Question

A straight segment $OC$ (of length $L$) of a circuit carrying a current $I$ is placed along the x-axis. Two infinitely long straight wires $A$ and $B$, each extending from $z=-\infty$ to $+\infty$, are fixed at $y=-a$ and $y=+a$ respectively, as shown in the figure. If the wires $A$ and $B$ each carry a current $I$ into the plane of the paper, obtain the expression for the force acting ont he segment $OC$. What will be the force on $OC$ if the current in the wire $B$ is reversed?

Answer 1

$B_{res}=Bcos\theta$

$F_{res}=I\int d\overrightarrow{l}\times B_{res}$

$\int I\times \frac{Rd\theta }{sin\theta }\frac{2{\mu }_{0}I}{2\pi R}cos\theta ,sin\alpha =\frac{a}{\sqrt{L^2+a^2}}$

$\frac{{\mu }_0{I}^2}{\pi }{\int }_{\pi /2}^{a}cot\theta d\theta =\frac{{\mu }_0{I}^2}{\pi }in\left(sin\theta \right){}_{90}^{\alpha }$

$\frac{{\mu }_0{I}^2}{\pi }in[\frac{a}{\sqrt{L^2+a^2}}-1]$