5.1 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°.
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 × 1022 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?

Solution:

(a) The three independent quantities conventionally used to specify Earth’s magnetic field are:

  • Declination (the angle between magnetic north and true north)
  • Inclination or dip (the angle made with the horizontal by the Earth’s magnetic field lines)
  • Horizontal component (the strength of the field in the horizontal direction)

(b) The dip angle in Britain would be greater than in southern India because Britain is closer to the magnetic poles, where the dip angle tends to be larger.

(c) In Melbourne, Australia, the magnetic field lines would appear to go into the ground because it is in the Southern Hemisphere, and magnetic field lines enter the Earth in that region.

(d) A compass placed right at the geomagnetic north pole would point vertically downward, and at the geomagnetic south pole, it would point vertically upward.

(e) The dipole magnetic moment of the Earth is approximately 8 × 10²² J T⁻¹. The order of magnitude of this number can be checked by comparing it to other magnetic moments, such as those of typical laboratory magnets, which are much smaller in comparison.

(f) The presence of local magnetic poles can be explained by irregularities in the distribution of magnetic minerals in the Earth’s crust, which can create localized magnetic fields that deviate from the overall dipole field of the Earth.

5.2 Answer the following questions:
(a) The earth’s magnetic field varies from point to point in space.
Does it also change with time? If so, on what time scale does it
change appreciably?
(b) The earth’s core is known to contain iron. Yet geologists do not
regard this as a source of the earth’s magnetism. Why?
(c) The charged currents in the outer conducting regions of the
earth’s core are thought to be responsible for earth’s magnetism.
What might be the ‘battery’ (i.e., the source of energy) to sustain
these currents?
(d) The earth may have even reversed the direction of its field several
times during its history of 4 to 5 billion years. How can geologists
know about the earth’s field in such distant past?
(e) The earth’s field departs from its dipole shape substantially at
large distances (greater than about 30,000 km). What agencies
may be responsible for this distortion?
(f) Interstellar space has an extremely weak magnetic field of the
order of 10–12 T. Can such a weak field be of any significant
consequence? Explain.
[Note: Exercise 5.2 is meant mainly to arouse your curiosity.
Answers to some questions above are tentative or unknown. Brief
answers wherever possible are given at the end. For details, you
should consult a good text on geomagnetism.]

Solution:

(a) Yes, the Earth’s magnetic field changes over time. This variation occurs on timescales ranging from seconds to millions of years, with significant changes, like geomagnetic reversals, occurring over hundreds of thousands of years.

(b) Although the Earth’s core contains iron, it is in a molten state and not in a form that can retain permanent magnetism. The Earth’s magnetism is instead believed to originate from currents in the molten outer core, driven by the motion of the liquid iron.

(c) The “battery” that sustains these currents is likely the heat generated by the decay of radioactive elements within the Earth, driving convection currents in the molten outer core.

(d) Geologists can know about the Earth’s magnetic field in the distant past through the study of paleomagnetism, where magnetic minerals in ancient rocks lock in the direction and strength of the Earth’s magnetic field at the time of their formation.

(e) The distortion of the Earth’s dipole field at large distances is caused by interactions with the solar wind, which shapes the Earth’s magnetosphere into an elongated structure on the side away from the Sun.

(f) Even though interstellar space has a very weak magnetic field, it can have significant effects on charged particles and on large-scale astrophysical processes, such as the formation of galaxies and stars.

5.3 A short bar magnet placed with its axis at 30° with a uniform external
magnetic field of 0.25 T experiences a torque of magnitude equal to
4.5 × 10–2 J. What is the magnitude of magnetic moment of the magnet?

Solution:


5.4 A short bar magnet of magnetic moment m = 0.32 JT–1 is placed in a
uniform magnetic field of 0.15 T. If the bar is free to rotate in the
plane of the field, which orientation would correspond to its (a) stable,
and (b) unstable equilibrium? What is the potential energy of the
magnet in each case?

5.5 A closely wound solenoid of 800 turns and area of cross section
2.5 × 10–4 m2carries a current of 3.0 A. Explain the sense in which
the solenoid acts like a bar magnet. What is its associated magnetic
moment?


5.6 If the solenoid in Exercise 5.5 is free to turn about the vertical
direction and a uniform horizontal magnetic field of 0.25 T is applied,
what is the magnitude of torque on the solenoid when its axis makes
an angle of 30° with the direction of applied field?


5.7 A bar magnet of magnetic moment 1.5 J T–1 lies aligned with thedirection of a uniform magnetic field of 0.22 T.
(a) What is the amount of work required by an external torque to
turn the magnet so as to align its magnetic moment: (i) normal
to the field direction, (ii) opposite to the field direction?
(b) What is the torque on the magnet in cases (i) and (ii)?


5.8 A closely wound solenoid of 2000 turns and area of cross-section
1.6 × 10–4 m2
, carrying a current of 4.0 A, is suspended through itscentre allowing it to turn in a horizontal plane.
(a) What is the magnetic moment associated with the solenoid?
(b) What is the force and torque on the solenoid if a uniform
horizontal magnetic field of 7.5 × 10–2 T is set up at an angle of
30° with the axis of the solenoid?

5.9 A circular coil of 16 turns and radius 10 cm carrying a current of
0.75 A rests with its plane normal to an external field of magnitude
5.0 × 10–2 T. The coil is free to turn about an axis in its plane
perpendicular to the field direction. When the coil is turned slightly
and released, it oscillates about its stable equilibrium with a
frequency of 2.0 s–1. What is the moment of inertia of the coil about
its axis of rotation?


5.10 A magnetic needle free to rotate in a vertical plane parallel to the
magnetic meridian has its north tip pointing down at 22° with the
horizontal. The horizontal component of the earth’s magnetic field
at the place is known to be 0.35 G. Determine the magnitude of the
earth’s magnetic field at the place.


5.11 At a certain location in Africa, a compass points 12° west of the
geographic north. The north tip of the magnetic needle of a dip circle
placed in the plane of magnetic meridian points 60° above the
horizontal. The horizontal component of the earth’s field is measured
to be 0.16 G. Specify the direction and magnitude of the earth’s field
at the location.


5.12 A short bar magnet has a magnetic moment of 0.48 J T–1. Give the
direction and magnitude of the magnetic field produced by the magnet
at a distance of 10 cm from the centre of the magnet on (a) the axis,
(b) the equatorial lines (normal bisector) of the magnet.


5.13 A short bar magnet placed in a horizontal plane has its axis aligned
along the magnetic north-south direction. Null points are found on
the axis of the magnet at 14 cm from the centre of the magnet. The
earth’s magnetic field at the place is 0.36 G and the angle of dip is
zero. What is the total magnetic field on the normal bisector of the
magnet at the same distance as the null–point (i.e., 14 cm) from the
centre of the magnet? (At null points, field due to a magnet is equal
and opposite to the horizontal component of earth’s magnetic field.)


5.14 If the bar magnet in exercise 5.13 is turned around by 180°, where
will the new null points be located?