# Retrieving the non-relativistic formulas for electric and magnetic fields from relativistic formulas

I was checking the formulas for the electric and magnetic fields components E and B given in this link from Wikipedia: https://en.wikipedia.org/wiki/Classical_electromagnetism_and_special_relativity#Transformation_of_the_fields_between_inertial_frames
By the end of the section, they convert these formulas from the relativistic to the non-relativistic aspect by just approximating that the Lorentz factor becomes very small, since $$v<. I was thinking if there is any way to derive these formulas by applying a Taylor expansion on the gamma-factor. Is this possible, and if yes, could someone write me how any of these vectors would transform to non-relativistic approach? I already know how to do the Taylor expansion of the gamma factor, but afterward, I cannot figure out the calculations. Any idea or help is very appreciated.

• Non-relativistic doesn’t mean $v/c$ is absent. The gamma factor gives $(v/c)^2$ terms, which you can ignore. – G. Smith Jan 14 at 18:39
• You’re being too literal. There is a $\mathbf{v}$ in each equation, but no $v^2$. Sometimes there is no $c$, and sometimes there is $c^2$; this is just because of the use of non-Gaussian units. In Gaussian units, every $v$ would be divided by $c$. – G. Smith Jan 14 at 19:12
• There is no point in using a Taylor series on gamma if you want equations accurate only to first order in $v$. That is why they approximated gamma as 1. – G. Smith Jan 14 at 19:17
• A nonrelativistic transformation usually keeps first order in $v$, but not higher, because this is similar to the Galilean transformation $x’=x-vt$. – G. Smith Jan 14 at 19:19
• In fact, taking the Galilean limit of electromagnetism is far more subtle. There are actually two different self-consistent ways to do it. To learn more just search it up on this site, it’s been discussed before. – knzhou Jan 14 at 22:19