Question

A metallic uniform rod $PQ$ is hinged at point $P$ and it can rotate about point $P$ in vertical plane and an infinite long wire $AB$ having current $i_1$ is placed as shown in the figure. If the mass of rod is $m$ and length $L$, then the current $i_2$ in rod $PQ$ such that it remains in equilibrium as shown is :
( separation between P and current carrying wire $AB$ is very small )

• A. $\frac{2mg\pi }{{\mu }_0{i}_1}$
• B. $\frac{mg\pi }{2{\mu }_0{i}_1}$
• C. $\frac{mg\pi }{4{\mu }_0{i}_1}$
• D. $\frac{mg\pi }{{\mu }_0{i}_1}$

Answer 1

The correct option is D $\frac{mg\pi }{{\mu }_0{i}_1}$

Consider a current element dx on $PQ$ at a distance of $x$ from $P$. Force on the element due to current $i_1$ is
$dF=\frac{i_{2}dx\mu i_1}{2\pi x}$
Further, torque $d\tau =\frac{i_{2}dx\mu i_1}{2\pi x}x$
Above torque shall be balanced by net torque due to gravitational force,
Or, ${\int }_0^L\frac{i_{2}dx\mu i_1}{2\pi }=\frac{mgL}{2}$
or $i_2=\frac{mg\pi }{\mu i_1}$