Chapter 28 Problem 24

Problem

Current loop and wire

A triangular loop of side length a carries a current $I$. If this loop is placed a distance $d$ away from a very long straight wire carrying a current $I^{\prime }$,determine the force on the loop.

Solution

Forces and distances

The current in the parallel branch flows in the same direction as the long straight wire, so the force is attractive with magnitude:

$F_{1}={\frac {\mu _{0}II^{\prime }}{2\pi d}}a$

By symmetry the magnetic force for the other two segments will be equal.

$F_{2}=\int _{0}^{a/2}{\frac {\mu _{0}II^{\prime }}{2\pi \left(d+{\sqrt {3}}x\right)}}dx={\frac {\mu _{0}II^{\prime }}{2\pi {\sqrt {3}}}}\left.\ln \left(d+{\sqrt {3}}x\right)\right|_{0}^{a/2}$ $={\frac {\mu _{0}II^{\prime }}{2\pi {\sqrt {3}}}}\ln \left(1+{\frac {{\sqrt {3}}a}{2d}}\right)$