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 $ \pm $ $45^{°}$

A magnet hung at $45^{\circ}$ with magnetic meridian makes an angle of $60^{\circ}$ with the horizontal. The actual value of the angle of dip is.

A very small magnet is placed in the magnetic meridian with its south pole pointing north. The null point is obtained $20\, cm $ away from the centre of the magnet. If the earth's magnetic field (horizontal component) at this point be $0.3$ $gauss$, the magnetic moment of the magnet is

A dip needle vibrates in the vertical plane perpendicular to the magnetic meridian. The time period of vibration is found to be $2$ seconds. The same needle is then allowed to vibrate in the horizontal plane and the time period is again found to be $2$ seconds. Then the angle of dip is.....$^o$

A long vertical wire carries a steady current of $5.0 \,A$. Asensitive magnetic compass is placed in a plane perpendicular to the wire and $10.0$ $cm$ south of it. It registers a deflection $60^{\circ}$ north of east. The magnitude of the horizontal component of the earth's magnetic field is (permeability of free space is $4 \pi \times 10^{-7} \,N/A{ }^2$ )

The magnetic needle of a tangent galvanometer is deflected at an angle $30^o$ due to a magnet. The horizontal component of earth’s magnetic field $0.34 \times {10^{ - 4}}\,T$ is along the plane of the coil. The magnetic intensity is