Assume the dipole model for Earth's magnetic field $B$,which is given by:
$B_{V} = \text{vertical component of magnetic field} = \frac{\mu_{0}}{4\pi} \frac{2m \cos \theta}{r^{3}}$
$B_{H} = \text{horizontal component of magnetic field} = \frac{\mu_{0}}{4\pi} \frac{m \sin \theta}{r^{3}}$
where $\theta = 90^{\circ} - \text{latitude}$ as measured from the magnetic equator.
$(a)$ Find the loci of points for which $|\vec{B}|$ is minimum.

(A) Given that,$B_{V} = \frac{\mu_{0}}{4\pi} \frac{2m \cos \theta}{r^{3}}$ and $B_{H} = \frac{\mu_{0}}{4\pi} \frac{m \sin \theta}{r^{3}}$.
The magnitude of the magnetic field $B$ is given by $B = \sqrt{B_{V}^{2} + B_{H}^{2}}$.
Substituting the expressions:
$B^{2} = \left(\frac{\mu_{0}}{4\pi} \frac{m}{r^{3}}\right)^{2} (4 \cos^{2} \theta + \sin^{2} \theta)$
Using $\sin^{2} \theta = 1 - \cos^{2} \theta$:
$B^{2} = \left(\frac{\mu_{0}}{4\pi} \frac{m}{r^{3}}\right)^{2} (3 \cos^{2} \theta + 1)$
$B = \frac{\mu_{0}}{4\pi} \frac{m}{r^{3}} \sqrt{3 \cos^{2} \theta + 1}$.
For $B$ to be minimum,the term $(3 \cos^{2} \theta + 1)$ must be minimum. This occurs when $\cos \theta = 0$,which implies $\theta = 90^{\circ}$.
Since $\theta = 90^{\circ} - \text{latitude}$,$\theta = 90^{\circ}$ corresponds to a latitude of $0^{\circ}$,which is the magnetic equator. Thus,the locus of points where $|\vec{B}|$ is minimum is the magnetic equator.