# Ampere's Law vs Biot-Savart Law on the field of the center of a current-carrying loop

Consider a current-carrying loop of radius R, and we want to find the field at the center of the loop on the same plane as the loop. Biot-Savart law tells me that it is $B_{center}=\frac{\mu_0I}{2R}$.

With Ampere's Law, I tried to consider the loop as a superposition of many infinitesimal arc segments of the loop , and each arc segment contributes to a field B towards the loop center. Then I applied Ampere's Law to find the B field due to the arc toward the loop center as $B=\frac{\mu_0I}{2\pi R}$. Further, as these arc segments combine to form the loop, I imagined that the contribution of each arc sums up. Then I did an integration over all these arc segments as $B_{center}=\int_0^{2\pi} \frac{\mu_0I}{2\pi R} Rd\theta=\mu_0I$ which looks very wrong. Indeed, I just redid the Amphere's Law.

I believe I applied the Amphere's Law in a wrong way and it has been bugging me, would you mind giving me some hints? It seems to me that it wasn't correct to say the arc segments have a circular field distribution, or superposition can't be done this way.

The issue is that the formula for $B$ from Ampere's law is valid only for infinite wires, not infinitesimal ones. Even parts of an infinite wire far from some point $P$ contribute to the field at $P$.