External inductance of a thin two-wire line ($L_{\mathrm{ext}}^{\prime }\simeq {\mu }_0\ln (d/a_w)/\pi$)

Two long round wires of radius $a_w\ll d$ carrying opposite electric currents form a loop with the displayed leading external self-inductance per unit length. Integrating their summed magnetic field across a connecting ribbon gives the logarithm; magnetic-energy integration gives the same leading value. The zero-radius limit diverges. Uniform direct current inside two nonmagnetic round wires contributes another ${\mu }_0/(4\pi )$, while an ideal surface current has no interior contribution.