Three infinite current carrying conductors are placed as shown in Fig. Two wires carry same current while current in third wire is unknown. The three wires 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?

Options

A. Magnetic field intensity at P is zero for all values of x, whatever is the current in the third wire.
B. If the current in the third wire is `(2I)/(sin alpha)`(left to right), then magnetic field will be zero at P for all values of x.
C. If the current in the third wire is `(2I)/(sin alpha)` (right to left), then magnetic field will be zero at P for all values of x.
D. none of these

Answer

C

Explanation

(c) Magnetic field at 'P' due to wires (1) and (2) is
`B_1=(mu_0I)/(2pi(xsinalpha)) +(mu_0I)/(2pi(xsinalpha))=(2mu_0I)/(2pi(xsinalpha))`
(outside the paper)

Now, if a current of `(2I)/(sin alpha)` is flowing in the third wire, then the
magnetic field due to the same will be `B_2=(mu_0)/(2pix)((2I)/(sin alpha))`,
which will cancel `B_1` if it is in the paper which is possible if
the current `(2I)/(sin alpha)` in the third wire is from right to left.