electromagnetism - Approximating magnetic field at center of current loop by solenoid of 1 turn

It's well known that by application of biot-savart law, the magnetic field at the center of a loop of current is $\frac{\mu_0 I}{2r}$.

Now when we consider a solenoid, isn't the resulting magnetic field at the center inside essentially just formed from a superposition of many current loops?

The magnetic field at the center of a solenoid is $\mu_0 N I$ where N is the number of turns per unit length. If we consider the case where there is only 1 turn, however, this doesn't seem to give the same result that manual integration via biot-savart did.

Similarly, a toroid with only 1 loop should be equivalent to a current loop, but we don't get the same result then either.

Why is it wrong to find the magnetic field at the center of a current loop in this way?

Answer

The formula for the solenoid or toroid assumes that the length of the solenoid is much greater than its radius, and that the pitch (distance between turns) is much smaller than the radius. These assumption do not hold for a single current loop.