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?

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

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.