Let $V$ and $H$ be the vertical and horizontal components of the Earth's magnetic field at any point on Earth. Near the North Pole:

At a point on the surface of the Earth,the value of the horizontal component of the Earth's magnetic field is equal to the value of the vertical component of the Earth's magnetic field. The angle of dip is:

$A$ compass needle oscillates $20$ times per minute at a place where the dip is $30^{\circ}$ and $30$ times per minute where the dip is $60^{\circ}$. The ratio of total magnetic field due to the earth at the two places respectively is $\frac{4}{\sqrt{x}}$. The value of $x$ is $............$.

If the dip circle is set at $45^{\circ}$ to the magnetic meridian,then the apparent dip is $30^{\circ}$. The true dip of the place is:

At a certain location in Africa,a compass points $12^{\circ}$ west of the geographic north. The north tip of the magnetic needle of a dip circle placed in the plane of magnetic meridian points $60^{\circ}$ above the horizontal. The horizontal component of the earth's field is measured to be $0.16 \; G$. Specify the direction and magnitude of the earth's field at the location.

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.