Question

Two long parallel straight wires $ A $ and $ B $ separated by a distance of $ 6\, cm $ carry currents of $ i $ and $ 2i $ respectively in the same direction. The distance from $ B $ at which the net field is zero, is

• A. $ 1\, cm $
• B. $ 2\, cm $
• C. $ 4 \, cm $
• D. $ 5\, cm $

Answer 1

Answer: (C)

The situation is shown in the figure

As the currents in both the wires are in the same direction, so the net magnetic field is zero in between them. Let it be zero at point $P$ at a distance $r$ from $B$. Then,
The magnetic field at $P$ due to current $i$ in $A$ is
$B_{1}=\frac{\mu_{0} I}{2\pi\left(6\,cm-r\right)} \otimes$
and that due to current $2i$ in $B$ is
$B_{2}=\frac{\mu_{0} 2i}{2\pi r} ⊙ $
Since $B_{1}$ and $B_{2}$ are in opposite directions, so the net field at $P$ is
$B=B_{2}-B_{1}=\frac{\mu_{0} 2i}{2\pi r}-\frac{\mu_{0}i}{2\pi\left(6\,cm-r\right)}$
But $B = 0$ (given)
$\therefore \frac{\mu_{0} 2i}{2\pi r}-\frac{\mu_{0} i}{2\pi\left(6\,cm-r\right)}=0$
or $\frac{\mu_{0} 2i}{2\pi r}=\frac{\mu_{0} i}{2\pi\left(6\,cm-r\right)} $
or $\frac{2}{r}=\frac{1}{\left(6\,cm-r\right)}$
or $ r=12\,cm -2r$ or $3r=12\, cm$
or $r=\frac{12\,cm}{3}$
$=4\,cm$