedance. Similarly, the circulation of the magnetic field around a wire is equal to the current multiplied by the impedance of free space (because that is what surrounds the wire).

Waves: In CGS, the electric and magnetic fields have the same dimensions, so the magnitudes of sinusoidal fields will also have the same dimensions and can be freely added and subtracted; in SI, this is not possible without extra prefactors. In addition, the Poynting vector is just equal to the cross product of the fields divided by the impedance of free space, which sort of shows how the vacuum can support a traveling electromagnetic wave; in SI, the prefactor is the reciprocal of the permeability of free space, but that begets the question of why the permittivity of free space is absent.

Material media: In CGS, all of the auxiliary fields, like the electric displacement field, the magnetizing field, the electric polarization density, and the magnetic polarization density have the same dimensions as the electric and magnetic fields, which themselves have the same dimensions. In SI, this is not the case, so the formulas become far more complicated and asymmetrical. This asymmetry is also present when considering the electric and magnetic susceptibilities in SI; these quantities are more consistently defined in CGS.

There's a lot more to the story than just this, and in a few days I hope to update this post with a more comprehensive writeup (probably done in LaTeX, though I have to figure out how to upload it for people online to see). It's sad to see CGS on the way out, but I suppose that is the way things must be.