Question

A conductor AB of length $l$ carrying a current $i$ is placed perpendicular to a long straight conductor XY carrying a current $I$ as shown. The force on AB will be :

• A. $\frac{{\mu }_{0}li}{2\pi }log2$
• B. $\frac{{\mu }_{0}li}{2\pi }log3$
• C. $\frac{3{\mu }_{0}li}{2\pi }$
• D. $\frac{2{\mu }_{0}li}{3\pi }$

Answer 1

The correct option is B $\frac{{\mu }_{0}li}{2\pi }log3$

$F=Bil$
$dF=\frac{{\mu }_{0}l}{2\pi x}idx$
$\int dF={\int }_{\frac{l}{2}}^{\frac{3l}{2}}\frac{{\mu }_{0}li}{2x\pi }dx$
$F=\frac{{\mu }_{0}li}{2\pi }{\left|lnx\right|}_{\frac{l}{2}}^{\frac{3l}{2}}$
$F=\frac{{\mu }_{0}li}{2\pi }ln\left(\frac{\frac{3l}{2}}{\frac{l}{2}}\right)$
$=\frac{{\mu }_{0}li}{2\pi }ln3$