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

Bv = vertical component of the magnetic field =μ04π2mcosθr3
BH = horizontal component of the magnetic field =μ04πmsinθr3
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=μ04π2mcosθr3...i
BH=μ04πmsinθr3...ii
Squaring both the equations and adding, we get;
BV2+BH2=μ04π2m2r64cos2θ+sin2θ...iii
B=BV2+BH2=μ04πmr33cos2θ+112
From Eq. (i), the value of B is minimum, if cosθ=π2.
Thus, the magnetic equator is the locus.
(b) Step 2: Angle of dip,
tanδ=BVBH=μ04π.2mcosθr3μ04πmsinθr3=2cotθ...ivtanδ=2cotθ
For dip angle is zero i.e., δ=0°
cotθ=0
θ=π2

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
2cotθ=1FromEq.iv
cotθ=12
tanθ=2
θ=tan1(2)
Thus, θ=tan1(2) is the locus.