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

Explanation

(a) Given that, `B_V = (mu_0)/( 4pi ) (2m cos theta)/( r^3) " "…(1)`
`B_H = (mu_0)/( 4pi ) (m sin theta )/( r^3) " "…(2)`
Squaring and adding equation (1) and (2),
`B_(V)^(2) +B_(H)^(2) = ((mu_0)/( 4pi ))^(2) (m^2)/( r^6) [ 4 cos ^(2) theta + sin^(2) theta]`
`therefore B^(2) = ((mu_0)/( 4pi ))^(2) (m^2)/( r^6) [ 4 cos^(2) theta +1 - cos^(2) theta ]`
`therefore B^(2) = sqrt((B_(V)^(2) + B_(H)^(2) ) `
`= sqrt(((mu_0)/( 4pi ))^(2) (m^2)/(r^6) [ 3 cos^(2) theta +1] )`
`= (mu_0)/( 4pi ) (m)/(r^3) [ 3 cos theta^(2) theta + 1]^(1//2) " "...(3)`
From above equation, the value of B is minimum if `cos theta = 0`, hence `theta = (pi)/(2)`
Thus, B is minimum at the magnetic equator.
(b) For angle of dip,
`tan delta = (B_V)/( B_H) = (2 ((mu_0)/( 4pi ) (m cos theta )/( r^3) ) )/( ( (mu_0)/( 4pi) (m sin theta )/( r^3) ) )`
`-2 ( cos theta)/(sin theta)`
`(B_V)/( B_H) = tan delta = 2 cot theta " "...(4)`
For dip angle `delta = 0`, then
`cot theta =0`
`therefore theta= (pi)/(2)`
Hence locus is on magnetic equator.
(c) `tan delta = (B_V)/( B_H)`
For `delta = pm 45^@`
`(B_V)/( B_H) = tan (pm 45^@)`
`(B_V)/( B_H) = 1`
`therefore 2 cot theta =1 " "` [From equation (4)]
`therefore cot theta = (1)/(2) `
` therefore tan theta = 2`
`therefore theta= tan^(-1) (2)` is the locus.