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.
 A: For a short wire of length $||dx||$, with current flowing in the direction of $d\vec{x}$, there will be a field at displacement $\vec{r}$ away from the wire, ($k\equiv \tfrac{\mu_0}{4\pi}$):
$$d\vec{B}= kI \frac{d \vec{x} \times \vec{r}}{||r||^3} $$
If we integrate for a full circle around its center, then $d\vec{x}$ and $\vec{r}$ will always be perpendicular, $d \vec{x} \times \vec{r}=rdx$, and all field contributions $d\vec{B}$ will be in the same direction by the right hand rule, crossing $d\vec{x}$ with $\vec{r}$. Therefore they will simply sum up (integrate). Each small length $dx$ will be $dx=rd\theta$, so that
$$d \vec{x} \times \vec{r}=rdx = r^2d\theta$$
We can integrate around the circle:
$$d\vec{B}= kI \frac{d \vec{x} \times \vec{r}}{||r||^3} $$ $$ dB= kI \frac{d \theta}{r} $$
$$\implies B = kI \int_{0}^{2\pi} \frac{d\theta}{r} =\frac{2\pi k I}{r} =\frac{\mu_0 I}{2r}$$
As $r$ is a constant. I don’t know what the answer is, but as you can see was pretty meticulous, and think I did it correctly. Edit: Oh I see is correct, misunderstood initial statement.
A: 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
A: Ampere's law uses a line integral of the B field around a closed loop.  For it to be useful you need a function for the component of the B field which is parallel to the loop at each point and which can conveniently integrated.  In this problem, if you could find the B field at each point on any chosen loop, you would not need ampere's law to find the B on the axis.
