Two concentric current loops lie in the same plane. The smaller loop has a radius of 2.7cm and a current of 12 A. The bigger loop has a current of 20 A. The magnetic field at the center of the loops is found to be zero. What is the radius of the bigger loop?

The B-field at the center of a circular loop of radius, r and current, I is;

Magnetic field, B = (μ * I) : 2pi * R

Given:

rs = 2.7 cm

= 0.027 m

Is = 12 A

Ib = 20 A

(μ * Is) / 2 * rs = (μ * Ib) / 2 * rb

Inputting values,

rb = (20 * 2 * 0.027) / 12 * 2

= 0.045 m

= 4.5 cm.

Radius of bigger loop (R) = 4.5cm

Explanation:

Consider a circular path of radius r around the wire. The magnetic field along that path is given by;

∫B*dl = k*I where I is the current enclosed. From symmetry, ∫B*dl = 2*π*r*B

B = K*I/r, so the magnetic field varies inversely as the loop radius and directly as the current.

The smaller loop current to radius ratio is 12/2.7

The bigger loop current to radius ratio is = 20/R

12/2.7 = 20/R

R = (20 * 2.7) / 12

R=54/12

R=4.5cm