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

`B_(V)=(mu_(0))/(4pi)(2mcos theta)/(r^(2))`
`B_(H)=(mu_(0))/(4pi)(sin thetam)/(r^(3))`
Squaring both the equations and adding we get
`B_(v)^(2)+B_(v)^(2)=((mu_(0))/(4pi))(m^(2))/(r^(2))(4cos^(2)theta+sin^(2)theta)`
`B=sqrt(B_(v)^(2)+B_(v)^(2))=(mu_(0))/(r_^(3))[3 cos ^(2)theta+1]^(1//2)`
From Eq. (ii) the value of B is minimum if `cos theta=(pi)/(2)` lt brgt `theta=(pi)/(2)` Thus, the magnetic equator is the locus.
(b) Angle of dip.
`tan delta=(B_(v))/(B_(H))=((mu_(2))/(4pi)(2mcos theta)/(3))/((mu_(2))/(4pi)(sin thetam)/(r^(2)))=2cot theta`
`tan delta=2 cot theta`
It means that locus is again magnetic equator.
(c) `tan delta=(B_(V))/(B_(H))`
Angle of dip i.e. `delta=+- 45`
`(B_(V))/(B_(H))=tan(+ -45^(@))`
`(B_(V))/(B_(H))=1` ltbr gt ` 2 cot theta= 1`
`cot theta=(1)/(2)`
`tan theta=2`
`theta=tan^(-1)(2)`