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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?

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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 unit's 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.

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