= Solution
The final unnumbered force question has this inferred third subpart. For either wire's exterior field,
$$
\mathbf E\cdot\mathbf B=0,\qquad
\boxed{E^2-c^2B^2=E^2(1-v^2/c^2)>0.}
$$
This <electromagnetic field invariants> is positive in every inertial frame. A purely magnetic nonzero field would instead have $E'^2-c^2B'^2<0$. 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 $u$, $\lambda'=\gamma_u\lambda(1-uv/c^2)$ cannot vanish when $|u|,|v|<c$.
For two identical wires separated by $d$, the laboratory transverse force per unit length is
$$
\boxed{\frac FL=\lambda E-IB
=\frac{\lambda^2}{2\pi\epsilon_0d}(1-v^2/c^2)
=\frac{\lambda_0^2}{2\pi\epsilon_0d}>0.}
$$
Electric 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.
Back to article page
