# All the Wrong Fuses and Splices

March 7, 2012

There are a few different conventions for electromagnetism, and keeping them straight can be a headache.

Here I collect the differences between the particle physics convention

and the… other convention

The particle physics convention is the same as SI if you replace $\varepsilon_{0}$, $\mu_{0}$, and $c \rightarrow 1$, so you can use any result in e.g. Griffiths easily.

$\mathcal{L}$ $-\frac{1}{4}F^2$ $-\frac{1}{16\pi}F^2$
Units   “Particle Physics” “Gaussian”
“Rationalized”
Lagrangian Density $\mathcal{L}$ $\frac{1}{2}\left(E^2-B^2\right)$ $\frac{1}{8\pi}\left(E^2-B^2\right)$
Hamiltonian Density $\mathcal{H}$ $\frac{1}{2}\left(E^2+B^2\right)$ $\frac{1}{8\pi}\left(E^2+B^2\right)$
Maxwell’s Equations $\partial_\mu F^{\mu\nu}$ $j^\nu$ $4\pi j^\nu$
Fine Structure Constant $\alpha$ $\frac{e^2}{4\pi}$ $e^2$
Flux quantum $\Phi_0$ $\frac{2\pi}{e}$ $\frac{1}{2e}$
Coulomb’s Law $\vec{E}$ $\frac{Q}{4\pi r^2}\hat{r}$ $\frac{Q}{r^2}\hat{r}$
Biot-Savart Law $\vec{B}$ $\frac{1}{4\pi}\oint \frac{I d\vec{\ell}\times \hat{r}}{r^2}$ $\oint \frac{I d\vec{\ell}\times \hat{r}}{r^2}$

You can see it’s just a matter of keeping track of where the $4\pi$ goes, which is what makes it so tough to remember.

March 7, 2012