A.12 Units in EM

There are several systems of units in electromagnetism. We are familiar with SI units, but Gaussian units are still very common and are used, for instance, in Shankar.

In SI units the force between two currents is used to define the unit of current, and hence the unit of charge. (Currents are much easier to calibrate and manipulate in the lab than charges.) The constant ${\mu }_0$ is defined as $4\pi \times 10^{-7}$ N A${}^{-2}$, with the magnitude chosen so that the Ampère is a “sensible” sort of size. Then Coulomb’s law reads

and ${\epsilon }_0$ has to be obtained from experiment. (Or, these days, as the speed of light is now has a defined value, ${\epsilon }_0$ is obtained from $1∕\left({\mu }_0{c}^2\right)$.)

However one could in principle equally decide to use Coulomb’s law to define charge. This is what is done in Gaussian units, where by definition

Then there is no separate unit of charge; charges are measured in N${}^{1∕2}$ m (or the non-SI equivalent): $e=4.803\times 10^{-10}$ g${}^{1∕2}$ cm${}^{3∕2}$ s${}^{-1}$. (You should never need that!) In these units, ${\mu }_0=4\pi ∕c^2$. Electric and magnetic fields are also measured in different units.

The following translation table can be used:

Note that $eE$ is the same in both systems of units, but $eB$ in SI units is replaced by $eB∕c$ in Gaussian units. Thus the Bohr magneton ${\mu }_B$ is $e\hslash ∕2m$ in SI units, but $e\hslash ∕2mc$ in Gaussian units, and ${\mu }_{B}B$ has dimesions of energy in both systems.

The fine-structure constant $\alpha$ is a dimensionless combination of fundamental units, and as such takes on the same value ($\approx 1∕137$) in all systems. In SI it is defined as $\alpha =e^2∕\left(4\pi {\epsilon }_0\hslash c\right)$, in Gaussian units as $\alpha =e^2∕\left(\hslash c\right)$. In all systems, therefore, Coulomb’s law between two particles of charge $z_{1}e$ and $z_{2}e$ can be written

and this is the form I prefer.

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