= 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 $\lambda_1'=\gamma_2[0-v_2I_1/c^2]=\gamma_2\lambda v_1v_2/c^2$. Its radial <electric field> is $\lambda_1'/(2\pi\epsilon_0d)$.
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 $\lambda/e$ gives
$$
f_\perp/\ell=-\frac{\lambda^2v_1v_2}{2\pi\epsilon_0c^2d}=-\frac{\mu_0I_1I_2}{2\pi d},
$$
with 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.
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