Question

The length of conductor ab carrying current $I_2$ is $l$. Find the force acting on it due to a long current carrying conductor having current $I_1$ as shown in Fig. The mid-point of wire ab is distance x apart from conductor.

Answer 1

Magnetic field at a distance $r$ produced by wire carrying current $I_1$ is

$B=\frac{{\mu }_0{I}_1}{2\pi r}$

Hence the total force acting on the wire carrying current $I_2={\int }_{x-\frac{l}{2}}^{x+\frac{l}{2}}B{I}_{2}dr$

$={\int }_{x-\frac{l}{2}}^{x+\frac{l}{2}}\frac{{\mu }_0{I}_1{I}_{2}dr}{2\pi r}$

$=\frac{{\mu }_0{I}_1{I}_2}{2\pi }ln\left(\frac{2x+l}{2x-l}\right)$