Force on a current carrying wire placed in a magnetic field

Question

A square loop ABCD carrying a current i,is placed near and coplanar with a long straight conductor XY carrying a current I, the net force on the loop will be

Difficult

Solution

$\text{Force on arm}AB\text{due to current in conductor}XY\text{is}$

${F}_{1}=\frac{{\mu}_{0}}{4\pi}\frac{2IiL}{(L/2)}=\frac{{\mu}_{0}Ii}{\pi}$

acting towards X Y in the plane of loop. Force on arm CD due to current in conductor X Y is

${F}_{2}=\frac{{\mu}_{0}}{4\pi}\frac{2IiL}{3(L/2)}=\frac{{\mu}_{0}Ii}{3\pi}$

acting away from X Y in the plane of loop.

therefore Net force on the loop $={F}_{1}-{F}_{2}$

$=\frac{{\mu}_{0}Ii}{\pi}\left[1-\frac{1}{3}\right]=\frac{2}{3}\frac{{\mu}_{0}Ii}{\pi}$

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