Verify the Ampere’s law for magnetic field of a point dipole of dipole moment ${\rm{\vec M = M\hat k}}$. Take $\mathrm{C}$ as the closed curve running clockwise along : the $\mathrm{z}$ - axis from $\mathrm{z} = \mathrm{a} \,>\, 0$ to $\mathrm{z = R}$;

From $P$ to $Q$, every point on the $z$-axis lies at the axial line of magnetic dipole of moment $M$. Due to this magnetic moment the point $(0,0, \mathrm{Z})$ at $z$ distance the magnetic field induction,

$\mathrm{B}=2\left(\frac{\mu_{0}}{4 \pi} \frac{\mathrm{M}}{z^{3}}\right)$

$\mathrm{B}=\frac{\mu_{0} \mathrm{M}}{2 \pi z^{3}}$

$(i)$ From Ampere's law, At point from $\mathrm{P}$ to $\mathrm{Q}$ along $z$-axis

$\int_{\mathrm{P}}^{\mathrm{Q}} \overrightarrow{\mathrm{B}} \cdot \overrightarrow{d l} =\int_{\mathrm{P}}^{\mathrm{Q}} \mathrm{B} d l \cos 0^{\circ}=\int_{a}^{\mathrm{R}} \mathrm{B} d z$

$=\int_{a}^{\mathrm{R}} \frac{\mu_{0}}{2 \pi} \frac{\mathrm{M}}{z^{3}} d z=\frac{\mu_{0} \mathrm{M}}{2 \pi}\left(-\frac{1}{2}\right)\left(\frac{1}{\mathrm{R}^{2}}-\frac{1}{a^{2}}\right)$

$=\frac{\mu_{0} \mathrm{M}}{4 \pi}\left(\frac{1}{a^{2}}-\frac{1}{\mathrm{R}^{2}}\right)$