Assume the dipole model for earth’s magnetic field $\mathrm{B}$ which is given by

${{\rm{B}}_{\rm{v}}} = $ vertical component of magnetic field

$ = \frac{{{\mu _0}}}{{4\pi }}\frac{{2m\,\cos \theta }}{{{r^3}}}$

${{\rm{B}}_H} = $ Horizontal component of magnetic field

${{\rm{B}}_H} = \frac{{{\mu _0}}}{{4\pi }}\frac{{m\,\sin \theta }}{{{r^3}}}$

$\theta $ $= 90^{°}$ -latitude as measured from magnetic equator.

$(a)$ Find loci of points for which : dip angle is zero;

A compass needle will show which one of the following directions at the earth's magnetic pole

At the magnetic north pole of the earth, the value of horizontal component of earth's magnetic field and angle of dip are, respectively

Assertion : If a compass needle be kept at magnetic north pole of the earth the compass needle may stay in any direction.

Reason : Dip needle will stay vertical at the north pole of earth

A magnetic needle oscillates in a horizontal plane with a period $T$ at a place where the angle of dip is $60^{\circ}$. When the same needle is made to oscillate in a vertical plane coinciding with the magnetic meridian, its period will be

The values of the apparent angles of dip in two planes at right angles to each other are $45^{\circ}$ and $30^{\circ}$ respectively. The true value of angle of dip at the place is ............