A square frame of side 'L' carries a current I. The magnetic field at its centre is B. The same current is passed through a circular coil having the same perimeter as the square. The magnetic field at the centre of the circular coil is B′. The ratio of  is:

Concept:

The magnetic field at the center of the circular coil is given by:

where B = strength of the magnetic field, I = current, and R = radius of the circular coil.

The intensity of the magnetic field due to a wire of finite length is​:

where μ0 = permittivity of free space, I = current in a wire, d = perpendicular distance between the wire and the centre.

Calculation:

Given:

The intensity of the magnetic field due to a wire of finite length is​:

Due to 4 sides of the square;

Here d = L/2 as can be seen from the diagram.

The magnetic field at the center of the circular coil is given by:

For circle

4L = 2πR (As perimeter is same)

According to the problem;