 Electrostatic-frame derivation of parallel-wire forces

Consider lab-neutral parallel wires with stationary positive line density $+\lambda$ and electron line density $-\lambda$. Let wire 1's electrons move at $v_1$ along the wire and wire 2's at $v_2$. The conventional currents are $I_j=-\lambda v_j$. In the rest frame of wire 2's electrons, wire 1's transformed total linear charge density is ${\lambda }_1^{\prime }={\gamma }_2[0-v_2{I}_1/c^2]={\gamma }_2\lambda v_1{v}_2/c^2$. Its radial electric field is ${\lambda }_1^{\prime }/(2\pi {ϵ}_{0}d)$.

A wire-2 electron at rest there feels only an electric force. Transforming its transverse force back to the lab divides it by ${\gamma }_2$, and multiplying by the lab electron count per length $\lambda /e$ giveswith positive radial direction away from wire 1. The lab-stationary positive charges feel zero net electric force because wire 1 is neutral in their frame. Equal current directions attract; opposite directions repel. Transforming force on each carrier and then counting carriers in the specified frame avoids an incorrect extra length-contraction factor in force per unit length.