Question1. 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 18º.
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?
(a) The three independent quantities conventionally used for specifying earth’s magnetic field are:
(i) Magnetic declination,
(ii) Angle of dip, and
(iii) 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°) 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.
Magnetic moment, m = 8 × 1022 JT-1
Radius of earth, r = 6.4 × 106m
Our task is to check the order of m, we shall attempt to do so by finding the magnetic field of earth using the available information. If we are close to the standard measured value then the order of m is correct.
We know that magnetic field strength,
(f)Yes, there are several local poles on earth’s surface oriented in different directions. A magnetised mineral deposit is an example of a local N-S pole.
Question2. 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.
(a) Earth’s magnetic field changes with time. It takes a few hundred years to change by an appreciable amount. The variation in earth’s magnetic field with the time cannot be neglected.
(b)Earth’s core contains molten iron. This form of iron is not ferromagnetic. Hence, this is not considered as a source of earth’s magnetism.
(c)Theradioa ctivity in earth’s interior is the source of energy that sustains the currents in the outer conducting regions of earth’s core. These charged currents are considered to be responsible for earth’s magnetism.
(d)Earth reversed the direction of its field several times during its history of 4 to 5 billion years. These magnetic fields got weakly recorded in rocks during their solidification. One can get clues about the geomagnetic history from the analysis of this rock magnetism.
(e)Earth’s field departs from its dipole shape substantially at large distances (greater than about 30,000 km) because of the presence of the ionosphere. In this region, earth’s field gets modified because of the field of single ions. While in motion, these ions produce the magnetic field associated with them.
(f)An extremely weak magnetic field can bend charged particles moving in a circle. This may not be noticeable for a large radius path. With reference to the gigantic interstellar space, the deflection can affect the passage of charged particles.
Question3. A short bar magnet placed with its axis at 30º 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?
Angle between axis of bar magnet and external magnetic field, θ = 30°
Strength of external magnetic field, B = 0.25 T
Torque experienced by bar magnet, Τ = 4.5 × 10-2J
We know that, when a Bar magnet is placed in a uniform magnetic field it experiences a Net torque given by,
T = M × B …(1)
T = M × B × Sinθ
M = magnetic moment
T = Torque on bar magnet
θ = angle between magnetic field and magnetic moment
Question4. A short bar magnet of magnetic moment m = 0.32 J T−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?
Magnetic moment of magnet, M = 0.32JT-1
Strength of external magnetic field, B = 0.15T
(a) When the magnetic moment is aligned (0°) with the magnetic field then we consider this as stable equilibrium, if the magnet is rotated, then it will have the tendency to come back in this position.
θ = 0°
We know that,
U = – M.B
U = -M × B × Cos θ …(1)
Where, U = potential energy
M = magnetic moment
B = magnetic field
θ = Angle between filed and magnetic moment
By putting the given values in the equation (1), we have,
U = -0.32 × 0.15 × Cos 0°
U = -0.048 J
Potential energy of the system in stable equilibrium is -0.048 J.
(b) Whereas, in case of unstable equilibrium the moment is at 180° with the magnetic field. If the magnet is rotated then it will never come back in the initial position.
θ = 180°
Now, by putting values in equation (1), we get,
U = -0.32 × 0.15 × Cos 180°
U = 0.048 J
Potential energy of the system in unstable equilibrium is 0.048J.
Note: The direction of the magnetic moment is from south pole to north pole inside the magnet.
Question5. A closely wound solenoid of 800 turns and area of cross section 2.5 × 10−4 m2 carries a current of 3.0 A. Explain the sense in which the solenoid acts like a bar magnet. What is its associated magnetic moment?
Number of turns in the solenoid, n = 800
Area of cross-section, A = 2.5 × 10−4 m2
Current in the solenoid, I = 3.0 A
A current-carrying solenoid behaves as a bar magnet because a magnetic field develops along its axis, i.e., along its length.
The magnetic moment associated with the given current-carrying solenoid is calculated as:
M = n I A
= 800 × 3 × 2.5 × 10−4
= 0.6 J T−1
Question6. 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?
Magnetic field strength, B = 0.5 T
Magnetic moment, M = 3.12 T−1
The angle θ, between the axis of the solenoid and the direction of the applied field is 60°.
Therefore, the torque acting on the solenoid is given as:
= 3.12× 0.5sin60 °
= 13.5×10−2 J
Question7. A bar magnet of magnetic moment 1.5 J T−1 lies aligned with the direction 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)?
(a) Magnetic moment, M = 1.5JT−1
Magnetic field strength, B = 0.22 T
(i) Initial angle between the axis and the magnetic field, θ1=0 °
Final angle between the axis and the magnetic field, θ2=90 °
The work required to make the magnetic moment normal to the direction of magnetic field is given as:
W = −MB(cos θ2 – cos θ1)
= −1.5 × 0.22 (cos90 ° – cos0 °
= – 0.33 (0 – 1)
= 0.33 J
(ii) Initial angle between the axis and the magnetic field, θ1=0°
Final angle between the axis and the magnetic field, θ2 =180 °
The work required to make the magnetic moment opposite to the direction of magnetic field is given as:
W = −MB (cosθ2 – cos θ1)
= −1.5 × 0.22 (cos180 ° – cos0 °)
= – 0.33 (– 1 – 1)
= 0.66 J
(b) For case (i): θ= θ2=90° ∴ Torque, τ=MBsinθ
= 0.33 J
For case (ii): θ= θ2=180 ° ∴ Torque, τ=MBsinθ
= 0 J
Question8. 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 its centre 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 30º with the axis of the solenoid?
Number of turns on the solenoid, n = 1000
Area of cross-section of the solenoid, A = 2×10−4m2
Current in the solenoid, I = 3.0 A
(a) The magnetic moment along the axis of the solenoid is calculated as:
M = nAI
= 1000 × 2 × 10−4×3
= 0.6 Am2
(b) Magnetic field, B = 5.5 × 10−2T
Angle between the magnetic field and the axis of the solenoid, θ=30°
Torque, τ=MB sinθ
= 0.6 × 5.5 ×10 − 2sin30°
= 1.65 × 10−2Nm
Since the magnetic field is uniform, the force on the solenoid is zero. The torque on the solenoid is 1.65×10−2Nm.
Question9. 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?
Solution :Number of turns in the circular coil, N = 16
Radius of the coil, r = 10 cm = 0.1 m
Cross-section of the coil, A = πr2 = π × (0.1)2 m2
Current in the coil, I = 0.75 A
Magnetic field strength, B = 5.0 × 10−2 T
Frequency of oscillations of the coil, f = 2.0 s−1
Magnetic moment, M = NIA = NI × πr2
= 16 × 0.75 A × π × (0.1)2 m2
= 0.377 J T−1
Now the frequency is given by,
Question10. A magnetic needle free to rotate in a vertical plane parallel to the magnetic meridian has its north tip pointing down at 22º 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.
Solution :Horizontal component of earth’s magnetic field, BH = 0.35 G
Angle made by the needle with the horizontal plane = Angle of dip = δ 22°
Earth’s magnetic field strength = B
Question11. At a certain location in Africa, a compass points 12º west of the geographic north. The north tip of the magnetic needle of a dip circle placed in the plane of magnetic meridian points 60º 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.
Solution :Angle of declination, θ = 12°
Angle of dip, δ = 60°
Horizontal component of earth’s magnetic field, BH = 0.16 G
Earth’s magnetic field at the given location = B
Question12. 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.
Solution :Magnetic moment of the bar magnet, M = 0.48 J T−1
(a) Distance, d = 10 cm = 0.1 m
The magnetic field at distance d, from the centre of the magnet on the axis is given by the relation:
Question13. 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.)
Solution :Earth’s magnetic field at the given place, B = 0.36 G
The magnetic field at a distance d, on the axis of the magnet is given as:
Question14. If the bar magnet in exercise 5.13 is turned around by 180º, where will the new null points be located?
Solution :The magnetic field on the axis of the magnet at a distance d1 = 14 cm, can be written as:
Question15. A short bar magnet of magnetic moment 5.25 × 10−2 J T−1 is placed with its axis perpendicular to the earth’s field direction. At what distance from the centre of the magnet, the resultant field is inclined at 45º with earth’s field on
(a) its normal bisector and (b) its axis. Magnitude of the earth’s field at the place is given to be 0.42 G. Ignore the length of the magnet in comparison to the distances involved.
Solution :Magnetic moment of the bar magnet, M = 5.25 × 10−2 J T−1
Magnitude of earth’s magnetic field at a place, H = 0.42 G = 0.42 × 10−4T
(a) The magnetic field at a distance R from the centre of the magnet on the normal bisector is given by the relation:
Question16. Answer the following questions:
(a) Why does a paramagnetic sample display greater magnetisation (for the same magnetising field) when cooled?
(b) Why is diamagnetism, in contrast, almost independent of temperature?
(c) If a toroid uses bismuth for its core, will the field in the core be (slightly) greater or (slightly) less than when the core is empty?
(d) Is the permeability of a ferromagnetic material independent of the magnetic field? If not, is it more for lower or higher fields?
(e) Magnetic field lines are always nearly normal to the surface of a ferromagnet at every point. (This fact is analogous to the static electric field lines being normal to the surface of a conductor at every point.) Why?
(f ) Would the maximum possible magnetisation of a paramagnetic sample be of the same order of magnitude as the magnetization of a ferromagnet?
(a) Owing to the random thermal motion of the molecules, the alignments of dipoles get disrupted at high temperatures. On cooling, this disruption is reduced. Hence, a paramagnetic sample displays greater magnetisation when cooled.
(b) Each molecule of the diamagnetic material is not a magnetic dipole in itself. Hence, the random thermal motion of the molecules of the diamagnetic material (which is related to the temperature) does not affect the diamagnetism of the material.
(c) Bismuth is a diamagnetic substance. Hence, a toroid with a bismuth core has a magnetic field slightly less than a toroid whose core is empty.
(d) The permeability of ferromagnetic materials is not independent of the applied magnetic field. It is greater for a lower field.
(e) The permeability of a ferromagnetic material is always greater than one. Hence, magnetic field lines are always nearly normal to the surface of such materials at every point. The proof of this fact is based on the boundary conditions of the magnetic fields at the interface of two media.
(f) Yes, the maximum possible magnetisation of a paramagnetic sample will be of the same order of magnitude as the magnetisation of a ferromagnet. This requires high magnetising fields for saturation.
Question17. Answer the following questions:
(a) Explain qualitatively on the basis of domain picture the irreversibility in the magnetisation curve of a ferromagnet.
(b) The hysteresis loop of a soft iron piece has a much smaller area than that of a carbon steel piece. If the material is to go through repeated cycles of magnetisation, which piece will dissipate greater heat energy?
(c) ‘A system displaying a hysteresis loop such as a ferromagnet, is a device for storing memory?’ Explain the meaning of this statement.
(d) What kind of ferromagnetic material is used for coating magnetic tapes in a cassette player, or for building ‘memory stores’ in a modern computer?
(e) A certain region of space is to be shielded from magnetic fields.
Suggest a method.
The hysteresis curve (B-H curve) of a ferromagnetic material is shown in the following figure.
(a) It can be observed from the given curve that magnetisation persists even when the external field is removed. This reflects the irreversibility of a ferromagnet.
(b)The dissipated heat energy is directly proportional to the area of a hysteresis loop. A carbon steel piece has a greater hysteresis curve area. Hence, it dissipates greater heat energy.
(c)The value of magnetisation is memory or record of hysteresis loop cycles of magnetisation. These bits of information correspond to the cycle of magnetisation. Hysteresis loops can be used for storing information.
(d)Ceramic is used for coating magnetic tapes in cassette players and for building memory stores in modern computers.
(e)A certain region of space can be shielded from magnetic fields if it is surrounded by soft iron rings. In such arrangements, the magnetic lines are drawn out of the region.
Question18. A long straight horizontal cable carries a current of 2.5 A in the direction 10º south of west to 10° north of east. The magnetic meridian of the place happens to be 10º west of the geographic meridian. The earth’s magnetic field at the location is 0.33 G, and the angle of dip is zero. Locate the line of neutral points (ignore the thickness of the cable). (At neutral points, magnetic field due to a current-carrying cable is equal and opposite to the horizontal component of earth’s magnetic field.)
Solution :Current in the cable (I) = 2.5 A
Earth’s magnetic field at the location (H) = 0.33 G = 0.33 × 10-4T
Angle of dip (δ) = 0
Let the line of neutral points be at a distance “r” meters from the horizontal cable.
The magnetic field at the neutral point due to current carrying cable,
Question19. A telephone cable at a place has four long straight horizontal wires carrying a current of 1.0 A in the same direction east to west. The earth’s magnetic field at the place is 0.39 G, and the angle of dip is 35º. The magnetic declination is nearly zero. What are the resultant magnetic fields at points 4.0 cm below the cable?
Solution :Current in the wires (I) = 1.0 A (east to west)
The earth’s magnetic field at the place (H) = 0.39G = 0.39 × 10-4T
The angle of dip (δ) = 35º
Magnetic declination (θ) ∼ 0o
Number of wires in the cable (n) = 4
The horizontal component of earth’s magnetic field (H’) = H × cosδ
Question20. A compass needle free to turn in a horizontal plane is placed at the centre of circular coil of 30 turns and radius 12 cm. The coil is in a vertical plane making an angle of 45º with the magnetic meridian. When the current in the coil is 0.35 A, the needle points west to east.
(a) Determine the horizontal component of the earth’s magnetic field at the location.
(b) The current in the coil is reversed, and the coil is rotated about its vertical axis by an angle of 90º in the anticlockwise sense looking from above. Predict the direction of the needle. Take the magnetic declination at the places to be zero.
Number of turns in the circular coil, N = 30
Radius of the circular coil, r = 12 cm = 0.12 m
Current in the coil, I = 0.35 A
Angle of dip, δ = 45°
(b) When the current in the coil is reversed and the coil is rotated about its vertical axis by an angle of 90 º, the needle will reverse its original direction. In this case, the needle will point from East to West.
Question21. A magnetic dipole is under the influence of two magnetic fields. The angle between the field directions is 60º, and one of the fields has a magnitude of 1.2 × 10−2 T. If the dipole comes to stable equilibrium at an angle of 15º with this field, what is the magnitude of the other field?
Solution :Magnitude of one of the magnetic fields (B1) = 1.2 × 10–2 T
The angle between the magnetic field directions (θ) = 60º
The angle between the dipole and the magnetic field B1 is θ1 = 15°
Let magnitude of second of the magnetic fields = B2
∴ The angle between the dipole and the magnetic field B2 is θ2 = θ-θ1 = 45°
Let magnetic moment of the dipole be M
Thus, at rotational equilibrium,
Torque due to field B1 = Torque due to field B2
⇒ M × B1sinθ1 = M × B2 sinθ2
Question22. A monoenergetic (18 keV) electron beam initially in the horizontal direction is subjected to a horizontal magnetic field of 0.04 G normal to the initial direction. Estimate the up or down deflection of the beam over a distance of 30 cm (me= 9.11 × 10−19 C). [Note: Data in this exercise are so chosen that the answer will give you an idea of the effect of earth’s magnetic field on the motion of the electron beam from the electron gun to the screen in a TV set.]
Solution :Energy of the electron beam(E) = 18keV = 18000eV = 18000 × 1.6 × 10-19 = 2.88 × 10-15J
Mass of electron (me) = 9.11 × 10–31 kg
Horizontal magnetic field (B) = 0.04 G = 0.04 × 10-4T
Distance of deflection from the beam (r) = 30 cm = 0.3m
Let velocity of electrons be v.
∴ Energy of the beam = Kinetic energy of electrons
⇒ E = ½ mev2
∵ The electron beam deflects along a circular path in a magnetic field.
Let the radius of the circular path be r m.
The force due to magnetic field on the electron = qvBsinθ = e × v × B × sin90°
Here, q = charge on the electron,
θ = angle between velocity and magnetic field.
∵ The force due to magnetic field on electron = centripetal force on electron
Question23. A sample of paramagnetic salt contains 2.0 × 1024 atomic dipoles each of dipole moment 1.5 × 10−23 J T−1. The sample is placed under a homogeneous magnetic field of 0.64 T, and cooled to a temperature of 4.2 K. The degree of magnetic saturation achieved is equal to 15%. What is the total dipole moment of the sample for a magnetic field of 0.98 T and a temperature of 2.8 K? (Assume Curie’s law)
Solution :Number of atomic dipoles (n) = 2 × 1024
Initially, Dipole moment of each atomic dipole (M) = 1.5 × 10-23JT-1
Magnetic field (B1) = 0.64 T and Temperature (θ1) = 4.2K
∵ Degree of magnetic saturation = 15%
∴ Effective Dipole moment (M1) = 15% of Total dipole moment
Question24. A Rowland ring of mean radius 15 cm has 3500 turns of wire wound on a ferromagnetic core of relative permeability 800. What is the magnetic field B in the core for a magnetising current of 1.2 A?
Solution :Radius of ring (r) = 15cm = 0.15m
Number of turns in the ring (n) = 3500
Relative permeability of the ferromagnetic core (μr) = 800
Current in the Rowland ring (I) = 1.2A
Question25. The magnetic moment vectors μsand μlassociated with the intrinsic spin angular momentum S and orbital angular momentum l, respectively, of an electron are predicted by quantum theory (and verified experimentally to a high accuracy) to be given by:
μs= –(e/m) S, μl = –(e/2m)l Which of these relations is in accordance with the result expected classically? Outline the derivation of the classical result.