Chapter 30 Problem 4

Problem

A long straight wire and a small rectangular wire loop lie in the same plane. Determine the mutual inductance in terms of $l_{1}$, $l_{2}$ and $w$. Assume the wire is very long compared to $l_{1}$, $l_{2}$ and $w$, and that the rest of its circuit is very far away compared to $l_{1}$, $l_{2}$ and $w$.

Solution

$M={\frac {\Phi _{12}}{I_{1}}}={\frac {1}{I_{1}}}\int _{l_{1}}^{l_{2}}{\frac {\mu _{0}I_{1}}{2\pi r}}wdr=$ $={\frac {\mu _{0}w}{2\pi }}\ln \left({\frac {l_{2}}{l_{1}}}\right)$