For the magnetic field at the center to be zero, the magnetic field due to the circular loop has to be exactly equal and opposite to the long straight wire at the center.

Given:

Current carried by circular loop of radius r = i

Current carried by long straight wire = 4i

Formula used:

Magnetic field at the center of a circular loop due to current in the loop(B_l) = , which points into the plane of the paper.

Here

μ₀ = magnetic permeability of vacuum,

i = current carried by the loop,

r = radius of the loop

Magnetic field due to an infinitely long wire at a distance x from it(B_w) = , which points out of the plane of the paper.

Here

μ0 = magnetic permeability of vacuum,

i = current carried by the wire,

x = distance of the center of the circle from the wire

Hence, from given information,

, and

where B_l = magnetic field due to loop, B_w = magnetic field due to wire, r = radius of loop, x = distance of the center of the circle from the wire

Hence, for the magnetic field at the center to be 0,

=> =>

Hence, for the magnetic field to be 0, the wire has to be placed at a distance of 4r/π from the center of the circular loop. (Answer)