5.5 MAGNETIC PROPERTIES OF MATERIALS The discussion in the previous section helps us to classify materials as diamagnetic, paramagnetic or ferromagnetic. In terms of the susceptibility χ, a material is diamagnetic if χ is negative, para- if χ is positive and small, and ferro- if χ is large and positive. A glance at Table 5.2 gives one a better feeling for these materials. Here ε is a small positive number introduced to quantify paramagnetic materials. Next, we describe these materials in some detail. TABLE 5.2 Diamagnetic Paramagnetic Ferromagnetic –1 ≤ χ < 0 0 < χ < ε χ >> 1 0 ≤ µr < 1 1< µr < 1+ ε µr >> 1 µ < µ0 µ > µ0 µ >> µ0 5.5.1 Diamagnetism Diamagnetic substances are those which have tendency to move from FIGURE 5.7 stronger to the weaker part of the external magnetic field. In other words, Behaviour of unlike the way a magnet attracts metals like iron, it would repel a magnetic field lines diamagnetic substance. near a Figure 5.7(a) shows a bar of diamagnetic material placed in an external (a) diamagnetic, magnetic field. The field lines are repelled or expelled and the field inside (b) paramagnetic the material is reduced. In most cases, this reduction is slight, being one substance. part in 105. When placed in a non-uniform magnetic field, the bar will tend to move from high to low field. 147 Reprint 2025-26 Physics The simplest explanation for diamagnetism is as follows. Electrons in an atom orbiting around nucleus possess orbital angular momentum. These orbiting electrons are equivalent to current-carrying loop and thus possess orbital magnetic moment. Diamagnetic substances are the ones in which resultant magnetic moment in an atom is zero. When magnetic field is applied, those electrons having orbital magnetic moment in the same direction slow down and those in the opposite direction speed up. This happens due to induced current in accordance with Lenz’s law which you will study in Chapter 6. Thus, the substance develops a net magnetic moment in direction opposite to that of the applied field and hence repulsion. Some diamagnetic materials are bismuth, copper, lead, silicon, nitrogen (at STP), water and sodium chloride. Diamagnetism is present in all the substances. However, the effect is so weak in most cases that it gets shifted by other effects like paramagnetism, ferromagnetism, etc. The most exotic diamagnetic materials are superconductors. These are metals, cooled to very low temperatures which exhibits both perfect conductivity and perfect diamagnetism. Here the field lines are completely expelled! χ = –1 and µr = 0. A superconductor repels a magnet and (by Newton’s third law) is repelled by the magnet. The phenomenon of perfect diamagnetism in superconductors is called the Meissner effect, after the name of its discoverer. Superconducting magnets can be gainfully exploited in variety of situations, for example, for running magnetically levitated superfast trains. 5.5.2 Paramagnetism Paramagnetic substances are those which get weakly magnetised when placed in an external magnetic field. They have tendency to move from a region of weak magnetic field to strong magnetic field, i.e., they get weakly attracted to a magnet. The individual atoms (or ions or molecules) of a paramagnetic material possess a permanent magnetic dipole moment of their own. On account of the ceaseless random thermal motion of the atoms, no net magnetisation is seen. In the presence of an external field B0, which is strong enough, and at low temperatures, the individual atomic dipole moment can be made to align and point in the same direction as B0. Figure 5.7(b) shows a bar of paramagnetic material placed in an external field. The field lines gets concentrated inside the material, and the field inside is enhanced. In most cases, this enhancement is slight, being one part in 105. When placed in a non-uniform magnetic field, the bar will tend to move from weak field to strong. Some paramagnetic materials are aluminium, sodium, calcium, oxygen (at STP) and copper chloride. For a paramagnetic material both χ and µr depend not only on the material, but also (in a simple fashion) on the sample temperature. As the field is increased or the temperature is lowered, the magnetisation increases until it reaches the saturation value at which point all the dipoles are perfectly aligned with the field. Reprint 2025-26 Magnetism and Matter 5.5.3 Ferromagnetism Ferromagnetic substances are those which gets strongly magnetised when placed in an external magnetic field. They have strong tendency to move from a region of weak magnetic field to strong magnetic field, i.e., they get strongly attracted to a magnet. The individual atoms (or ions or molecules) in a ferromagnetic material possess a dipole moment as in a paramagnetic material. However, they interact with one another in such a way that they spontaneously align themselves in a common direction over a macroscopic volume called domain. The explanation of this cooperative effect requires quantum mechanics and is beyond the scope of this textbook. Each domain has a net magnetisation. Typical domain size is 1mm and the domain contains about 1011 atoms. In the first instant, the magnetisation varies randomly from domain to domain and there is no bulk magnetisation. This is shown in Fig. 5.8(a). When we apply an external magnetic field B0, the domains FIGURE 5.8 orient themselves in the direction of B0 and simultaneously the domain (a) Randomly oriented in the direction of B0 grow in size. This existence of domains and oriented domains, their motion in B0 are not speculations. One may observe this under a (b) Aligned domains. microscope after sprinkling a liquid suspension of powdered ferromagnetic substance of samples. This motion of suspension can be observed. Fig. 5.8(b) shows the situation when the domains have aligned and amalgamated to form a single ‘giant’ domain. Thus, in a ferromagnetic material the field lines are highly concentrated. In non-uniform magnetic field, the sample tends to move towards the region of high field. We may wonder as to what happens when the external field is removed. In some ferromagnetic materials the magnetisation persists. Such materials are called hard magnetic materials or hard ferromagnets. Alnico, an alloy of iron, aluminium, nickel, cobalt and copper, is one such material. The naturally occurring lodestone is another. Such materials form permanent magnets to be used among other things as a compass needle. On the other hand, there is a class of ferromagnetic materials in which the magnetisation disappears on removal of the external field. Soft iron is one such material. Appropriately enough, such materials are called soft ferromagnetic materials. There are a number of elements, which are ferromagnetic: iron, cobalt, nickel, gadolinium, etc. The relative magnetic permeability is >1000! The ferromagnetic property depends on temperature. At high enough temperature, a ferromagnet becomes a paramagnet. The domain structure disintegrates with temperature. This disappearance of magnetisation with temperature is gradual. SUMMARY 1. The science of magnetism is old. It has been known since ancient times that magnetic materials tend to point in the north-south direction; like 149 Reprint 2025-26 Physics magnetic poles repel and unlike ones attract; and cutting a bar magnet in two leads to two smaller magnets. Magnetic poles cannot be isolated. 2. When a bar magnet of dipole moment m is placed in a uniform magnetic field B, (a) the force on it is zero, (b) the torque on it is m × B, (c) its potential energy is –m.B, where we choose the zero of energy at the orientation when m is perpendicular to B. 3. Consider a bar magnet of size l and magnetic moment m, at a distance r from its mid-point, where r >>l, the magnetic field B due to this bar is, µ0 m B = 2 π r 3 (along axis) µ0 m = – 3 (along equator) 4 π r 4. Gauss’s law for magnetism states that the net magnetic flux through any closed surface is zero B B i S 0 all area elements S 5. Consider a material placed in an external magnetic field B0. The magnetic intensity is defined as, B0 H = µ0 The magnetisation M of the material is its dipole moment per unit volume. The magnetic field B in the material is, B = m0 (H + M) 6. For a linear material M = c H. So that B = m H and c is called the magnetic susceptibility of the material. The three quantities, c, the relative magnetic permeability mr, and the magnetic permeability m are related as follows: m = m0 mr mr = 1+ c 7. Magnetic materials are broadly classified as: diamagnetic, paramagnetic, and ferromagnetic. For diamagnetic materials c is negative and small and for paramagnetic materials it is positive and small. Ferromagnetic materials have large c. 8. Substances, which at room temperature, retain their ferromagnetic property for a long period of time are called permanent magnets. Reprint 2025-26 Magnetism and Matter Physical quantity Symbol Nature Dimensions Units Remarks Permeability of µ0 Scalar [MLT–2 A–2] T m A–1 µ0/4π = 10–7 free space Magnetic field, B Vector [MT–2 A–1] T (tesla) 104 G (gauss) = 1 T Magnetic induction, Magnetic flux density Magnetic moment m Vector [L–2 A] A m2 Magnetic flux φB Scalar [ML2T–2 A–1] W (weber) W = T m2 Magnetic momentMagnetisation M Vector [L–1 A] A m–1 Volume Magnetic intensity H Vector [L–1 A] A m–1 B = µ0 (H + M) Magnetic field strength Magnetic χ Scalar - - M = χH susceptibility Relative magnetic µr Scalar - - B = µ0 µr H permeability Magnetic permeability µ Scalar [MLT–2 A–2] T m A–1 µ = µ0 µr N A–2 B = µ H POINTS TO PONDER 1. A satisfactory understanding of magnetic phenomenon in terms of moving charges/currents was arrived at after 1800 AD. But technological exploitation of the directional properties of magnets predates this scientific understanding by two thousand years. Thus, scientific understanding is not a necessary condition for engineering applications. Ideally, science and engineering go hand-in-hand, one leading and assisting the other in tandem. 2. Magnetic monopoles do not exist. If you slice a magnet in half, you get two smaller magnets. On the other hand, isolated positive and negative charges exist. There exists a smallest unit of charge, for example, the electronic charge with value |e| = 1.6 ×10–19 C. All other charges are integral multiples of this smallest unit charge. In other words, charge is quantised. We do not know why magnetic monopoles do not exist or why electric charge is quantised. 3. A consequence of the fact that magnetic monopoles do not exist is that the magnetic field lines are continuous and form closed loops. In contrast, the electrostatic lines of force begin on a positive charge and terminate on the negative charge (or fade out at infinity). 4. A miniscule difference in the value of χ, the magnetic susceptibility, yields radically different behaviour: diamagnetic versus paramagnetic. For diamagnetic materials χ = –10–5 whereas χ = +10–5 for paramagnetic materials. 151 Reprint 2025-26 Physics 5. There exists a perfect diamagnet, namely, a superconductor. This is a metal at very low temperatures. In this case χ = –1, µr = 0, µ = 0. The external magnetic field is totally expelled. Interestingly, this material is also a perfect conductor. However, there exists no classical theory which ties these two properties together. A quantum-mechanical theory by Bardeen, Cooper, and Schrieffer (BCS theory) explains these effects. The BCS theory was proposed in1957 and was eventually recognised by a Nobel Prize in physics in 1970. 6. Diamagnetism is universal. It is present in all materials. But it is weak and hard to detect if the substance is para- or ferromagnetic. 7. We have classified materials as diamagnetic, paramagnetic, and ferromagnetic. However, there exist additional types of magnetic material such as ferrimagnetic, anti-ferromagnetic, spin glass, etc. with properties which are exotic and mysterious. EXERCISES 5.1 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? 5.2 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? 5.3 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? 5.4 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? 5.5 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)? 5.6 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. Reprint 2025-26 Magnetism and Matter (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?

14.2 Which of the statements given in Exercise 14.1 is true for p-type semiconductos.

5.19 [Cr(NH3)6] 3+ is paramagnetic while [Ni(CN)4] 2– is diamagnetic. Explain why? 5.20 A solution of [Ni(H2O)6] 2+ is green but a solution of [Ni(CN)4] 2– is colourless. Explain. 5.21 [Fe(CN)6] 4– and [Fe(H2O)6] 2+ are of different colours in dilute solutions. Why?