The following is the equation which, I want to know, if it is valid in relativistic domain. Consider two equal charges moving in same direction with velocity $v$ and charge $q$ at a separation of $d$. The magnetic force acting between them is $(1/4\pi\epsilon_0 c^2)v^2q^2/d^2$ and the electrostatic force is $(1/4\pi\epsilon_0 )q^2/d^2$.the ratio of the magnetic (attractive) force and the electrostatic (repelling) force is $v^2/c^2$ and therefore the conclusion provided was that two individual charges with same (or any) velocity can never attract. Is this equation valid relativistically. Moreover, since magnetic field is just the relativistic counterpart of electrostatic force, can we find a frame of reference wherein in the case stated, all the force is magnetic and must be attractive? I am comfortable only with basic special relativistic mechanics but an intuitive understanding would be better.
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asked Aug 18, 2013 at 3:38
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1$\begingroup$ Related: physics.stackexchange.com/q/71378 $\endgroup$– MichaelCommented Aug 18, 2013 at 4:17
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2$\begingroup$ Also relevant to the latter part of the question is the fact that $\vec{E}\cdot\vec{B}$ and $\vec{E}^2 - \vec{B}^2 c^2$ are relativistic invariants of the electromagnetic field. That is they are the same in all reference frames. In particular, if $\vec{E}^2 - \vec{B}^2 c^2$ is positive in one frame there is no other frame in which it is negative. Think on that. :) $\endgroup$– MichaelCommented Aug 18, 2013 at 4:19
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