# Dimensional Analysis in Electromagnetism (SI vs Gaussian-cgs)

Looking at Konopinski's formula for conjugate momentum (in the comment after equation 3 of "What the Vector Potential Describes"):

$$\mathbf{p}= M \mathbf{v} + q\mathbf{A} /c$$

it is plain enough that $$M \mathbf{v}$$ is the momentum, but if we naively take the usual notion of the magnetic vector potential $$\mathbf{A}$$to have dimensions of magnetic flux/length or, equivalently, momentum/charge (weber/meter = (kg $$\cdot$$ m/s)/coulomb) the following dimensional expression obtains for the right hand side of the $$+$$:

charge $$\cdot$$ ((mass $$\cdot$$ velocity)/charge)/velocity

So in a naive dimensional analysis this simplifies to mass; but this is not commensurable with momentum (mass $$\cdot$$ velocity) on the left side of the $$+$$.

Likewise, in that same comment, Konopinski describes the "interaction energy" as:

$$q[\phi-\mathbf{v}\cdot\mathbf{A}/c]$$

a naive dimensional analysis would notice a similar discrepancy -- this time in that the electric scalar potential, phi is energy/charge whereas the dimensional expression of the right hand side of the $$-$$ is:

velocity $$\cdot$$ (momentum/charge)/velocity

which, again, naively, simplifies to momentum/charge rather than the required energy/charge.

Obviously, naive dimensional analysis doesn't work here.

• – Qmechanic Sep 6 '14 at 23:39

## 1 Answer

You seem to be doing dimensional analysis in SI units. The paper seems to using Gaussian units. The magnetic field differs between these units by a factor $c$. In SI units we have $$\mathbf F = q(\mathbf E + \mathbf v \times \mathbf B \tag{SI})$$ but in Gaussian units $$\mathbf F = q(\mathbf E + \frac{\mathbf{v}}{c} \times \mathbf B \tag{G}).$$ The latter definition makes electric and magnetic fields have the same unit and is the form used in the paper you linked.

• Thanks. In order to make this question findable by others with the same problem, I should probably change the title of the question. For instance "Dimensional Analysis and Electromagnetism" since Gaussian units are so often used in the history of papers on electromagnetism. Its truly horrifying that different "units" systems differ not only in their units but in their underlying dimensions. Enough damage is done simply by people failing to get their units consistent and commensurable in arithmetic expressions. Inconsistent dimensionality between different systems multiplies confusion. – James Bowery Sep 7 '14 at 0:21