Assume the dipole model for the earth's magnetic field B which is given by,

${\mathrm{B}}_{\mathrm{v}}$ = vertical component of the magnetic field
${\mathrm{B}}_{\mathrm{H}}$ = horizontal component of the magnetic field
= lattitude as measured from the magnetic equator.

Find loci of points for which
(a) |B| is minimum
(b) dip angle is zero and
(c) dip angle is 45°

Hint: Use the formula of the magnetic field due to bar magnet.
(a) Step 1:

Squaring both the equations and adding, we get;
$\begin{array}{l}{\mathrm{B}}_{\mathrm{V}}^{2}+{\mathrm{B}}_{\mathrm{H}}^{2}={\left(\frac{{\mathrm{\mu }}_{0}}{4\mathrm{\pi }}\right)}^{2}\end{array}\frac{{\mathrm{m}}^{2}}{{\mathrm{r}}^{6}}\left(4{\mathrm{cos}}^{2}\mathrm{\theta }+{\mathrm{sin}}^{2}\mathrm{\theta }\right)$                        ...iii
$\mathrm{B}=\begin{array}{l}\sqrt{{\mathrm{B}}_{\mathrm{V}}^{2}+{\mathrm{B}}_{\mathrm{H}}^{2}}=\left(\frac{{\mathrm{\mu }}_{0}}{4\mathrm{\pi }}\right)\end{array}\frac{\mathrm{m}}{{\mathrm{r}}^{3}}{\left(3{\mathrm{cos}}^{2}\mathrm{\theta }+1\right)}^{1}{2}}$
From Eq. (i), the value of B is minimum, if .
Thus, the magnetic equator is the locus.
(b) Step 2: Angle of dip,
For dip angle is zero i.e., $\delta =0$$°$

$\mathrm{\theta }=\frac{\mathrm{\pi }}{2}$

It means that the locus is again the magnetic equator.
(c) Step 3: $\mathrm{tan}\delta =\frac{{B}_{V}}{{B}_{H}}$

Thus,  is the locus.