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}} $