Question

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

• A. $\frac{{\mu }_0}{2\pi }\ln \left(\frac{d-a}{a}\right)$
• B. $\frac{{\mu }_0}{\pi }\ln \left(\frac{d-a}{a}\right)$
• C. $\frac{3{\mu }_0}{\pi }\ln \left(\frac{d-a}{a}\right)$
• D. $\frac{{\mu }_0}{3\pi }\ln \left(\frac{d-a}{a}\right)$

Answer 1

Answer: (B)

Since, the wires are infinite, so the system of these two wires can be considered as closed rectangle of infinite length and breadth equal to $d$.
Flux through strip
$\varphi =\underset{a}{\overset{d-a}{\int }}\frac{{\mu }_{0}I}{2\pi r}(ldr)=\frac{{\mu }_{0}II}{2\pi }\ln \left(\frac{d-a}{a}\right)$

The other wire produces the same result, so the total flux through the rectangle $ABCD$ is
${\varphi }_{\text{total }}=\frac{{\mu }_{0}I}{\pi }\ln \left(\frac{d-a}{a}\right)$
The total inductance of length 1
$L=\frac{I_{\text{total }}}{I}=\frac{{\mu }_{0}I}{\pi }\ln \left(\frac{d-a}{a}\right)$
Inductance per unit length $=\frac{L}{1}=\frac{{\mu }_0}{\pi }\ln \left(\frac{d-a}{a}\right)$