Constant magnetic field > The magnetic field of a moving charge. Induction of the magnetic field from a linear current > Problem 9.2.18

Statement

$9.2.18.$ The magnetic field of a flat contour with current at large distances from it is determined by the magnetic moment of the contour and does not depend on its shape. Prove it.

Solution

By dividing the contour into a set of small magnetic dipoles in the shape of a square, with the current in the touching sides directed in opposite directions, this combination, with the side of the square tending to zero, will give us a field equivalent to the field of the contour. At large field distances from such a sum of squares, there will be an equidistant dipole in the field of one dipole, and the number of dipoles in the contour will depend only on its area.

Answer

Q.E.D.