# Can we make a formula that describes the magnetic force as a function of electric field using length contraction equation ?

I know that the magnetism is just electricity and due to length contraction and the change in charge density moving charges experience what we call magnetic force. so my question is can we calculate that force using electricity stuff (like electric field and the permittivity of free space) instead of using magnetic field?

• Yes, we can and you know this. Commented Dec 21, 2018 at 10:51
• @VladimirKalitvianski Not always. I challenge you to find a reference frame where an ellipse-shaped current loop can be described only in terms of an electric field. Commented Dec 21, 2018 at 11:12
• Apply the Lorentz transformation to the electromagnetic field tensor, or the 4-potential (phi, A_i).
– user196418
Commented Dec 21, 2018 at 12:23
• Next to a electrically neutral wire with a current $I$ the magnetic field $B$ is big. It may act on a moving charge according to the Lorentz force $F\propto v\times B$. In a co-moving reference frame the charge is still ($v'=0$) so the force from the wire is purely electric $F'\propto E'$ which is possible due to non zero charge density $\rho'$ in this reference frame. Commented Dec 21, 2018 at 16:43

I dont think you can, because $$E^2-B^2$$ is an invariant, and if $$|B|>|E|$$ in one reference frame it will be the same in all reference frames, so you cannnot always eliminate $$B$$.
• The quantity $E^2-B^2$ is a local function and vary from one place to another. The statement to prove is simple: can one find a reference frame where the magnetic force is absent and there is an electric force solely? The answer is yes. Instantly the force on a charge can be reduced to an electric one even if there is some magnetic field too. One makes the magnetic force zero in any co-moving RF due to $v'=0$. Commented Dec 21, 2018 at 17:20
• No, one cannot eliminate the magnetic field in all the space, even instantly, so you are right. In case of a straight neutral wire with $I$, the wire looks charged in a moving RF and the charge density $\rho'$ depends on velocity and velocity sign ;-) Commented Dec 21, 2018 at 17:29