# Magnetic Field and the Speed of Light

Is it just a historical choice that both magnetic field and the Lorentz force equation include the speed of light? I figure that whoever wrote up the equations (in cgs!) could have put both factors of $c$ in either the force equation, or have define the magnetic field as being smaller by a factor of c- but they didn't. Any ideas why?

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Related question by OP: physics.stackexchange.com/q/63584/2451 –  Qmechanic May 7 '13 at 18:17

## 1 Answer

There were historically several systems of units (ancestors to modern SI, CGS electric, CGS magnetic, CGS Gaussian, CGS by Heaviside), and the ultimate choice in favour of Gaussian CGS was made when Special Relativity has united electric and magnetic fields into one electromagnetic field tensor. Only in Gaussian (and Heavisidian) versions, these fields take no additional factors and make components of the field tensor immediately. Any other choice just looks ugly.

For the reference, the electromagnetic field tensor in CGS has a form $$F_{\mu\nu}=\left(\begin{array}{cccc}\hphantom{-}0&\hphantom{-}E_x&\hphantom{-}E_y&\hphantom{-}E_z\\-E_x&\hphantom{-}0&-B_z&\hphantom{-}B_y\\-E_y&\hphantom{-}B_z&\hphantom{-}0&-B_x\\-E_z&-B_y&\hphantom{-}B_x&\hphantom{-}0\end{array}\right)$$

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But why is it that the c in the Lorentz Force Equation is isn't just absorbed into the magnetic field? –  Anthony May 7 '13 at 18:28
Just because if we absorb it into the magnetic field, then the field tensor $F_{\mu\nu}$ would consist of values $E_{x,\,y,\,z}$ and $cB_{x,\,y,\,z}$. And we prefer to make it of $E_{x,\,y,\,z}$ and $B_{x,\,y,\,z}$, without any factors. –  firtree May 7 '13 at 18:39
I don't actually know what a field tensor is-I'll go look that up-but from what you wrote it seems like the cBx,y,z is only again a consequence of the fact that c wasn't absorbed into B. –  Anthony May 7 '13 at 18:41
Then you may start with en.wikipedia.org/wiki/Electromagnetic_tensor (there is actually a $1/c$ factor in that article, because it is written in SI). A CGS survey can be found in Landau and Lifshitz volume 2 "The Classical Theory of Fields". –  firtree May 7 '13 at 18:50