Assume the dipole model of earth's magnetic field B which is given by `B_V`= vertical component of magnetic field `=(mu_0)/(4pi)(2Mcos theta)/(r^3)`, `B_H`=Horizontal component of magnetic field `=(mu_0)/(4pi)(sin theta M)/(r^3)`, `theta=90^@`-latitude as measured from magnetic equator.
Find loci of points for which (i) `|vecB|` is minimum, (ii) dip angle is zero, and (iii) dip angle is `+-45^@`.

Explanation

(i) Given, vertical component of earth's magnetic field, `B_V=(mu_0)/(4pi)(2Mcostheta)/(r^3)`
Horizontal component of earth's magnetic field `B_H=(mu_0)/(4pi)(Msintheta)/(r^3)`
`:. B=sqrt(B_V^2+B_H^2)=(mu_0)/(4pi)M/r^3sqrt(4cos^2theta+sin^2theta)=(mu_0)/(4pi)M/r^3sqrt(3cos^2theta+1)`
From (i) , we note that B is minimum if `cos theta=0` or `theta=pi/2` which is so, at magnetic equator. Thus, B is minimum at magnetic equator which is the loci of points.
(ii) If `delta` is the angle of dip, then `tan delta=(B_V)/(B_H)=(2costheta)/(sin theta)=2cottheta`.
Angle of dip `delta` is zero if `cot theta=0`. It will be so if `theta=pi//2`. So, the angle of dip is zero on magnetic equator which is the loci of points.
(iii) If `delta` is `45^@`, then `tan 45^@=2 cot theta=2//tan theta` or `tan theta=2` or `theta=tan^-1` (2) is the loci of points.