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 the dip angle is zero.

(A) The dip angle $\delta$ is defined by the relation $\tan \delta = \frac{B_v}{B_H}$.
For the dip angle to be zero,we must have $\tan \delta = 0$,which implies $B_v = 0$.
Substituting the given expression for $B_v$:
$B_v = \frac{\mu_0}{4\pi} \frac{2m \cos \theta}{r^3} = 0$
Since $\frac{\mu_0}{4\pi}$,$m$,and $r^3$ are non-zero,we must have $\cos \theta = 0$.
This implies $\theta = 90^\circ$.
Given that $\theta = 90^\circ - \text{latitude}$,we have $90^\circ = 90^\circ - \text{latitude}$,which means $\text{latitude} = 0^\circ$.
Therefore,the locus of points where the dip angle is zero is the magnetic equator.