The magnetic field lines are shown at right. There is not enough symmetry in this situation to invoke Ampere’s law easily for the calculation of the field.
Since the final effect of each dB is that only its horizontal component contributes to the total b, our final integral need involve only that component from the Biot-Savart law. Any infinitesimal segment ds contributes the same horizontal component regardless of its position on the loop.
Since ds is perperdicular to r for all differential current elements, regardless of their position on the loop:
Thus the magnitude of the contribution to the magnetic field from the current segment of arc length ds is:
Integrating over all current segments reduces to evaluation of the circumference of the loop since all contributions dB have the same magnitude:
The denominator involving x and a above reduces to a3 in this limit. Thus magnetic field at the center of the loop is
The denominator involving x and a above reduces to x3 in this limit. Thus the magnetic field behaves as a dipolar field, as expected!