Verify Ampere's law for the magnetic field of a point dipole with dipole moment $\vec{M} = M\hat{k}$. Take $C$ as the closed curve running clockwise along the quarter circle of radius $R$ and center at the origin,in the first quadrant of the $x-z$ plane.

(A) The magnetic field of a point dipole $\vec{M} = M\hat{k}$ at a position $\vec{r}$ is given by $\vec{B}(\vec{r}) = \frac{\mu_0}{4\pi} \left[ \frac{3(\vec{M} \cdot \hat{r})\hat{r} - \vec{M}}{r^3} \right]$.
Ampere's law states that $\oint_C \vec{B} \cdot d\vec{l} = \mu_0 I_{enclosed}$.
For a point dipole,the magnetic field is conservative in regions excluding the origin,meaning $\nabla \times \vec{B} = 0$ for $\vec{r} \neq 0$.
Since the path $C$ is a closed loop that does not enclose any current source (the dipole is a point source at the origin,and the path is in the $x-z$ plane),the total current enclosed $I_{enclosed} = 0$.
Therefore,$\oint_C \vec{B} \cdot d\vec{l} = 0$,which verifies Ampere's law as $0 = \mu_0(0)$.