Write the equation for the magnetic field due to a circular current-carrying loop at a point on the axis of the loop. Give its special cases.

(N/A) The magnetic field $\overrightarrow{B}$ due to a circular loop of radius $R$ carrying current $I$ at a distance $x$ from its center along the axis is given by:
$B = \frac{\mu_{0} I R^{2}}{2(x^{2} + R^{2})^{3/2}}$
Case $1$: When the loop has $N$ turns,the magnetic field is:
$B = \frac{\mu_{0} N I R^{2}}{2(x^{2} + R^{2})^{3/2}}$
Case $2$: Magnetic field at the center of the loop $(x = 0)$:
$B = \frac{\mu_{0} N I}{2R}$
Case $3$: When the point is far away from the loop $(x >> R)$:
$B = \frac{\mu_{0} N I R^{2}}{2x^{3}}$
Case $4$: At a distance $x = R$ from the center:
$B = \frac{\mu_{0} N I R^{2}}{2(R^{2} + R^{2})^{3/2}} = \frac{\mu_{0} N I R^{2}}{2(2R^{2})^{3/2}} = \frac{\mu_{0} N I}{2^{5/2} R}$