In the figure below a long circular pipe with outside radius

In the figure below, a long circular pipe with outside radius R = 2.48 cm carries a (uniformly distributed) current i = 11.1 mA into the page. A wire runs parallel to the pipe at a distance of 3.00R from center to center. Find the (a) magnitude and (b) direction (into or out of the page) of the current in the wire such that the ratio of the magnitude of the net magnetic field at point P to the magnitude of the net magnetic field at the center of the pipe is 1.45, but it has the opposite direction.

Solution

B(P) = [ [mu₀ i_pipe /  2pi (2 R)] - [mu₀ i_wire /  2 pi R ] ]

B(Pipe ctr) = [mu₀ i_wire /  2pi (3 R)]

ratio B(P) / B(Pipe ctr) = x = [ [mu₀ i_pipe /  2pi (2 R)] - [mu₀ i_wire /  2 pi R ] ] / [mu₀ i_wire /  2pi (3 R)]

i_wire = i_pipe / 2(x/3 + 1)

= 11.1 mA / 2(1.45/3 + 1)

= 3.74 mA

The direction is into the page