Past exam of the symmetry and relativity course of the University of Oxford / 2016 / 4 / d / iii / Solution

The final unnumbered force question has this inferred third subpart. For either wire's exterior field,This electromagnetic field invariants is positive in every inertial frame. A purely magnetic nonzero field would instead have $E^{\prime 2}-c^2{B}^{\prime 2}<0$. Thus no physical boost makes the interaction purely magnetic. There is a comoving frame with a purely electric interaction, which is consistent with the invariant's sign.

The convected charge current gives the same conclusion. Its effective line-current invariant is ${\lambda }^2-I^2/c^2={\lambda }_0^2>0$, so no inertial frame can neutralize the charge while leaving only a current. In particular for a longitudinal boost of speed $u$, ${\lambda }^{\prime }={\gamma }_u\lambda (1-uv/c^2)$ cannot vanish when $|u|,|v|<c$.

For two identical wires separated by $d$, the laboratory transverse force per unit length isElectric repulsion exceeds the magnetic attraction. This assumes each moving wire has uncompensated convected charge; a neutral wire carrying conduction current is a different charge-current system and has a different invariant classification.