Question

Three infinite straight current carrying conductors are placed as shown in figure. Two wires carry same current while current in the third wire is unknown. The three are electrically insulated from each other and all of them are in the plane of paper. Which of the following is correct about point $P$ which is also in the same plane?

• A. Magnetic field intensity at $P$ is zero for all values of $x$, whatever is the current in the third wire.
• B. Magnetic field intensity at $P$ is zero if the current in the third wire is $\frac{2I}{\sin a}$ (left to right).
• C. Magnetic field intensity at $P$ is zero if the current in the third wire is $\frac{2I}{\sin a}$ (right to left).
• D. None of these

Answer 1

The correct option is C Magnetic field intensity at $P$ is zero if the current in the third wire is $\frac{2I}{\sin a}$ (right to left).

Magnetic field at ${}^{\prime }{P}^{\prime }$ due to wires (1) and (2) is
$B_1=\frac{{\mu }_{0}I}{2\pi \left(x\sin a\right)}+\frac{{\mu }_{0}I}{2\pi xsina}=\frac{{\mu }_{0}I}{\pi \left(x\sin a\right)}$ (outside paper)

For magnetic field at P to be zero, magnetic field due to wire 3 should oppose $B_1$ and hence directed from right to left using thumb rule.
Magnetic field due to third wire is $B_3=\frac{{\mu }_o{I}^{\prime }}{2\pi x}$ (inside paper).

$B_1+B_3=0$

$⟹I^{\prime }=\frac{2I}{sina}$