Considering the earth as a short magnet with its centre coinciding with the centre of earth, show that the angle of dip `phi` is related to magnetic latitude `lambda` through the relation `tan phi=2 tan lambda`

Explanation

Considering the situation for dipole, at position `(r,theta)` we have

`B_(r )=(mu_(0))/(4pi)(2M costheta)/(r^(3))` and `B_(theta)=(mu_(0))/(4pi)(M sin theta)/(r^(3))`
and as `tan phi=(B_(V))/(B_(H))=-(B_(r ))/(B_(theta))`, so in the light of Eq. `(1)`
`tan phi=-2 cot theta`, But from firgure `theta=90^(@)+lambda`
So, `tan phi=-2 cot(90^(@)+lambda),i.e, tan phi=2 tan lambda`