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

Bv = vertical component of the magnetic field =μ04π2m cos θr3
BH = horizontal component of the magnetic field =μ04πmsin θr3
θ = 90° = 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:
BV=μ04π2m cos θr3                         ...i
BH=μ04πmsinθr3                              ...ii
Squaring both the equations and adding, we get;
BV2+BH2=μ04π2m2r64cos2θ+sin2θ                        ...iii
From Eq. (i), the value of B is minimum, if cos θ=π2.
Thus, the magnetic equator is the locus.
(b) Step 2: Angle of dip,
tan δ=BVBH=μ04π.2mcos θr3μ04πmsin θr3=2cot θ                        ...ivtan δ=2cot θ
For dip angle is zero i.e., δ=0°
cot θ=0

It means that the locus is again the magnetic equator.
(c) Step 3: tanδ=BVBH
 Angle of dip i.e., δ = ±45°
                         BVBH = tan(±45)
                         BVBH = 1
                    2cot θ = 1                          From Eq. iv
                       cot θ = 12
                       tan θ = 2
                        θ = tan1 (2)
Thus, θ = tan1 (2) is the locus.