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?

(N/A) The three independent quantities conventionally used for specifying the earth's magnetic field are:
$(i)$ Magnetic declination,
$(ii)$ Magnetic inclination or angle of dip,and
$(iii)$ Horizontal component of the earth's magnetic field.
$(b)$ The angle of dip at a point depends on its latitude. Britain is closer to the magnetic North Pole than southern India,so the angle of dip would be greater in Britain (approximately $70^o$).
$(c)$ The earth's magnetic field lines emanate from the magnetic South Pole (near the geographic North Pole) and terminate at the magnetic North Pole (near the geographic South Pole). Since Melbourne is in the Southern Hemisphere,the field lines would seem to come out of the ground.
$(d)$ At the geomagnetic poles,the earth's magnetic field is purely vertical. $A$ compass free to move in the vertical plane would point vertically downwards at the North Pole and vertically upwards at the South Pole.
$(e)$ Given magnetic moment $M = 8 \times 10^{22} \, J \, T^{-1}$ and radius $r = 6.4 \times 10^6 \, m$. The magnetic field $B = \frac{\mu_0 M}{4 \pi r^3}$. Substituting $\mu_0 = 4 \pi \times 10^{-7} \, T \, m \, A^{-1}$,we get $B = \frac{10^{-7} \times 8 \times 10^{22}}{(6.4 \times 10^6)^3} \approx 3 \times 10^{-5} \, T = 0.3 \, G$. This matches the observed order of magnitude of the earth's magnetic field.
$(f)$ Local magnetic poles exist due to the presence of magnetized mineral deposits or geological formations containing ferromagnetic materials,which create localized magnetic field anomalies.