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
$\phi=\int\limits_{a}^{d-a} \frac{\mu_{0} I}{2 \pi r}(l d r)=\frac{\mu_{0} I I}{2 \pi} \ln \left(\frac{d-a}{a}\right)$

The other wire produces the same result, so the total flux through the rectangle $A B C D$ is
$\phi_{\text {total }}=\frac{\mu_{0} I}{\pi} \ln \left(\frac{d-a}{a}\right)$
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 $=\frac{L}{1}=\frac{\mu_{0}}{\pi} \ln \left(\frac{d-a}{a}\right)$