Given two infinite parallel charged rods with equal charge density $\lambda$. They are moving with same constant velocity $\vec v$ parallel to the rods. Find the speed $v$ for which the magnetic attraction is equal to the electrostatic repulsion.
Well, I know how to solve this problem: we first find out the magnetic field created by one rod on the other using Biot's and Savart's law, then we use the definition of $\vec B$ ( $d\vec F=\vec vdq\times\vec B$ ) to find the magnetic force, then equate magnetic and electrostatic forces to find $v$, which will be greater than or equal to $c$, thus conclude it is impossible for the forces to be equal.
However, one can argue as the following:
We all know that "same laws of physics apply in all inertial frames". With a constant velocity $\vec v$,the rest frame of the rods is an inertial frame. Therefore, if Biot-Savart law applies in our frame, it has to apply in the rest frame. If so, none of the rods will feel a magnetic field from the other one because their relative speed is zero, and there will be no magnetic force between the rods.
I've seen this question several times before in references, exams, exercise sheets,and in many different forms (parallel planes, beam of electrons ...),but no one ever used this argument.What is the problem in it ? Is it something related to Maxwell's equations or special relativity ? Or what else ?
I know a similar question was asked before, but the answers weren't satisfying. Please provide your answers with necessary mathematics.