Consider the rectangular loop carrying current as shown in the figure. Point P is located a distance x from the centre of the loop. We have to find an expression for the magnetic field at P due to the current loop, assuming that P is very far away.

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We will calculate the magnetic field at P due to opposite sides of the rectangular taken in pairs. We first consider the sides 1 and 3. In the coordinate system fixed as shown, let the coordinates of the point P be . We assume that , and the distance to point P from the sides of the loop can be approximated as nearly equal to x.

From the Ampere’s law we note that the magnetic field at P due to line element at will be

We note that the components of due to current elements in sides 1 and 3 will cancel each other in pairs. Therefore, the component of in the direction at P due to current in the sides 1 and 3 of the loop will be

and the combined field at P due to current flows in sides 1 and 3 of the loop will be

We can calculate the magnetic field at P due to the current flow in the sides 2 and 4 of the loop and we will find that

Therefore, the magnetic field at P due to current flow in the loop will be

The magnetic dipole moment of the planar loop of area with current is

Therefore,

This is the expression of field due to a magnetic dipole at a distance x on its axis far away from the dipole.