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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.

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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 $P$.

See, for instance, the first answer to Ampère's law applied on a "short" current-carrying wire

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  • $\begingroup$ Thanks for the link, I was reading it also. For a broken current path, I could relate that to the complication in apply Stroke’s theorem, where there will be some surface integral (a deep dome) that does not cross any current at all, so it is not well defined. Apply that back to a loop, however, they will always cross each other, and that confuses me. I will read it again to see if I missed something. $\endgroup$ – Sandbo Mar 20 '18 at 4:50
  • $\begingroup$ @Sandbo, I think the issue is that a "broken current path" would break conservation of charge, this is more completely discussed in the first four paragraphs of the linked answer. $\endgroup$ – Munthe Mar 20 '18 at 22:16

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