Why does the formula for magnetic field force include the speed of light in the denominator in cgs units? Where does the extra $c$ go in SI units?


The natural way to write it in this notation is

$$F = q(E + \beta \times B)$$

where $\beta$ is the velocity measured in natural units - the velocity as a fraction of the speed of light.

In the CGS system, we instead write $\beta = \frac{v}{c}$ and the equation becomes

$$F = q(E + \frac{v}{c} \times B)$$

That's not silly enough, though, so we go really crazy and invent two new constants and give them obscure names: "permittivity of free space" ($\epsilon_0$) and "permeability of free space" ($\mu_0$) and relate them so $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$.

If that's where it ended the equation would be

$$F = q(E + v \sqrt{\mu_0 \epsilon_0} \times B)$$

But that would still not be silly enough. Instead, we then decided to invent a new magnetic field given by $B' = \sqrt{\mu_0/4\pi} B$ and a new electric field given by $E' = \frac{E}{\sqrt{4\pi \epsilon_0}}$. This would give us the new formula

$$F = q(\sqrt{4\pi \epsilon_0}E' + v \times \sqrt{4\pi \epsilon_0} B')$$

but we need to be more sillier, so we define a new measure of charge by $q' = q\sqrt{4\pi\epsilon_0}$. At last we get

$$F = q'(E' + v \times B')$$

and there you have the SI Lorentz force law. So the answer is that the speed of light appears only because of complicated ways of changing the units.

It's a bit instructive to look at the energy densities of the fields. In the original units they're just $\frac{E^2}{8\pi}$ and $\frac{B^2}{8\pi}$. But we made a bunch of units changes, so these become $\frac{4\pi \epsilon_0 E'^2}{8\pi} = \frac{\epsilon_0 E'^2}{2}$ and $\frac{4\pi B'^2}{8 \pi\mu_0} = \frac{B'^2}{2\mu_0}$

So the cost of making the speed of light go away in the Lorentz force law is that we pick up these strange $\mu_0$ and $\epsilon_0$ constants that hide the speed of light and flop around through all the subsequent formulas.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy