Question

Find the inductance of a unit length of two long 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

• 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 a closed rectangle of infinite length and breadth equal to $d$ Flux through the strip of area $l\, dr$, due to current flowing in one wire is given by
$\phi=\int_{a}^{d-a} \frac{\mu_{0}I}{2\pi r}\left(ldr\right)$
$=\frac{\mu_{0}Il}{2\pi} ln \left(\frac{d-a}{a}\right)$
The other wire produces the same result, so the total flux through the dotted rectangle is
$\phi_{total}=\frac{\mu_{0}Il}{\pi}ln \left(\frac{d-a}{a}\right)$
The total inductance of length $l$,
$L=\frac{\phi_{total}}{I}=\frac{\mu_{0}l}{\pi} ln \left(\frac{d-a}{a}\right)$
Inductance per unit length $=\frac{L}{l}=\frac{\mu_{0}}{\pi}$ ln $\left(\frac{d-a}{a}\right)$