Find the inductance of a unit length of two parallel wires, each of radius a, whose centers are a distance d apart and carry equal currents in opposite directions. Neglect the flux within the wire.

Since the wires are infinite, so the system of these two wires can be considered as a closed rectangle of infinite length and breadth equal to d. Flux through the strip:

$\varphi ={\int }_a^{d-a}\frac{{\mu }_{0}I}{2\pi r}\left(Idr\right)=\frac{{\mu }_{0}Il}{2\pi }In\left(\frac{d-a}{a}\right)$

The other wire produces the same result, so the total flux through the dotted rectangle is

${\varphi }_{total}=\frac{{\mu }_{0}Il}{\pi }In\left(\frac{d-a}{a}\right)$

The total inductance of length /

$L=\frac{{\varphi }_{total}}{I}=\frac{{\mu }_{0}I}{\pi }In\left(\frac{d-a}{a}\right)$

Inductance per unit length $=\frac{L}{l}=\frac{{\mu }_0}{\pi }In\left(\frac{d-a}{a}\right)$