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 :
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 :
A.
$$\frac{{2{\mu _0}Ii}}{{3\pi }}$$
B.
$$\frac{{{\mu _0}Ii}}{{2\pi }}$$
C.
$$\frac{{2{\mu _0}IiL}}{{3\pi }}$$
D.
$$\frac{{{\mu _0}IiL}}{{2\pi }}$$
Answer :
$$\frac{{2{\mu _0}Ii}}{{3\pi }}$$
Solution :
The net force on the loop
$$\eqalign{ & {F_{{\text{net}}}} = {F_{AB}} + {F_{BC}} + {F_{CD}} + {F_{AD}} \cr & \Rightarrow {F_{{\text{net}}}} = \frac{{{\mu _0}iI}}{\pi } - \frac{{{\mu _0}iI}}{{3\pi }} = \frac{{2{\mu _0}iI}}{{3\pi }} \cr} $$
The net force on the loop
$$\eqalign{ & {F_{{\text{net}}}} = {F_{AB}} + {F_{BC}} + {F_{CD}} + {F_{AD}} \cr & \Rightarrow {F_{{\text{net}}}} = \frac{{{\mu _0}iI}}{\pi } - \frac{{{\mu _0}iI}}{{3\pi }} = \frac{{2{\mu _0}iI}}{{3\pi }} \cr} $$