(a) A wire in the form of a regular polygon of sides is just enclosed by a circle of radius . If the current in the wire is , we have to show that the magnetic field at the centre of the circle is given in magnitude by

(b) We have to show that as this result approaches that of a circular loop. (c) We have to find the dipole moment of the polygon.

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(a)

A wire in the form of a regular polygon of sides is just enclosed by a circle of radius . The angle subtended by each side of the polygon with the centre of the circle of radius will be . Length of each side of the polygon will be

.

We have already shown that the magnetic field associated with a current carrying straight line segment of length at a distance from the segment along a perpendicular bisector is

For our problem we use that contribution to the magnetic field at the point P by each side of the polygon will be the same and can be obtained from the above result for the values

Therefore, the magnitude of the magnetic field at P by each side of the polygon will be

As the magnetic field at the centre of the polygon sue to each side will be perpendicular to the plane of the polygon, the total magnetic field will be n times the field due to one side. Therefore,

(b)

As

It is same as the magnetic field at the centre of a circular loop of radius carrying current .

(c)

We calculate next the dipole moment of the polygon. The magnetic dipole moment of a planar closed loop in current times the area enclosed. The area of each triangle formed by joining the side of a polygon with the centre will be

And, the magnetic dipole moment of the polygon will be