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

$\mathcal{L}=-\frac{1}{4}F^2$

and the… other convention

$\mathcal{L}=-\frac{1}{16\pi}F^2.$

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 →

Natural Wonder

Evan Berkowitz

© Evan Berkowitz

All the Wrong Fuses and Splices