Answer the following questions regarding earth’s magnetism:

$(a)$ A vector needs three quantities for its specification. Name the three independent quantities conventionally used to specify the earth’s magnetic field.

$(b)$ The angle of dip at a location in southern India is about $18^o$. Would you expect a greater or smaller dip angle in Britain?

$(c)$ If you made a map of magnetic field lines at Melbourne in Australia, would the lines seem to go into the ground or come out of the ground?

$(d)$ In which direction would a compass free to move in the vertical plane point to, if located right on the geomagnetic north or south pole?

$(e)$ The earth’s field, it is claimed, roughly approximates the field due to a dipole of magnetic moment $8 \times 10^{22}\; J\, T^{-1}$ located at its centre. Check the order of magnitude of this number in some way.

$(f)$ Geologists claim that besides the main magnetic $N-S$ poles, there are several local poles on the earth’s surface oriented in different directions. How is such a thing possible at all?

$(a)$ The three independent quantities conventionally use for specifying earth’s magnetic field are:

$(i)$ Magnetic declination,

$(ii)$ Magnetic inclination or angle of dip, and

$(ii)$ Horizontal component of earth’s magnetic field

$(b)$ The angle of dip at a point depends on how far the point is located with respect to the North Pole or the South Pole. The angle of dip would be greater in Britain (it is about $70^o$) than in southern India because the location of Britain on the globe is closer to the magnetic North Pole.

$(c)$ It is hypothetically considered that a huge bar magnet is dipped inside earth with its north pole near the geographic South Pole and its south pole near the geographic North Pole. Magnetic field lines emanate from a magnetic north pole and terminate at a magnetic south pole. Hence, in a map depicting earth’s magnetic field lines, the field lines at Melbourne, Australia would seem to come out of the ground.

$(d)$ If a compass is located on the geomagnetic North Pole or South Pole, then the compass will be free to move in the horizontal plane while earth’s field is exactly vertical to the magnetic poles. In such a case, the compass can point in any direction.

$(e)$ Magnetic moment, $M=8 \times 10^{22} \,J\,T ^{-1}$

Radius of earth, $r=6.4 \times 10^{6} \,m$

Magnetic field strength, $B=\frac{\mu_{0} M}{4 \pi r^{3}}$

Where,

$\mu_{0}=$ Permeability of free space $=4 \pi \times 10^{-7} \,T\,m\, A ^{-1}$

$\therefore B=\frac{4 \pi \times 10^{-7} \times 8 \times 10^{22}}{4 \pi \times\left(6.4 \times 10^{6}\right)^{3}}=0.3\, G$

This quantity is of the order of magnitude of the observed field on earth.

$(f)$ Yes, there are several local poles on earth's surface oriented in different directions. A magnetized mineral deposit is an example of a local $N-S$ pole.