4.10 THE MOVING COIL GALVANOMETER Currents and voltages in circuits have been discussed extensively in Chapters 3. But how do we measure them? How do we claim that current in a circuit is 1.5 A or the voltage drop across a resistor is 1.2 V? Figure 4.20 exhibits a very useful instrument for this purpose: the moving coil galvanometer (MCG). It is a device whose principle can be understood on the basis of our discussion in Section 4.9. The galvanometer consists of a coil, with many turns, free to rotate about a fixed axis (Fig. 4.20), in a uniform radial magnetic field. There is a cylindrical soft iron core which not only makes the field radial but also increases the strength of the magnetic field. When a current flows through the coil, a torque acts on it. This torque is given by Eq. (4.20) to be 129 τ = NI AB Reprint 2025-26 Physics where the symbols have their usual meaning. Since the field is radial by design, we have taken sin θ = 1 in the above expression for the torque. The magnetic torque NIAB tends to rotate the coil. A spring Sp provides a counter torque kφ that balances the magnetic torque NIAB; resulting in a steady angular deflection φ. In equilibrium kφ = NI AB where k is the torsional constant of the spring; i.e. the restoring torque per unit twist. The deflection φ is indicated on the scale by a pointer attached to the spring. We have  NAB  φ =  k  I (4.26) The quantity in brackets is a constant for a given galvanometer. The galvanometer can be used in a number of ways. It can be used as a detector to check if a current is FIGURE 4.20 The moving coil flowing in the circuit. We have come across this usage galvanometer. Its elements are in the Wheatstone’s bridge arrangement. In this usage described in the text. Depending on the neutral position of the pointer (when no current is the requirement, this device can be flowing through the galvanometer) is in the middle of used as a current detector or for the scale and not at the left end as shown in Fig.4.20. measuring the value of the current Depending on the direction of the current, the pointer’s (ammeter) or voltage (voltmeter). deflection is either to the right or the left. The galvanometer cannot as such be used as an ammeter to measure the value of the current in a given circuit. This is for two reasons: (i) Galvanometer is a very sensitive device, it gives a full- scale deflection for a current of the order of µA. (ii) For measuring currents, the galvanometer has to be connected in series, and as it has a large resistance, this will change the value of the current in the circuit. To overcome these difficulties, one attaches a small resistance rs, called shunt resistance, in parallel with the galvanometer coil; so that most of the current passes through the shunt. The resistance of this arrangement is, ≃ RG rs / (RG + rs) rs if RG >> rs If rs has small value, in relation to the resistance of the rest of the circuit Rc, the effect of introducing the measuring instrument is also small and negligible. This arrangement is schematically shown in Fig. 4.21. FIGURE 4.21 The scale of this ammeter is calibrated and then graduated to read off Conversion of a the current value with ease. We define the current sensitivity of the galvanometer (G) to galvanometer as the deflection per unit current. From Eq. (4.26) this an ammeter by the current sensitivity is, introduction of a NAB φshunt resistance rs of = (4.27) very small value in I k parallel. A convenient way for the manufacturer to increase the sensitivity is to increase the number of turns N. We choose galvanometers having 130 sensitivities of value, required by our experiment. Reprint 2025-26 Moving Charges and Magnetism The galvanometer can also be used as a voltmeter to measure the voltage across a given section of the circuit. For this it must be connected in parallel with that section of the circuit. Further, it must draw a very small current, otherwise the voltage measurement will disturb the original set up by an amount which is very large. Usually we like to keep the disturbance due to the measuring device below one per cent. To ensure this, a large resistance R is connected in series with the galvanometer. This arrangement is schematically depicted in Fig.4.22. Note that the resistance of the voltmeter is now, FIGURE 4.22 RG + R ≃ R : large Conversion of a The scale of the voltmeter is calibrated to read off the voltage value galvanometer (G) to a with ease. We define the voltage sensitivity as the deflection per unit voltmeter by the introduction of avoltage. From Eq. (4.26), resistance R of large φ  NAB  I  NAB  1 value in series. = = (4.28) V  k  V  k  R An interesting point to note is that increasing the current sensitivity may not necessarily increase the voltage sensitivity. Let us take Eq. (4.27) which provides a measure of current sensitivity. If N → 2N, i.e., we double the number of turns, then φ φ → 2 I I Thus, the current sensitivity doubles. However, the resistance of the galvanometer is also likely to double, since it is proportional to the length of the wire. In Eq. (4.28), N →2N, and R →2R, thus the voltage sensitivity, φ φ → V V remains unchanged. So in general, the modification needed for conversion of a galvanometer to an ammeter will be different from what is needed for converting it into a voltmeter. Example 4.12 In the circuit (Fig. 4.23) the current is to be measured. What is the value of the current if the ammeter shown (a) is a galvanometer with a resistance RG = 60.00 Ω; (b) is a galvanometer described in (a) but converted to an ammeter by a shunt resistance rs = 0.02 Ω; (c) is an ideal ammeter with zero resistance? EXAMPLE FIGURE 4.23 4.12 131 Reprint 2025-26 Physics Solution (a) Total resistance in the circuit is, RG + 3 = 63 Ω. Hence, I = 3/63 = 0.048 A. (b) Resistance of the galvanometer converted to an ammeter is, R G rs 60 Ω × 0. 02 Ω = ≃ 0.02Ω R G + rs ( 60 + 0 .02 )Ω 4.12 Total resistance in the circuit is, 0.02 Ω+ 3 Ω= 3.02 Ω. Hence, I = 3/3.02 = 0.99 A. (c) For the ideal ammeter with zero resistance, EXAMPLE I = 3/3 = 1.00 A SUMMARY 1. The total force on a charge q moving with velocity v in the presence of magnetic and electric fields B and E, respectively is called the Lorentz force. It is given by the expression: F = q (v × B + E) The magnetic force q (v × B) is normal to v and work done by it is zero. 2. A straight conductor of length l and carrying a steady current I experiences a force F in a uniform external magnetic field B, F = I l × B where|l| = l and the direction of l is given by the direction of the current. 3. In a uniform magnetic field B, a charge q executes a circular orbit in a plane normal to B. Its frequency of uniform circular motion is called the cyclotron frequency and is given by: q B νc = 2 π m This frequency is independent of the particle’s speed and radius. This fact is exploited in a machine, the cyclotron, which is used to accelerate charged particles. 4. The Biot-Savart law asserts that the magnetic field dB due to an element dl carrying a steady current I at a point P at a distance r from the current element is: r d l µ × 0 I dB = 3 r 4 π To obtain the total field at P, we must integrate this vector expression over the entire length of the conductor. 5. The magnitude of the magnetic field due to a circular coil of radius R carrying a current I at an axial distance x from the centre is Reprint 2025-26 Moving Charges and Magnetism 2 IR µ 0 B = 2 2 2( x + R )3/2 At the centre this reduces to µ0 I B = 2 R 6. Ampere’s Circuital Law: Let an open surface S be bounded by a loop B.d l = µ0 I where I refers to C. Then the Ampere’s law states that ∫CÑ the current passing through S. The sign of I is determined from the right-hand rule. We have discussed a simplified form of this law. If B is directed along the tangent to every point on the perimeter L of a closed curve and is constant in magnitude along perimeter then, BL = µ0 Ie where Ie is the net current enclosed by the closed circuit. 7. The magnitude of the magnetic field at a distance R from a long, straight wire carrying a current I is given by: µ0 I B = 2 π R The field lines are circles concentric with the wire. 8. The magnitude of the field B inside a long solenoid carrying a current I is B = µ0nI where n is the number of turns per unit length. 9. Parallel currents attract and anti-parallel currents repel. 10. A planar loop carrying a current I, having N closely wound turns, and an area A possesses a magnetic moment m where, m = N I A and the direction of m is given by the right-hand thumb rule : curl the palm of your right hand along the loop with the fingers pointing in the direction of the current. The thumb sticking out gives the direction of m (and A) When this loop is placed in a uniform magnetic field B, the force F on it is: F = 0 And the torque on it is, τ = m × B In a moving coil galvanometer, this torque is balanced by a counter- torque due to a spring, yielding kφ = NI AB where φ is the equilibrium deflection and k the torsion constant of the spring. 11. A moving coil galvanometer can be converted into a ammeter by introducing a shunt resistance rs, of small value in parallel. It can be converted into a voltmeter by introducing a resistance of a large value in series. 133 Reprint 2025-26 Physics Physical Quantity Symbol Nature Dimensions Units Remarks Permeability of free µ0 Scalar [MLT –2A–2] T m A–1 4π × 10–7 T m A–1 space Magnetic Field B Vector [M T –2A–1] T (telsa) Magnetic Moment m Vector [L2A] A m2 or J/T Torsion Constant k Scalar [M L2T –2] N m rad–1 Appears in MCG POINTS TO PONDER 1. Electrostatic field lines originate at a positive charge and terminate at a negative charge or fade at infinity. Magnetic field lines always form closed loops. 2. The discussion in this Chapter holds only for steady currents which do not vary with time. When currents vary with time Newton’s third law is valid only if momentum carried by the electromagnetic field is taken into account. 3. Recall the expression for the Lorentz force, F = q (v × B + E) This velocity dependent force has occupied the attention of some of the greatest scientific thinkers. If one switches to a frame with instantaneous velocity v, the magnetic part of the force vanishes. The motion of the charged particle is then explained by arguing that there exists an appropriate electric field in the new frame. We shall not discuss the details of this mechanism. However, we stress that the resolution of this paradox implies that electricity and magnetism are linked phenomena (electromagnetism) and that the Lorentz force expression does not imply a universal preferred frame of reference in nature. 4. Ampere’s Circuital law is not independent of the Biot-Savart law. It can be derived from the Biot-Savart law. Its relationship to the Biot-Savart law is similar to the relationship between Gauss’s law and Coulomb’s law. EXERCISES 4.1 A circular coil of wire consisting of 100 turns, each of radius 8.0 cm carries a current of 0.40 A. What is the magnitude of the magnetic field B at the centre of the coil? 4.2 A long straight wire carries a current of 35 A. What is the magnitude of the field B at a point 20 cm from the wire? 4.3 A long straight wire in the horizontal plane carries a current of 50 A in north to south direction. Give the magnitude and direction of B 134 at a point 2.5 m east of the wire. Reprint 2025-26 Moving Charges and Magnetism

4.9 A square coil of side 10 cm consists of 20 turns and carries a current of 12 A. The coil is suspended vertically and the normal to the plane of the coil makes an angle of 30º with the direction of a uniform horizontal magnetic field of magnitude 0.80 T. What is the magnitude of torque experienced by the coil?

4.13 (a) A circular coil of 30 turns and radius 8.0 cm carrying a current of 6.0 A is suspended vertically in a uniform horizontal magnetic field of magnitude 1.0 T. The field lines make an angle of 60° with the normal of the coil. Calculate the magnitude of the counter torque that must be applied to prevent the coil from turning. (b) Would your answer change, if the circular coil in (a) were replaced by a planar coil of some irregular shape that encloses the same area? (All other particulars are also unaltered.) 135 Reprint 2025-26 Physics Chapter Five MAGNETISM AND MATTER 5.1 INTRODUCTION Magnetic phenomena are universal in nature. Vast, distant galaxies, the tiny invisible atoms, humans and beasts all are permeated through and through with a host of magnetic fields from a variety of sources. The earth’s magnetism predates human evolution. The word magnet is derived from the name of an island in Greece called magnesia where magnetic ore deposits were found, as early as 600 BC. In the previous chapter we have learned that moving charges or electric currents produce magnetic fields. This discovery, which was made in the early part of the nineteenth century is credited to Oersted, Ampere, Biot and Savart, among others. In the present chapter, we take a look at magnetism as a subject in its own right. Some of the commonly known ideas regarding magnetism are: (i) The earth behaves as a magnet with the magnetic field pointing approximately from the geographic south to the north. (ii) When a bar magnet is freely suspended, it points in the north-south direction. The tip which points to the geographic north is called the north pole and the tip which points to the geographic south is called 136 the south pole of the magnet. Reprint 2025-26 Magnetism and Matter (iii) There is a repulsive force when north poles ( or south poles ) of two magnets are brought close together. Conversely, there is an attractive force between the north pole of one magnet and the south pole of the other. (iv) We cannot isolate the north, or south pole of a magnet. If a bar magnet is broken into two halves, we get two similar bar magnets with somewhat weaker properties. Unlike electric charges, isolated magnetic north and south poles known as magnetic monopoles do not exist. (v) It is possible to make magnets out of iron and its alloys. We begin with a description of a bar magnet and its behaviour in an external magnetic field. We describe Gauss’s law of magnetism. We next describe how materials can be classified on the basis of their magnetic properties. We describe para-, dia-, and ferromagnetism. 5.25.25.25.25.2 TTTHETTHEHEHEHE BBBARBBARARARAR MMMAGNETMMAGNETAGNETAGNETAGNET We begin our study by examining iron filings sprinkled on a sheet of glass placed over a short bar magnet. The arrangement of iron filings is shown in Fig. 5.1. The pattern of iron filings suggests that the magnet has two poles similar to the positive and negative charge of an electric dipole. As mentioned in the introductory section, one pole is designated the North pole and the other, the South pole. When suspended freely, these poles point approximately towards the geographic north and south poles, respectively. A similar pattern of iron filings is observed around a current carrying solenoid. FIGUREFIGUREFIGUREFIGUREFIGURE 5.15.15.15.15.1 The arrangement 5.2.15.2.15.2.15.2.15.2.1 TheTheTheTheThe magneticmagneticmagneticmagneticmagnetic fieldfieldfieldfieldfield lineslineslineslineslines of iron filings surrounding a bar magnet. The pattern mimics The pattern of iron filings permits us to plot magnetic field lines. The pattern the magnetic field lines*. This is shown both suggests that the bar magnet is for the bar-magnet and the current- a magnetic dipole. carrying solenoid in Fig. 5.2. For comparison refer to the Chapter 1, Figure 1.14(d). Electric field lines of an electric dipole are also displayed in Fig. 5.2(c). The magnetic field lines are a visual and intuitive realisation of the magnetic field. Their properties are: (i) The magnetic field lines of a magnet (or a solenoid) form continuous closed loops. This is unlike the electric dipole where these field lines begin from a positive charge and end on the negative charge or escape to infinity. * In some textbooks the magnetic field lines are called magnetic lines of force. This nomenclature is avoided since it can be confusing. Unlike electrostatics the field lines in magnetism do not indicate the direction of the force on a 137 (moving) charge. Reprint 2025-26 Physics FIGURE 5.2 The field lines of (a) a bar magnet, (b) a current-carrying finite solenoid and (c) electric dipole. At large distances, the field lines are very similar. The curves labelled i and ii are closed Gaussian surfaces. (ii) The tangent to the field line at a given point represents the direction of the net magnetic field B at that point. (iii) The larger the number of field lines crossing per unit area, the stronger is the magnitude of the magnetic field B. In Fig. 5.2(a), B is larger around region ii than in region i . (iv) The magnetic field lines do not intersect, for if they did, the direction of the magnetic field would not be unique at the point of intersection. One can plot the magnetic field lines in a variety of ways. One way is to place a small magnetic compass needle at various positions and note its orientation. This gives us an idea of the magnetic field direction at various points in space. 5.2.2 Bar magnet as an FIGURE 5.3 Calculation of (a) The axial field of a equivalent solenoid finite solenoid in order to demonstrate its In the previous chapter, we have similarity to that of a bar magnet. (b) A magnetic explained how a current loop acts as a needle in a uniform magnetic field B. The arrangement may be used to determine either B magnetic dipole (Section 4.9). We or the magnetic moment m of the needle. mentioned Ampere’s hypothesis that all magnetic phenomena can be explained in 138 terms of circulating currents. Reprint 2025-26 Magnetism and Matter The resemblance of magnetic field lines for a bar magnet and a solenoid suggest that a bar magnet may be thought of as a large number of circulating currents in analogy with a solenoid. Cutting a bar magnet in half is like cutting a solenoid. We get two smaller solenoids with weaker magnetic properties. The field lines remain continuous, emerging from one face of the solenoid and entering into the other face. One can test this analogy by moving a small compass needle in the neighbourhood of a bar magnet and a current-carrying finite solenoid and noting that the deflections of the needle are similar in both cases. To make this analogy more firm we may calculate the axial field of a finite solenoid depicted in Fig. 5.3 (a). We can demonstrate that at large distances this axial field resembles that of a bar magnet. The magnitude of the field at point P due to the solenoid is µ0 2 m B = 3 (5.1) 4π r This is also the far axial magnetic field of a bar magnet which one may obtain experimentally. Thus, a bar magnet and a solenoid produce similar magnetic fields. The magnetic moment of a bar magnet is thus equal to the magnetic moment of an equivalent solenoid that produces the same magnetic field. 5.2.3 The dipole in a uniform magnetic field Let’s place a small compass needle of known magnetic moment m allowing it to oscillate in the magnetic field. This arrangement is shown in Fig. 5.3(b). The torque on the needle is [see Eq. (4.23)], τ = m × B (5.2) In magnitude τ = mB sinθ Here τ is restoring torque and θ is the angle between m and B. An expression for magnetic potential energy can be obtained on lines similar to electrostatic potential energy. The magnetic potential energy Um is given by (θ)d θ U m = ∫τ sin θ d θ = −mB cos θ = ∫mB = –m.B (5.3) We have emphasised in Chapter 2 that the zero of potential energy can be fixed at one’s convenience. Taking the constant of integration to be zero means fixing the zero of potential energy at θ = 90°, i.e., when the needle is perpendicular to the field. Equation (5.3) shows that potential energy is minimum (= –mB) at θ = 0° (most stable position) and maximum (= +mB) at θ = 180° (most unstable position). Example 5.1 (a) What happens if a bar magnet is cut into two pieces: (i) transverse to its length, (ii) along its length? (b) A magnetised needle in a uniform magnetic field experiences a EXAMPLE torque but no net force. An iron nail near a bar magnet, however, experiences a force of attraction in addition to a torque. Why? 5.1 139 Reprint 2025-26 Physics (c) Must every magnetic configuration have a north pole and a south pole? What about the field due to a toroid? (d) Two identical looking iron bars A and B are given, one of which is definitely known to be magnetised. (We do not know which one.) How would one ascertain whether or not both are magnetised? If only one is magnetised, how does one ascertain which one? [Use nothing else but the bars A and B.] Solution (a) In either case, one gets two magnets, each with a north and south pole. (b) No force if the field is uniform. The iron nail experiences a non- uniform field due to the bar magnet. There is induced magnetic moment in the nail, therefore, it experiences both force and torque. The net force is attractive because the induced south pole (say) in the nail is closer to the north pole of magnet than induced north pole. (c) Not necessarily. True only if the source of the field has a net non-zero magnetic moment. This is not so for a toroid or even for a straight infinite conductor. (d) Try to bring different ends of the bars closer. A repulsive force in some situation establishes that both are magnetised. If it is always attractive, then one of them is not magnetised. In a bar magnet the intensity of the magnetic field is the strongest at the two ends (poles) and weakest at the central region. This fact may be used to determine whether A or B is the magnet. In this case, to see which one of the two bars is a magnet, pick up one, (say, A) and lower one of its ends; first on one of the ends of the 5.1 other (say, B), and then on the middle of B. If you notice that in the middle of B, A experiences no force, then B is magnetised. If you do not notice any change from the end to the middle of B, EXAMPLE then A is magnetised. 5.2.4 The electrostatic analog Comparison of Eqs. (5.1), (5.2) and (5.3) with the corresponding equations for electric dipole (Chapter 1), suggests that magnetic field at large distances due to a bar magnet of magnetic moment m can be obtained from the equation for electric field due to an electric dipole of dipole moment p, by making the following replacements: 1 µ0 E → B , p → m , → 4 πε0 4 π In particular, we can write down the equatorial field (BE) of a bar magnet at a distance r, for r >> l, where l is the size of the magnet: µ0 m B E = − 4 π r 3 (5.4) Likewise, the axial field (BA) of a bar magnet for r >> l is: µ0 2 m B A =140 4 π r 3 (5.5) Reprint 2025-26 Magnetism and Matter Equation (5.5) is just Eq. (5.1) in the vector form. Table 5.1 summarises the analogy between electric and magnetic dipoles. TABLE 5.1 THE DIPOLE ANALOGY Electrostatics Magnetism 1/ε0 µ0 Dipole moment p m Equatorial Field for a short dipole –p/4πε0r 3 – µ0 m / 4π r 3 Axial Field for a short dipole 2p/4πε0r 3 µ0 2m / 4π r 3 External Field: torque p × E m × B External Field: Energy –p.E –m.B Example 5.2 Figure 5.4 shows a small magnetised needle P placed at a point O. The arrow shows the direction of its magnetic moment. The other arrows show different positions (and orientations of the magnetic moment) of another identical magnetised needle Q. (a) In which configuration the system is not in equilibrium? (b) In which configuration is the system in (i) stable, and (ii) unstable equilibrium? (c) Which configuration corresponds to the lowest potential energy among all the configurations shown? FIGURE 5.4 Solution Potential energy of the configuration arises due to the potential energy of one dipole (say, Q) in the magnetic field due to other (P). Use the result that the field due to P is given by the expression [Eqs. (5.4) and (5.5)]: µ0 m P B P = − 3 (on the normal bisector) 4π r µ0 2 m P B P = 3 (on the axis) 4π r where mP is the magnetic moment of the dipole P. EXAMPLE Equilibrium is stable when mQ is parallel to BP, and unstable when it is anti-parallel to BP. 5.2 141 Reprint 2025-26 Physics For instance for the configuration Q3 for which Q is along the perpendicular bisector of the dipole P, the magnetic moment of Q is parallel to the magnetic field at the position 3. Hence Q3 is stable. 5.2 Thus, (a) PQ1 and PQ2 (b) (i) PQ3, PQ6 (stable); (ii) PQ5, PQ4 (unstable) EXAMPLE (c) PQ6 5.3 MAGNETISM AND GAUSS’S LAW In Chapter 1, we studied Gauss’s law for electrostatics. In Fig 5.2(c), we see that for a closed surface represented by i , the number of lines leaving the surface is equal to the number of lines entering it. This is consistent with the fact that no net charge is enclosed by the surface. However, in the same figure, for the closed surface ii , there is a net outward flux, since it does include a net (positive) charge.1855) The situation is radically different for magnetic fields – which are continuous and form closed loops. Examine the Gaussian surfaces represented by i or ii in Fig 5.2(a) or Karl Friedrich Gauss Fig. 5.2(b). Both cases visually demonstrate that the(1777 (1777 – 1855) He was a number of magnetic field lines leaving the surface is child prodigy and was gifted balanced by the number of lines entering it. The net in mathematics, physics, magnetic flux is zero for both the surfaces. This is true engineering, astronomy for any closed surface.GAUSS and even land surveying. The properties of numbers fascinated him, and in his work he anticipated major mathematical development of later times. Along with Wilhelm Welser, he built theFRIEDRICH first electric telegraph in 1833. His mathematical theory of curved surface laid the foundation for theKARL later work of Riemann. FIGURE 5.5 Consider a small vector area element ∆S of a closed surface S as in Fig. 5.5. The magnetic flux through ÄS is defined as ∆φB = B.∆S, where B is the field at ∆S. We divide S into many small area elements and calculate the individual flux through each. Then, the net flux φB is, B.∆S = 0 (5.6) φB = ∑ ∆ φB = ∑ ’ all ’ ’ all ’ where ‘all’ stands for ‘all area elements ∆S′. Compare this with the Gauss’s law of electrostatics. The flux through a closed surface in that case is given by E.∆S =142 ∑ q ε0 Reprint 2025-26 Magnetism and Matter where q is the electric charge enclosed by the surface. The difference between the Gauss’s law of magnetism and that for electrostatics is a reflection of the fact that isolated magnetic poles (also called monopoles) are not known to exist. There are no sources or sinks of B; the simplest magnetic element is a dipole or a current loop. All magnetic phenomena can be explained in terms of an arrangement of dipoles and/or current loops. Thus, Gauss’s law for magnetism is: The net magnetic flux through any closed surface is zero. Example 5.3 Many of the diagrams given in Fig. 5.6 show magnetic field lines (thick lines in the figure) wrongly. Point out what is wrong with them. Some of them may describe electrostatic field lines correctly. Point out which ones. EXAMPLE FIGURE 5.6 5.3 143 Reprint 2025-26 Physics Solution (a) Wrong. Magnetic field lines can never emanate from a point, as shown in figure. Over any closed surface, the net flux of B must always be zero, i.e., pictorially as many field lines should seem to enter the surface as the number of lines leaving it. The field lines shown, in fact, represent electric field of a long positively charged wire. The correct magnetic field lines are circling the straight conductor, as described in Chapter 4. (b) Wrong. Magnetic field lines (like electric field lines) can never cross each other, because otherwise the direction of field at the point of intersection is ambiguous. There is further error in the figure. Magnetostatic field lines can never form closed loops around empty space. A closed loop of static magnetic field line must enclose a region across which a current is passing. By contrast, electrostatic field lines can never form closed loops, neither in empty space, nor when the loop encloses charges. (c) Right. Magnetic lines are completely confined within a toroid. Nothing wrong here in field lines forming closed loops, since each loop encloses a region across which a current passes. Note, for clarity of figure, only a few field lines within the toroid have been shown. Actually, the entire region enclosed by the windings contains magnetic field. (d) Wrong. Field lines due to a solenoid at its ends and outside cannot be so completely straight and confined; such a thing violates Ampere’s law. The lines should curve out at both ends, and meet eventually to form closed loops. (e) Right. These are field lines outside and inside a bar magnet. Note carefully the direction of field lines inside. Not all field lines emanate out of a north pole (or converge into a south pole). Around both the N-pole, and the S-pole, the net flux of the field is zero. (f ) Wrong. These field lines cannot possibly represent a magnetic field. Look at the upper region. All the field lines seem to emanate out of the shaded plate. The net flux through a surface surrounding the shaded plate is not zero. This is impossible for a magnetic field. The given field lines, in fact, show the electrostatic field lines around a positively charged upper plate and a negatively charged 5.3 carefullylower plate.grasped.The difference between Fig. [5.6(e) and (f)] should be (g) Wrong. Magnetic field lines between two pole pieces cannot be precisely straight at the ends. Some fringing of lines is inevitable. Otherwise, Ampere’s law is violated. This is also true for electric EXAMPLE field lines. Example 5.4 (a) Magnetic field lines show the direction (at every point) along which a small magnetised needle aligns (at the point). Do the magnetic field lines also represent the lines of force on a moving charged particle at every point? (b) If magnetic monopoles existed, how would the Gauss’s law of 5.4 magnetism be modified? (c) Does a bar magnet exert a torque on itself due to its own field? Does one element of a current-carrying wire exert a force on another EXAMPLE element of the same wire? Reprint 2025-26 Magnetism and Matter (d) Magnetic field arises due to charges in motion. Can a system have magnetic moments even though its net charge is zero? Solution (a) No. The magnetic force is always normal to B (remember magnetic force = qv × B). It is misleading to call magnetic field lines as lines of force. (b) Gauss’s law of magnetism states that the flux of B through any closed surface is always zero ∫s B .∆s = 0 . If monopoles existed, the right hand side would be equal to the monopole (magnetic charge) qm enclosed by S. [Analogous to Gauss’s law of electrostatics, B.∆s = µ0qm where qm is the ∫S (monopole) magnetic charge enclosed by S.] (c) No. There is no force or torque on an element due to the field produced by that element itself. But there is a force (or torque) on an element of the same wire. (For the special case of a straight wire, this force is zero.) (d) Yes. The average of the charge in the system may be zero. Yet, the mean of the magnetic moments due to various current loops may not be zero. We will come across such examples in connection EXAMPLE with paramagnetic material where atoms have net dipole moment 5.4 through their net charge is zero.