a) By careful and thorough application of the Biot-Savart law, evaluate
B at the center of the circular current loop.
b) 2 point bonus -- What is B at the center of a split-ring of current?
Select an arbitrary current element and identify vectors involved:
Since ds and r are perpendicular for all current elements, regrdless of position on loop, ds x r = ds. The direction of each contribution to dB will be out of the page (screen) for each element. Also r = a for all elements. So:
B at the center of a circular current loop points out of the page (screen).
b) The direction of current in the top half of the loop has reversed direction. Thus the contribution of the top half of the loop to B will also reverse direction. Since it is exactly equal in magnitude to the contribution from the bottom half of the loop, but opposite in direction, B = 0 for this current configuration.