The angle of dip is the angle

The relation between $Z_E, H_E$,and $B_E$ 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 $0.16\, G$. The magnitude of the earth's field at the location will be

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 $\pm 45^{\circ}$.

If a magnet is suspended at an angle $30^o$ to the magnetic meridian,it makes an angle of $45^o$ with the horizontal. The real dip is

At some location on Earth,the horizontal component of Earth's magnetic field is $18 \times 10^{-6} \ T$. At this location,a magnetic needle of length $0.12 \ m$ and pole strength $1.8 \ A \ m$ is suspended from its mid-point using a thread. It makes a $45^{\circ}$ angle with the horizontal in equilibrium. To keep this needle horizontal,the vertical force that should be applied at one of its ends is: