Question

Two wires $A$ & $B$ are carrying currents $I_1$ & $I_2$ as shown in the figure. The separation between them is $d$. A third wire $C$ carrying a current $I$ is to be kept parallel to them at a distance $x$ from A such that the net force acting on it is zero. The possible values of $x$ are :

• A. $x=\left(\frac{I_1}{I_1-I_2}\right)d$ and $x=\frac{I_2}{\left(I_1+I_2\right)}d$
• B. $x=\pm \frac{I_{1}d}{\left(I_1+I_2\right)}$
• C. $x=\left(\frac{I_1}{I_1+I_2}\right)d$ and $x=\frac{I_2}{\left(I_1-I_2\right)}d$
• D. $x=\left(\frac{I_1}{I_1+I_2}\right)d$ and $x=(\frac{I_2}{I_1+I_2})d$

Answer 1

Answer: (B)

Net force on wire carrying current I per unit length is
$\frac{{\mu }_0{I}_{1}I}{2\pi x}+\frac{{\mu }_0{I}_{2}I}{2x\left(d-x\right)}=0$
$\frac{I_1}{x}=\frac{I_2}{x-d}$
$\Rightarrow x=\frac{I_{1}d}{I_1-I_2}$