# Why are there two different SI units of magnetic charge?

Taking a look at the wiki on magnetic charge, there are two different forms of the Dirac Quantization Condition in SI units: the Weber form

$$\frac{q_e q_m}{2 \pi \hbar} \in \mathbb{Z}$$

and the Ampere*Meter form

$$\frac{q_e q_m}{2\pi \epsilon_0 \hbar c^2} \in \mathbb{Z}$$

Can someone please elucidate why the SI system would provide two possible values of the magnetic charge? Unless I am missing something, they are numerically different, as $\epsilon_0 c^2 \neq 1$. I have had trouble finding sources on this topic. Which one would be used in the Coulomb force equation for magnetic monopoles, and under which circumstances?

The 'Weber' interpretation is due to Kennelly, comes from $F=Q_mQ_m/4\pi\mu r^2$, symmetrically with Coulomb's Law. While there is no magnetic charge, the terminology is balanced if one supposes that F = HQ, and B is defined as D.

The 'Ampere-metre' version is due to Sommerfield (of 'fine structure constant' fame). This can be rewritten as C.(m/s), such as in Qv. An electric current is presented as Qv/l, = C.(m/s)/m = C/s = A.

Both interpretations are to be found in various texts. Other variations are things like m=µIA (Wb.m) or m=IA (Am²). Since this is moment of magnetism, the unit of magnetism is m/length = Wb, or A.m.

The A.m system gives numbers closer to unity than the Weber system, which is why it has some currency.

The two equations are not both in SI. The upper equation is in CGS units, a system closely related to SI that was a favorite of spectroscopists because of the simple form of Coulomb's law in it. That lead to it being the unit system of choice in standard texts, Purcell and Jackson, which were updated only recently, and in astronomy. Some theorists also like CGS because in it electric and magnetic fields have the same units.

I'm not, personally, a fan of CGS for two reasons. First, $\mathbf{E}$ and $\mathbf{B}$ should only have the same units when the speed of light, $c$, is 1. Just like the units for energy versus momentum. Second, the $4\pi$ belongs in Coulomb's law because it gives the denominator physical meaning. With $$F = \frac{q_1 q_2}{\epsilon_0 4\pi r^2}$$ it is clear that the denominator is the area of a sphere, as it should be. When the $4\pi$ is moved to the field equations, they just seem arbitrary.

Here's a wiki page on E&M equations in different unit systems.

• I'm don't think that's true, there are three equations listed on wiki: the first is the CGS equation, and the second two (which I copied in my post) are labeled as SI. I do find the answer interesting though and am happy to learn more. – doublefelix Sep 20 '16 at 18:41
• The decision that most differentiates what is usually called "cgs" from SI isn't actually the choice of centimeters versus meters and grams versus kilograms, but the choice of how to construct the units and constants of E&M. The size of base-units thing is a bit of a distraction. – dmckee --- ex-moderator kitten Sep 20 '16 at 18:44
• I'm not the one who dubbed them CGS, @dmckee. Sorry I didn't look the your question closely enough, user. I've never heard of "Weber convention" and searching for that exact term leads only to Wikipedia and sources derived from it. My copy of Jackson isn't handy, which is the source they cite, so I can't comment on the origin of it. What they're doing, though, is absorbing $\mu_0 = 1/(c^2\epsilon_0)$ into the definition of the unit of magnetic charge. Since monopoles are purely hypothetical, there's no SI standard for their units. – Sean E. Lake Sep 20 '16 at 18:54
• "I'm not the one who dubbed them CGS" No rebuke intended, just a comment for anyone new to the muddle that E&M units can seem the first (or indeed, third) time you see them. For a while in my education I didn't notice that the difference between "cgs" and "SI" was more than just the base units of length and mass which left my head swirling when I tried to reconcile the expressions found in different texts. Possible just a sign that I am particularly dense, but in case anyone else was in that same boat... – dmckee --- ex-moderator kitten Sep 20 '16 at 18:58
• @SeanLake the absorbing of $\mu_0$ I think answers my question, thank you. If I understand correctly, then with the Weber convention, the magnetic Coulomb force law would change from $F=qB$ (Amp*m) to $F=qB/\mu_0$ (Weber)? – doublefelix Sep 21 '16 at 0:16