Question

Two infinitely long conductors kept along XX' and YY' axes are carrying current I₁ and I₂ along the X-axis and the Y-axis, respectively. Find the magnitude and direction of the net magnetic field produced at point P(X, Y).

Answer 1

A long, straight wire produces a magnetic field given by:

B = `(mu_0 I)/(2 pi r)`

The magnetic field generated by a wire positioned along the X-axis at y = 0, with point P(X, Y) located at a perpendicular distance Y from the wire.

According to the right-hand rule, the magnetic field at P(X, Y) generated by this wire is directed into the plane (negative Z-direction).

Thus, the magnitude is:

B₁ = `(mu_0 I)/(2 pi Y)`

Its direction is into the plane (negative Z-direction).

Magnetic field due to the wire along the Y-axis, where x = 0, and the point P(X, Y) is at a perpendicular distance X from this wire.

Using the right-hand rule, the magnetic field at P(X, Y) due to this wire is also into the plane (negative Z-direction).

Thus, the magnitude is:

B₂ = `(mu_0 I)/(2 pi X)`

Its direction is also into the plane (negative Z-direction).

Net magnetic field at P(X, Y):

B_net = B₁ + B₂

= `(mu_0 I)/(2 pi Y) + (mu_0 I)/(2 pi X)`

= `(mu_0 I)/(2 pi) (1/X + 1/Y)`

The direction of the net magnetic field is into the plane (negative Z-direction).