Chapter 11: Magnetic Materials
In the middle of the nineteenth century, Michael Faraday was working in his London laboratory with one of the most powerful electromagnets available at the time. Everyone already knew that iron was attracted to a magnet. Faraday wanted to know something different: does a magnetic field do anything at all to ordinary, non-magnetic substances?
He hung a small piece of heavy glass by a thread between the poles of his electromagnet and switched on the current. The glass did not stay still. It slowly turned and settled itself across the field, sitting at right angles to the line joining the poles, as though it were being gently pushed out of the strongest part of the field rather than pulled into it.
Faraday repeated the test with bismuth, with wood, with water — and found the same quiet pushing-away. He gave such substances a new name: diamagnetic. Materials that behaved the opposite way, drawing themselves into the field, he called paramagnetic.
The importance of that experiment was not the tiny motion of a glass rod. It was the conclusion Faraday drew from it: every substance responds to a magnetic field in some way. Magnetism is not a rare property belonging only to lodestone and iron. It is a property of matter itself — and its explanation lies inside the atoms of the material.
Diagram
Fig. 11.0 – A small rod suspended by a thread between the poles of a large electromagnet, shown setting itself at right angles to the field.
As you work through this chapter, you will find the answer to each of these questions.
11.1 Introduction
In your earlier classes you have already met the idea of a magnetic dipole. A short bar magnet is the simplest example of one: it has a north pole and a south pole separated by a small distance, and these two poles always occur together. You have also studied the magnetic dipole moment \(\vec{m}\) of such a magnet — a vector quantity, directed from the south pole to the north pole inside the magnet, whose magnitude tells us how strong the magnet is. Its SI unit is ampere metre squared (\(\text{A m}^2\)).
You have also learnt how to find the magnetic field produced by a short bar magnet at any point in its vicinity, both along its axis and along its equator.
The Magnet That Points North
One familiar property of such a magnet is worth recalling carefully. When a small bar magnet is suspended freely, so that it is able to turn without any hindrance, it does not settle in a random direction. It comes to rest along the geographic north–south direction, as shown in Fig. 11.1.
Diagram
Fig. 11.1 – A short bar magnet suspended freely by an inextensible string, resting along the north–south direction.
This happens because the Earth itself behaves as a huge magnet, and the suspended magnet aligns itself with the Earth’s magnetic field. The fact that the direction it chooses is fixed and repeatable is what makes this property so useful.
What This Chapter Covers
Those simple observations are the starting point of this chapter. We shall first study the behaviour of a short bar magnet placed in a uniform magnetic field — the torque it experiences, the potential energy it stores in different orientations, and the oscillations it performs when it is disturbed from its resting position.
We shall then ask a deeper question: what makes a material magnetic in the first place? To answer it, we will go inside the atom and look at the magnetic moment of the revolving and spinning electron.
That understanding will let us classify materials by how they respond to an external magnetic field, into diamagnetic, paramagnetic and ferromagnetic substances, and to study the properties and examples of each. Finally, we shall look at how this knowledge is used in practice — in permanent magnets, in electromagnets, and in magnetic shielding.
11.2 Torque Acting on a Magnetic Dipole in a Uniform Magnetic Field
In the previous chapter you learnt that a rectangular current-carrying coil placed in a uniform magnetic field experiences a torque. Torque is the turning effect of a force; it is what makes a body rotate, just as a force makes a body move in a straight line. The torque on such a coil is given by
\[\vec{\tau} = \vec{m} \times \vec{B}\]
\[\tau = mB \sin\theta \qquad \text{--- (11.1)}\]
Here \(\vec{m}\) is the magnetic dipole moment of the coil, measured in \(\text{A m}^2\); \(\vec{B}\) is the external applied uniform magnetic field, measured in tesla (T); and \(\theta\) is the angle between \(\vec{m}\) and \(\vec{B}\). The torque \(\tau\) is measured in newton metre (N m). The arrangement is shown in Fig. 11.2.
Diagram
Fig. 11.2 – A short bar magnet placed at an angle θ in a uniform magnetic field, with equal and opposite forces on its two poles forming a couple.
Why a Bar Magnet Behaves the Same Way
Exactly the same behaviour is observed when a small bar magnet, instead of a coil, is placed in a uniform magnetic field. Look carefully at Fig. 11.2. The magnetic field pushes the north pole of the magnet one way and the south pole the opposite way, so the two forces are equal in magnitude but opposite in direction.
These two forces do not act along the same line — their lines of action are different. A pair of equal, opposite, non-collinear forces of this kind is called a couple. As you have studied earlier, a couple produces no translational motion at all; it produces pure rotation. This is why a bar magnet in a uniform field turns but does not drift away.
So the bar magnet, being a magnetic dipole just like the current loop, obeys the same Eq. (11.1).
Magnetic Potential Energy of the Dipole
Because of this torque the bar magnet turns. Whenever a displacement takes place — linear or angular — work is being done. That work does not disappear: it is stored in the new position in the form of potential energy, exactly as you studied in Chapter 8. When an electric dipole is kept in an electric field, the energy stored in this way is called the electrostatic potential energy. For a magnetic dipole in a magnetic field, it is the magnetic potential energy.
To rotate the magnet through a small angle \(d\theta\) against the torque, the work done is \(\tau \, d\theta\). Adding up all such small contributions gives the magnetic potential energy:
\[U_m = \int_0^\theta \tau \, d\theta \qquad \text{--- (11.2)}\]
\[U_m = \int_0^\theta mB \sin\theta \, d\theta\]
\[U_m = -\, mB \cos\theta \qquad \text{--- (11.3)}\]
Here \(U_m\) is in joule (J). Note that the zero of potential energy has been chosen at \(\theta = 90^\circ\), since \(\cos 90^\circ = 0\). Potential energy is always measured with respect to some chosen reference position, and this is the convenient choice for a dipole.
Reading the Three Special Positions
Let us now consider various positions of the magnet and find the potential energy in each.
Case 1 — When \(\theta = 0^\circ\), \(\cos 0^\circ = 1\), so \(U_m = -mB\). This is the position in which \(\vec{m}\) and \(\vec{B}\) are parallel. The bar magnet possesses minimum potential energy here and is in its most stable state.
Case 2 — When \(\theta = 180^\circ\), \(\cos 180^\circ = -1\), so \(U_m = mB\). This is the position in which \(\vec{m}\) and \(\vec{B}\) are antiparallel. The bar magnet possesses maximum potential energy and is therefore in its most unstable state.
Case 3 — When \(\theta = 90^\circ\), \(\cos 90^\circ = 0\), so \(U_m = 0\). This is the position in which the bar magnet is aligned perpendicular to the direction of the magnetic field.
The physical meaning is easy to see. A system left to itself always tends towards the state of least potential energy, which is why the magnet swings round until it lines up with the field. The antiparallel position is like a pencil balanced on its tip: it is an equilibrium position, but the slightest disturbance sends the magnet turning right round.
The potential energy as a function of \(\theta\) is shown in Fig. 11.3.
Diagram
Fig. 11.3 – Graph of magnetic potential energy \(U_m\) against angular position θ, a cosine curve with minimum \(-mB\) at 0°, zero at 90° and maximum \(+mB\) at 180°.
Oscillations of a Suspended Magnet
Now return to the activity described earlier, in which the bar magnet is suspended by an inextensible string. When we disturb it, we observe that the magnet rotates through an angle and then becomes stationary rather than spinning on for ever. This happens because of a restoring torque, of the kind you studied in Chapter 10, set up in the string in a direction opposite to the deflecting torque. In equilibrium the two torques balance each other.
For any rotating body, torque is related to angular acceleration in the same way that force is related to linear acceleration:
\[\tau = I \frac{d^2\theta}{dt^2} \qquad \text{--- (11.4)}\]
where \(I\) is the moment of inertia of the bar magnet, measured in \(\text{kg m}^2\), and \(\dfrac{d^2\theta}{dt^2}\) is its angular acceleration in \(\text{rad s}^{-2}\). The moment of inertia plays the part that mass plays in linear motion — it measures how hard it is to change the rotational state of the body.
As seen from Fig. 11.2, the direction of the torque acting on the magnet is clockwise. Now if one tries to rotate the magnet anticlockwise through an angle \(d\theta\), then a restoring torque acts in the opposite direction. Thus we write the restoring torque as
\[\tau = -\, mB \sin\theta \qquad \text{--- (11.5)}\]
The minus sign is the whole point of the equation: it says that the torque always acts so as to bring the magnet back towards its equilibrium position.
From the two equations we get
\[I \frac{d^2\theta}{dt^2} = -\, mB \sin\theta\]
When the angular displacement \(\theta\) is very small, \(\sin\theta \approx \theta\) (with \(\theta\) in radian), and the above equation can be written as
\[I \frac{d^2\theta}{dt^2} = -\, mB\,\theta\]
\[\frac{d^2\theta}{dt^2} = -\left(\frac{mB}{I}\right)\theta \qquad \text{--- (11.6)}\]
This small-angle step is what makes the motion simple. Without it, the equation would contain \(\sin\theta\) and the motion would not be simple harmonic at all.
The above equation has angular acceleration on the left-hand side and angular displacement \(\theta\) on the right-hand side, with \(m\), \(B\) and \(I\) being held constant. This is the angular simple harmonic motion (S.H.M.) that you studied in Chapter 5, analogous to linear S.H.M. governed by the equation
\[\frac{d^2x}{dt^2} = -\,\omega^2 x\]
Comparing the two equations term by term,
\[\omega^2 = \frac{mB}{I}\]
\[\therefore\ \omega = \sqrt{\frac{mB}{I}} \qquad \text{--- (11.7)}\]
where \(\omega\) is the angular frequency in \(\text{rad s}^{-1}\). The time period of angular oscillations of the bar magnet will be
\[T = \frac{2\pi}{\omega} = 2\pi \sqrt{\frac{I}{mB}} \qquad \text{--- (11.8)}\]
Equation (11.8) is worth reading physically. A stronger magnet or a stronger field means a bigger restoring torque, so the magnet returns faster and \(T\) is smaller. A heavier, bulkier magnet has a larger moment of inertia, resists being turned, and so oscillates more slowly.
Location of Magnetic Poles of a Current-Carrying Loop
You have studied that a current-carrying conductor produces a magnetic field, and that if you bend the conductor into the form of a loop, this loop behaves like a bar magnet, as discussed in Chapter 10. This is the reason a current loop and a bar magnet obey the very same torque equation — both are magnetic dipoles.
But which face of the loop is the north pole and which is the south? The answer depends only on the sense in which the current goes round, as illustrated in Fig. 11.3a.
Diagram
Fig. 11.3a – A circular current loop viewed from both sides, showing the face with clockwise current as the south pole and the face with anticlockwise current as the north pole.
11.3 Origin of Magnetism in Materials
In order to understand magnetism in materials, we have to use the basic concepts we already have — magnetic poles and magnetic dipole moment. In Std. XI you studied the magnetic properties of a short bar magnet, such as the magnetic field it produces along its axial direction and along its equatorial direction.
But that leaves the central question untouched. A bar of iron and a bar of copper look much the same and are both metals, yet one can be made into a magnet and the other cannot. What makes some materials behave like a magnet while others do not?
Looking Inside the Material
To answer this, one must consider the building blocks of any material, that is, its atoms. In an atom, negatively charged electrons are revolving about the nucleus, which consists of protons and neutrons. The details of atomic structure will be studied in Chapter 15.
This picture immediately gives us a link to what we already know. An electron revolving in a closed orbit is a charge in motion, and a charge in motion is a current. A current going round a closed path is a current loop — and we saw in the previous section that a current loop behaves exactly like a tiny bar magnet. So every atom carries within it the possibility of being a small magnet, without any battery or external source of current.
Diagram
Fig. 11.3b – A block of material shown magnified to reveal individual atoms, each represented as a tiny current loop with its own magnetic moment.
Why the Outermost Electrons Matter Most
You have studied in the periodic table in Chemistry in Std. XI that the chemical properties of an element are dominated by the electrons orbiting in the outermost orbit of the atom. The inner orbits are completely filled, and their electrons are, in a sense, locked into fixed arrangements that do not respond easily to the outside world.
The same is true for magnetic properties, as described in the sections that follow. It is the outermost electrons that decide whether an atom has a net magnetic moment of its own, and therefore whether the material built out of those atoms can behave magnetically.
11.3.1 Magnetic Moment of an Electron Revolving Around the Nucleus of an Atom
Let us consider, as a simple model, an electron revolving around a positively charged nucleus in a circular orbit. A negatively charged electron revolving in an orbit is equivalent to a tiny current loop.
The magnetic dipole moment due to a current loop is given by \(m_{orb} = IA\), where \(A\) is the area enclosed by the loop (in \(\text{m}^2\)) and \(I\) is the current (in ampere). This is only a part of the magnetic moment of an atom, and is referred to as the orbital magnetic moment.
An electron also possesses an intrinsic angular momentum of its own, called the spin angular momentum. A further contribution, the spin magnetic dipole moment, results from this intrinsic spin. We shall take up the orbital part first, and return to spin later in this section.
Finding the Orbital Magnetic Moment
Consider an electron moving with constant speed \(v\) in a circular orbit of radius \(r\) about the nucleus, as shown in Fig. 11.4.
Diagram
Fig. 11.4 – A single electron revolving in a circular orbit of radius r around the nucleus, with the direction of the equivalent current and the orbital magnetic moment marked.
If the electron travels a distance \(2\pi r\), the circumference of the circle, in time \(T\), then its orbital speed is \(v = 2\pi r/T\).
Current means charge crossing a point per unit time. The whole charge \(e\) of the electron passes any chosen point on the orbit once in every revolution, that is, once every \(T\) seconds. Thus the current \(I\) associated with this orbiting electron of charge \(e\) is
\[I = \frac{e}{T}\]
Using \(T = \dfrac{2\pi}{\omega}\), where \(\omega = \dfrac{v}{r}\) is the angular speed in \(\text{rad s}^{-1}\),
\[I = \frac{e\omega}{2\pi} = \frac{ev}{2\pi r} \qquad \text{--- (11.9)}\]
The orbital magnetic moment associated with this orbital current loop is obtained by multiplying the current by the area \(\pi r^2\) enclosed by the orbit:
\[m_{orb} = IA = \frac{ev}{2\pi r} \times \pi r^2 = \frac{1}{2} evr \qquad \text{--- (11.10)}\]
Its SI unit is \(\text{A m}^2\).
Linking the Moment to Angular Momentum
For this electron, the orbital angular momentum is
\[L = m_e v r\]
where \(m_e\) is the mass of the electron in kg, so that \(L\) is in \(\text{kg m}^2\text{ s}^{-1}\) (or \(\text{N m s}\)). Be careful not to confuse the symbol \(m_e\) for the electron’s mass with the symbol \(m\) used for magnetic moment.
Notice that Eq. (11.10) already contains the product \(vr\). If we multiply and divide it by \(m_e\), the angular momentum appears on its own. Hence the orbital magnetic moment, Eq. (11.10), can be written as
\[m_{orb} = \left(\frac{e}{2m_e}\right) (m_e v r) = \left(\frac{e}{2m_e}\right) L \qquad \text{--- (11.11)}\]
This equation shows that the orbital magnetic moment is proportional to the angular momentum. But as the electron bears a negative charge, the vectors \(\vec{m}_{orb}\) and \(\vec{L}\) are in opposite directions, both being perpendicular to the plane of the orbit. Hence, vectorially,
\[\vec{m}_{orb} = -\frac{e}{2m_e} \vec{L}\]
The same result is obtained using quantum mechanics, which is an important check — the simple circular-orbit model, crude though it is, gives the correct relation.
The ratio \(\dfrac{e}{2m_e}\) is called the gyromagnetic ratio.
The Bohr Magneton
For an electron revolving in a circular orbit, the angular momentum cannot take just any value. It is an integral multiple \(n\) of \(h/2\pi\), where \(h\) is Planck’s constant. This follows from the second postulate of Bohr’s theory of the hydrogen atom.
\[L = m_e v r = \frac{nh}{2\pi}\]
Substituting the value of \(L\) in Eq. (11.11) we get
\[m_{orb} = \frac{enh}{4\pi m_e}\]
For the first orbit, \(n = 1\), giving
\[m_{orb} = \frac{eh}{4\pi m_e}\]
The quantity \(\dfrac{eh}{4\pi m_e}\) is called the Bohr magneton. Its value is \(9.274 \times 10^{-24}\ \text{A m}^2\). The magnetic moment of an atom is stated in terms of Bohr magnetons (B.M.).
The Bohr magneton is therefore a natural “unit of atomic magnetism”. Saying that an ion has a moment of about 5 B.M. is far more informative to a physicist than quoting a long decimal number, in the same way that atomic sizes are quoted in ångström rather than in metre.
| Ion | Configuration | Effective magnetic moment in terms of Bohr magneton (B.M.) (experimental values) |
|---|---|---|
| Fe³⁺ | 3d⁵ | 5.9 |
| Fe²⁺ | 3d⁶ | 5.4 |
| Co²⁺ | 3d⁷ | 4.8 |
| N²⁺ | 3d⁸ | 3.2 |
(Courtesy: Introduction to Solid State Physics by Charles Kittel, pg. 306.) These magnetic moments are calculated from the experimental value of magnetic susceptibility. In several ions the magnetic moment is due to both orbital and spin angular momenta.
The Contribution of Electron Spin
As stated earlier, apart from the orbital motion, electron spin also contributes to the resultant magnetic moment of an atom. The vector sum of these two moments is the total magnetic moment of the atom.
Diagram
Fig. 11.4a – Two possible orientations of the spin angular momentum \(\vec{s}\) of an electron in an external field \(\vec{B}\), with the spin magnetic moment \(\vec{m}_s\) drawn opposite to \(\vec{s}\) in each case.
Why Paired Electrons Cancel
You have studied Pauli’s exclusion principle. According to it, no two electrons can have the same set of quantum numbers, namely \(n\), \(l\), \(m_l\) and \(m_s\), defining a state. Therefore, for atoms in which the electrons occur in pairs sharing the same \(n\), \(l\) and \(m_l\), the two electrons of the pair must differ in \(m_s\) — that is, their spins must be opposite. Their magnetic moments then point in opposite directions and cancel, so the resultant magnetic dipole moment for such a pair will be zero.
The atoms with an odd number of electrons in their outermost orbit will possess a nonzero resultant magnetic moment, since one electron is left without a partner to cancel its contribution. Inner orbits are completely filled and hence do not contribute to the total magnetic moment of the atom.
This single idea — paired electrons cancel, unpaired electrons do not — is the key that opens up the whole of the next few sections. Whether a material turns out to be diamagnetic, paramagnetic or ferromagnetic depends first of all on whether its atoms have unpaired electrons.
Example — Chlorine (Cl), which has 17 electrons in all. 1. Electronic configuration: \(1s^2\ 2s^2\ 2p^6\ 3s^2\ 3p^5\) 2. Ignore the inner completely filled orbitals and consider only the valence electrons, filling the 3s and 3p boxes. 3. There is one unpaired electron.
In a similar manner, study Fe, Zn, He, B and Ni, and draw your conclusions.
11.4 Magnetization and Magnetic Intensity
In the earlier section you have seen that atoms with unpaired electrons have a net magnetic dipole moment. A bulk material is made up of a very large number of such atoms, each having an inherent magnetic moment of its own.
In most materials these magnetic dipole moments are randomly oriented, pointing every which way, and hence the net dipole moment of the bulk material is zero. For some materials, such as \(\text{Fe}_3\text{O}_4\), the vector addition of all these magnetic dipole moments may not be zero. Such materials have a net magnetic moment.
Magnetization
A large lump of magnetised iron obviously has a greater total magnetic moment than a small piece cut from it, though the two are made of exactly the same material. To describe the material itself, and not merely the size of the sample, we use the magnetic moment per unit volume. The ratio of magnetic moment to the volume of the material is called magnetization.
\[M = \frac{m_{net}}{\text{volume}} \qquad \text{--- (11.12)}\]
\(M\) is a vector quantity having dimensions \([\text{L}^{-1}\text{A}]\) and SI units \(\text{A m}^{-1}\).
A Rod Inside a Solenoid
To study what happens when such a material is exposed to a magnetic field, consider a rod of the material, with some net magnetization, placed inside a solenoid having \(n\) turns per unit length and carrying a current \(I\), as shown in Fig. 11.4b.
Diagram
Fig. 11.4b – A material rod placed inside a current-carrying solenoid of n turns per unit length, showing the field of the solenoid and the additional field due to the magnetised rod.
The magnetic field inside the solenoid, produced by the current alone, is given by
\[B_0 = \mu_0 n I\]
where \(\mu_0\) is the permeability of free space. Let us denote the magnetic field due to the material kept inside the solenoid by \(B_m\). The rod is itself magnetised, so it contributes a field of its own, over and above the field of the coil. Thus the net magnetic field inside the rod can be expressed as
\[B = B_0 + B_m \qquad \text{--- (11.13)}\]
It has been observed that \(B_m\) is proportional to the magnetisation \(M\) of the material:
\[B_m = \mu_0 M\]
\[\therefore\ B = \mu_0 n I + \mu_0 M \qquad \text{--- (11.14)}\]
Magnetic Intensity
Here we introduce one more quantity, called the magnetic field intensity \(H\), where
\[H = nI\]
The noticeable difference between the expressions for \(B\) and \(H\) is that \(H\) does not depend at all on the material rod which is placed inside the solenoid. It is decided purely by the current we pass and the way the coil is wound — quantities under the experimenter’s control. This is exactly why the separation is useful: \(H\) describes the cause we apply from outside, while \(M\) describes the response of the material and \(B\) is the total result inside it.
Substituting \(H = nI\) in Eq. (11.14),
\[B = \mu_0 H + \mu_0 M\]
\[B = \mu_0 (H + M) \qquad \text{--- (11.15)}\]
\[\therefore\ H = \frac{B}{\mu_0} - M \qquad \text{--- (11.16)}\]
From the above expression we conclude that \(H\) and \(M\) have the same unit, that is, ampere per metre, and also have the same dimensions. Thus the magnetic field induced in the material, \(B\), depends on both \(H\) and \(M\).
Magnetic Susceptibility
Further, it is observed that if \(H\) is not too strong, the magnetization \(M\) induced in the material is proportional to the magnetic intensity:
\[M = \chi H \qquad \text{--- (11.17)}\]
where \(\chi\) is called the magnetic susceptibility. It is a measure of the magnetic behaviour of the material in an external applied magnetic field. Since \(\chi\) is the ratio of two quantities with the same units (\(\text{A m}^{-1}\)), it is a dimensionless constant.
Permeability
From Eq. (11.15) and Eq. (11.17) we get
\[B = \mu_0 (H + \chi H)\]
\[B = \mu_0 H (1 + \chi)\]
\[B = \mu_0 \mu_r H \qquad \text{--- (11.18)}\]
where
\[\mu_r = 1 + \chi\]
Writing \(\mu = \mu_0 \mu_r\), this becomes simply \(B = \mu H\), with
\[\mu = \mu_0 (1 + \chi) \qquad \text{--- (11.19)}\]
Here \(\mu\) is the magnetic permeability of the material, analogous to the permittivity \(\varepsilon\) in electrostatics, and \(\mu_r\) is the relative magnetic permeability of the substance, which is a pure number with no unit.
Notice how compactly these relations tie together. Measure \(\chi\) for a substance and you at once know \(\mu_r\), and hence how much field will be set up inside it for a given magnetising field.
Magnetic susceptibility \(\chi\) is the indicator of the response of a given material to the external applied magnetic field. In other words, it indicates how much magnetization will be produced in a given substance when it is kept in an external magnetic field. Again it is analogous to electrical susceptibility. When the substance is kept in a magnetic field, the atomic dipole moments either align with the external magnetic field or oppose it. If the atomic dipole moments oppose the field, \(\chi\) is observed to be negative; if they align themselves in the direction of the field, \(\chi\) is observed to be positive. The number of atomic dipole moments getting aligned in the direction of the applied magnetic field is proportional to \(\chi\). It is large for soft iron (\(\chi > 1000\)).
Susceptibility Values of Some Common Substances
The table below lists measured values of \(\chi\) for a few substances. Two things are worth noticing before we go further: some values are negative and some positive, and all of them are extremely small numbers. The sign turns out to be the key to classifying materials, which is the subject of the next section.
Table 11.1: Magnetic susceptibility of some materials
| Diamagnetic substance | \(\chi\) | Paramagnetic substance | \(\chi\) |
|---|---|---|---|
| Silicon | \(-4.2 \times 10^{-6}\) | Aluminium | \(2.3 \times 10^{-5}\) |
| Bismuth | \(-1.66 \times 10^{-5}\) | Calcium | \(1.9 \times 10^{-5}\) |
| Copper | \(-9.8 \times 10^{-6}\) | Chromium | \(2.7 \times 10^{-4}\) |
| Diamond | \(-2.2 \times 10^{-5}\) | Lithium | \(2.1 \times 10^{-5}\) |
| Gold | \(-3.6 \times 10^{-5}\) | Magnesium | \(1.2 \times 10^{-5}\) |
| Lead | \(-1.7 \times 10^{-5}\) | Niobium | \(2.6 \times 10^{-5}\) |
| Mercury | \(-2.9 \times 10^{-5}\) | Oxygen (STP) | \(2.1 \times 10^{-6}\) |
| Nitrogen | \(-5.0 \times 10^{-9}\) | Platinum | \(2.9 \times 10^{-4}\) |
11.5 Magnetic Properties of Materials
The behaviour of a material in the presence of an external magnetic field classifies materials broadly into three groups: diamagnetic, paramagnetic and ferromagnetic materials.
The quantity that separates them is the magnetic susceptibility \(\chi\), which we defined in the previous section as the magnetization produced per unit magnetic intensity applied. Magnetic susceptibility for a diamagnetic material is negative, and for a paramagnetic material it is positive but small. For ferromagnetic materials it is positive and large.
The sign and the size of \(\chi\) each carry a separate piece of information, and it is worth reading them separately. The sign tells us the direction of the material’s response: a negative \(\chi\) means the magnetization set up inside the material opposes the applied field, while a positive \(\chi\) means it lies along the applied field. The magnitude tells us how strong that response is: a value of the order of \(10^{-5}\) is a barely detectable effect, whereas a value of the order of \(10^{3}\) means the material multiplies the field enormously.
This is why a piece of iron sticks firmly to a magnet, a bar of aluminium is drawn towards it far too feebly for us to notice, and a piece of bismuth is actually pushed away. All three are responding to the field — but by amounts and in directions that differ hugely. Fig. 11.4c compares the three behaviours side by side.
Diagram
Fig. 11.4c – Three identical rods (diamagnetic, paramagnetic, ferromagnetic) suspended near the pole of a magnet, showing weak repulsion, weak attraction and strong attraction respectively.
Table 11.2: Range of magnetic susceptibility of diamagnetic, paramagnetic and ferromagnetic materials
| Type of material | \(\chi\) |
|---|---|
| Diamagnetic | \(-10^{-3}\) to \(-10^{-5}\) |
| Paramagnetic | \(0\) to \(10^{-3}\) |
| Ferromagnetic | \(10^{2}\) to \(10^{3}\) |
The three classes are taken up one by one in the sections that follow, and in each case the explanation goes back to the same starting point established earlier: whether the atoms of the substance have paired or unpaired electrons, and what happens to those atomic moments when a field is switched on.
11.5.1 Diamagnetism
In an earlier section it was discussed that atoms or molecules with completely filled electron orbits possess no net magnetic dipole moment, because every electron is paired with another whose magnetic effect exactly cancels its own. Such materials behave as diamagnetic materials.
When these materials are placed in an external magnetic field, they move from the stronger part of the magnetic field to the weaker part. Unlike ordinary magnets, which attract magnetic material, these materials are repelled by a magnet.
Why an Atom With No Moment Still Responds
The simplest explanation for diamagnetism can be given using a pair of electron orbits. As discussed earlier, the net magnetic moment of such atoms is zero, because the two electrons of the pair go round in opposite senses — one clockwise, the other anticlockwise — and their equal moments cancel.
Now the material is brought into an external magnetic field. As the field is switched on, the magnetic flux through each of these tiny orbits changes, and a changing flux induces a current, as per Lenz’s law, which you will study in detail in Chapter 12. Lenz’s law states that the induced effect always opposes the change that produced it. The direction of the induced current \(I_i\) is therefore such that the magnetic field created by it is opposite to the direction of the applied external magnetic field. The two cases are shown in Fig. 11.5.
Diagram
Fig. 11.5 – (a) Electron orbit in which the induced current opposes the loop current, and (b) electron orbit in which the induced current adds to the loop current, when the pair is brought into a magnetic field.
Out of the pair of orbits, consider first the loop in which the direction of the induced current is the same as that of the loop current, shown in Fig. 11.5 (b). Here the total current \(I\) increases, and so the magnetic dipole moment of that loop increases.
For the other current loop, in which the direction of the induced current is opposite to the direction of the loop current, shown in Fig. 11.5 (a), the net current is reduced, effectively reducing its magnetic dipole moment.
The perfect cancellation is therefore spoiled. One moment has grown, the other has shrunk, and they no longer add up to zero. This results in a net magnetic dipole moment in the direction opposite to that of the applied external magnetic field — which is precisely what a negative susceptibility means.
Examples of Diamagnetic Materials
Examples of diamagnetic materials are copper, gold metal, bismuth and many metals, lead, silicon, glass, water, wood, plastics and so on.
An important point follows from the explanation above: the diamagnetic property is present in all materials, since every material contains orbiting electrons whose orbits are modified when a field is applied. In some materials it is simply weaker than other properties such as paramagnetism or ferromagnetism, which mask the diamagnetism completely.
Behaviour in Uniform and Non-Uniform Fields
When a diamagnetic material is placed in an external magnetic field, the induced magnetic field inside the material repels the magnetic lines of force, resulting in a reduction of the magnetic field inside the material. This is shown in Fig. 11.6, where the lines of force are seen to be pushed apart and made sparser inside the specimen.
Diagram
Fig. 11.6 – A diamagnetic substance placed in a uniform magnetic field, with field lines bending away so that fewer lines pass through the specimen.
Similarly, when a diamagnetic material is placed in a non-uniform magnetic field, it moves from the stronger to the weaker part of the field. If a diamagnetic liquid is filled in a U-tube and one arm of the U-tube is placed in an external magnetic field, the liquid is pushed down in that arm and rises in the arm which is outside the field. In general, these materials try to move to a place of weaker magnetic field.
If a rod of diamagnetic material is suspended freely in a magnetic field, it aligns itself in the direction perpendicular to the direction of the external magnetic field. This is because there is a torque acting on the induced dipole.
To see why it settles perpendicular, recall the potential energy result from Section 11.2. In a diamagnetic material the induced dipole points opposite to the field. As studied earlier, when the dipole is in a direction opposite to the direction of the magnetic field it has a large magnetic potential energy, and so it is unstable. It will try to go into a position where the magnetic potential energy is least, and for this induced dipole that position is when the rod lies perpendicular to the direction of the magnetic field.
The magnetic susceptibility of a diamagnetic material is negative.
11.5.2 Paramagnetism
In an earlier section you have studied that atoms or molecules having unpaired electrons possess a net magnetic dipole moment. Materials made of such atoms are called paramagnetic materials.
At first sight one would expect every such material to behave as a magnet, since each of its atoms is already a tiny magnet. That is not what is observed. As these dipole moments are randomly oriented, the resultant magnetic dipole moment of the material is, however, zero.
The Role of Thermal Agitation
The reason for the random orientation is heat. At any temperature above absolute zero the atoms of a material are in constant thermal motion, and this incessant jostling keeps knocking the atomic magnets out of any alignment they might have. This is what is meant by thermal agitation.
The three situations are compared in Fig. 11.7.
Diagram
Fig. 11.7 – Paramagnetic substance: (a) no external magnetic field, arrows random; (b) weak external magnetic field, arrows partly aligned; (c) strong external magnetic field, arrows largely aligned.
As seen from Fig. 11.7 (a) and (b), each atom or molecule of a paramagnetic material possesses a net magnetic dipole moment, but because of thermal agitation these are randomly oriented. Due to this, the net magnetic dipole moment of the material is zero in the absence of an external magnetic field.
When such materials are placed in a sufficiently strong external magnetic field at low temperature — the low temperature being needed to reduce thermal excitation — as in Fig. 11.7 (c), most of the magnetic dipoles align themselves in the direction of the applied field. They do so because that is the orientation corresponding to less potential energy, as we found in Section 11.2, where the parallel position gave \(U_m = -mB\), the minimum possible value.
There is thus a competition going on inside the material: the applied field tries to line the atomic magnets up, and thermal agitation tries to scatter them. Which side wins decides how much magnetization appears.
The field lines get closer together inside such a material, as shown in Fig. 11.8.
Diagram
Fig. 11.8 – A paramagnetic substance in an external magnetic field, with field lines drawn in towards the specimen and crowded inside it.
Behaviour in a Non-Uniform Field
When a paramagnetic material is placed in a non-uniform magnetic field, it tends to move itself from the weaker region towards the stronger region. In other words, these materials are attracted by an external magnetic field — the exact opposite of the diamagnetic behaviour described in the previous section.
This can be shown with a liquid. When a paramagnetic liquid is placed in a U-tube manometer with a magnet kept in close vicinity of one of the arms, it is observed that the liquid rises into the arm close to the magnet.
When the external magnetic field is removed, these magnetic dipoles arrange themselves again in random directions. Thus the net magnetic dipole moment is reduced to zero in the absence of a magnetic field. A paramagnetic material therefore cannot retain any magnetism of its own.
Metals such as magnesium, lithium, molybdenum and tantalum, salts such as \(\text{MnSO}_4 \cdot \text{H}_2\text{O}\), and oxygen gas are examples of paramagnetic materials.
Curie’s Law
\[M = C\,\frac{B}{T} \qquad \text{--- (11.20)}\]
Here \(C\) is a constant, known as the Curie constant, and \(T\) is the absolute temperature in kelvin.
To express the same law in terms of the magnetic intensity, use \(B = \mu_0 H\):
\[M = C\,\frac{\mu_0 H}{T}\]
\[\frac{M}{H} = \chi = C\,\frac{\mu_0}{T}\]
\[\chi = \mu_r - 1 = C\,\frac{\mu_0}{T} \qquad \text{--- (11.21)}\]
\[\chi \propto \frac{1}{T}\]
Thus when we increase the applied magnetic field and reduce the temperature, a greater number of magnetic moments align themselves in the direction of the magnetic field, resulting in an increase in magnetization.
This is the competition described earlier, written as an equation. A stronger \(B\) strengthens the aligning influence; a higher \(T\) strengthens the disordering influence. Note also that \(T\) must always be in kelvin — using degrees Celsius in a Curie’s law numerical is one of the commonest mistakes in examinations.
11.5.3 Ferromagnetism
Atoms or molecules of a ferromagnetic material possess magnetic moments similar to those in paramagnetic materials — that is, they too have unpaired electrons. However, such substances exhibit strong magnetic properties and can become permanent magnets, which no paramagnetic substance can.
Metals and rare earths such as iron, cobalt, nickel, and some transition metal alloys exhibit ferromagnetism.
Something extra must therefore be present in these materials, over and above the mere existence of atomic moments. That something is the way neighbouring atoms influence one another.
Domain Theory
In ferromagnetic materials there is a strong interaction, called exchange coupling or exchange interaction, between neighbouring magnetic dipole moments.
Due to this interaction, small regions are formed in which all the atoms have their magnetic moments aligned in the same direction. Such a region is called a domain, and the common direction of magnetic moment within it is called the domain axis.
A domain is small on the scale of everyday objects but enormous on the scale of atoms. Domain size can be a fraction of a millimetre, of the order of \(10^{-6}\) to \(10^{-4}\ \text{m}\), and a single domain contains about \(10^{10}\) to \(10^{17}\) atoms. The boundary between adjacent domains having different orientations of magnetic moment is called a domain wall.
Magnetised and Unmagnetised States
In the unmagnetized state, the domain axes of different domains are oriented randomly, resulting in the net magnetic moment of the whole material being zero, even though the magnetic moments of the individual domains are nonzero. This is shown in Fig. 11.9 (a).
This is worth pausing over. An ordinary iron nail is not “non-magnetic” internally; it is intensely magnetised in each tiny region, but the regions point in all directions and cancel out.
When an external magnetic field is applied, domains try to align themselves along the direction of the applied magnetic field. The number of domains aligning in that direction increases as the magnetic field is increased. This process is referred to as flipping or domain rotation.
When a sufficiently high magnetic field is applied, all the domains coalesce together to form a giant domain, as shown in Fig. 11.9 (b). At this stage the material is as strongly magnetised as it can possibly be.
Diagram
Fig. 11.9 – (a) Unmagnetised ferromagnetic material with randomly oriented domains, and (b) magnetised material in which the domains have merged into one giant domain aligned with the applied field B.
In a non-uniform magnetic field, a ferromagnetic material tends to move from the weaker part to the stronger part of the field — the same direction as a paramagnetic material, but far more forcefully.
Retaining Magnetization
When the strong external magnetic field is completely removed, it does not set the domain boundaries back to their original positions. The net magnetic moment is still nonzero, and the ferromagnetic material is said to retain magnetization. Such materials are used in preparing permanent magnets.
This is the property that makes ferromagnetism technologically important: the material remembers the field it was exposed to. We shall study this memory effect in detail in the next section but one, when we take up hysteresis.
11.5.4 Effect of Temperature
Ferromagnetism depends upon temperature. The strong magnetism of iron is not a fixed, unchangeable property of the metal — it survives only as long as the ordering inside the material survives.
As the temperature of a ferromagnetic material is increased, the domain structure starts distorting, because the exchange coupling between neighbouring moments weakens. The thermal energy given to the atoms works against the exchange interaction that was holding all the moments of a domain parallel to each other.
At a certain temperature, depending upon the material, the domain structure collapses totally and the material behaves like a paramagnetic material. The atomic moments are still there, but they are no longer locked together in domains — they are free to be jostled into random directions, exactly as in a paramagnetic substance.
The temperature at which a ferromagnetic material transforms into a paramagnetic substance is called the Curie temperature (\(T_C\)) of that material.
Susceptibility Above the Curie Temperature
Once the material has acquired paramagnetic properties, its susceptibility again falls with rising temperature, but the relation is not quite the simple Curie’s law of Section 11.5.2. The relation between the magnetic susceptibility of such a material and the temperature \(T\) is given by
\[\chi = \frac{C}{T - T_C} \qquad \text{for } T > T_C \qquad \text{--- (11.22)}\]
where \(C\) is a constant. This behaviour is shown in Fig. 11.10.
Diagram
Fig. 11.10 – Graph of magnetic susceptibility against temperature for a ferromagnetic material, rising steeply as the temperature falls towards the Curie temperature \(T_C\).
Table 11.3: Curie temperature of some materials
| Material | \(T_C\) (K) |
|---|---|
| Metallic cobalt | 1394 |
| Metallic iron | 1043 |
| \(\text{Fe}_2\text{O}_3\) | 893 |
| Metallic nickel | 631 |
| Metallic gadolinium | 317 |
Read the last entry carefully. Gadolinium has a Curie temperature of 317 K, which is only about 44 °C. A piece of gadolinium is ferromagnetic in a cold room but loses its ferromagnetism if warmed slightly above body temperature.
11.6 Hysteresis
The behaviour of a ferromagnetic material when placed in an external magnetic field is quite interesting and informative. It is non-linear, and it provides information about the magnetic history of the sample — that is, the material’s response depends not only on the field applied now, but also on what fields it has been through before.
To understand this, let us consider an unmagnetized ferromagnetic material in the form of a rod placed inside a solenoid. On passing a current through the solenoid, a magnetic field is generated which magnetises the rod.
Knowing the value of \(\chi\) of the material of the rod, the magnetization \(M\) can be easily calculated from \(M = \chi H\). From Eq. (11.16),
\[H = \frac{B}{\mu_0} - M\]
Knowing the value of \(H\ (= nI)\) and \(M\), one can calculate the corresponding magnetic field \(B\) from the above equation. In an experiment, then, we control \(H\) through the current in the solenoid and observe how \(B\) inside the rod responds.
Figure 11.11 shows the behaviour of the material as we take it through one complete cycle of magnetisation.
Diagram
Fig. 11.11 – Hysteresis cycle (loop): B plotted against H, with the initial dotted magnetisation curve from O to a, and the closed loop through points a, b, c, d, e, f back to a.
Going Round the Loop
At point O in the graph, the material is in a non-magnetised state. As the strength of the external magnetic intensity \(H\) is increased, \(B\) also increases, as shown by the dotted line. But the increase is non-linear.
Near point a, the magnetic field is at its maximum value, which is the saturation magnetization condition of the rod. This represents the complete alignment and merger of the domains, as we saw in the previous section. If \(H\) is increased further, by increasing the current flowing through the solenoid, there is no increase in \(B\) — there are simply no more domains left to align.
This process is not reversible. At this stage, if the current in the solenoid is reduced, the earlier path of the graph is not retraced, because the earlier domain structure is not recovered. The domain walls, once moved, do not slide back neatly to where they started.
When \(H = 0\), that is, when the current through the solenoid is made zero — point b in the figure — we do not get \(B = 0\). The value of \(B\) when \(H = 0\) is called retentivity or remanence. This means some domain alignment is still retained even when \(H = 0\).
Next, when the current in the solenoid is increased in the reverse direction, point c in the graph is reached, where \(B = 0\) at a certain value of \(H\). This value of \(H\) is called coercivity. At this point the domain axes are randomly oriented with respect to each other, so their contributions once again cancel out.
If the current is further increased in the reverse direction, \(B\) increases in the reverse sense and again reaches a saturation state, at point d. Here, if \(H\) is increased further, \(B\) does not increase.
From this point d onwards, when \(H\) is reduced, \(B\) also reduces along the path de. At point e, again \(H = 0\) but \(B\) is not zero. It means the domain structure is present but the direction of magnetisation is reversed.
A further increase in the current gives the curve efa. On reaching point a, one loop is complete.
This loop is called the hysteresis loop, and the process of taking a magnetic material through the loop once is called a hysteresis cycle.
What the Loop Costs Us
Notice that the material never returns to its starting condition without work being done on it. Energy has to be supplied by the source in order to push the domain walls back and forth, and that energy does not come back out — it appears as heat in the specimen.
11.7 Permanent Magnet and Electromagnet
The hysteresis loop of the previous section is not merely a curiosity. Its shape decides what a given ferromagnetic material is good for, and the two extremes of that shape give us the two most useful magnetic devices — the electromagnet and the permanent magnet.
The Electromagnet
Soft iron, having a large permeability (\(\mu_r > 1000\)) and only a small amount of retained magnetization, is used to make electromagnets.
For this purpose, a soft iron rod, or a rod of some other soft ferromagnetic material, is placed as the core of a solenoid. On passing a current through the solenoid, the magnetic field associated with the solenoid increases a thousandfold, because the core itself becomes strongly magnetised and adds its own field to that of the current.
When the current through the solenoid is switched off, the associated magnetic field effectively becomes zero, since a soft material retains almost nothing. This is exactly the behaviour we want in a device that must be switched on and off at will.
These electromagnets are used in electric bells, loudspeakers, circuit breakers, and also in research laboratories. Giant electromagnets are used in cranes to lift heavy loads made of iron. Superconducting magnets are used to prepare very high magnetic fields of the order of a few tesla, and such magnets are used in NMR (Nuclear Magnetic Resonance) spectroscopy.
The Permanent Magnet
Permanent magnets are prepared by using a hard ferromagnetic rod instead of the soft one used in the earlier case.
When the current is switched on, the magnetic field of the solenoid magnetises the rod. As a hard ferromagnetic material has the property of retaining its magnetization to a large extent, the material remains magnetised even after switching off the current through the solenoid.
A good permanent magnet therefore needs both high retentivity, so that it keeps a strong field of its own, and high coercivity, so that stray fields and rough handling cannot easily demagnetise it. Modern permanent magnets made of alloys of neodymium, iron and boron are so strong that a small disc a few millimetres across is enough to drive the speaker in an earphone.
Diagram
Fig. 11.11a – Hysteresis loops of B against H drawn side by side for a hard magnetic material (wide loop) and a soft magnetic material (narrow loop).
11.8 Magnetic Shielding
When a soft ferromagnetic material is kept in a uniform magnetic field, a large number of magnetic lines crowd up inside the material, leaving only a few outside it. This follows directly from the high permeability of such materials: the lines of force always prefer the easiest path available, and a soft ferromagnetic material offers a far easier path than air.
Now suppose we have a closed structure made of this material — say a spherical shell of iron — kept in a magnetic field. Very few lines of force then pass through the enclosed space, because most of the lines are crowded into the iron shell itself, as shown in Fig. 11.12. This effect is known as magnetic shielding.
Diagram
Fig. 11.12 – An iron ring placed between the N and S poles of a magnet, with the field lines crowding into the ring and the enclosed shielded area left almost free of lines.
It is worth being precise about what is happening here. The magnetic field is not destroyed, absorbed or blocked by the iron; there is no known material that simply stops a magnetic field the way an opaque sheet stops light. The field is merely guided around the protected region, taking the easier route through the iron.
Where Shielding Is Used
In practice, the instrument which needs to be protected from a magnetic field is completely surrounded by a soft ferromagnetic substance. This technique is being used in spacecraft, where sensitive electronics must work reliably in the presence of external fields.
Some scientific experiments also require the apparatus to be protected from magnetic fields in the laboratory. There, the high magnetic fields of nearby magnets need to be shielded by providing a case made up of soft ferromagnetic material.
11.9 Summary
Magnetic Dipole in a Uniform Field
- A magnetic dipole of moment \(\vec{m}\) placed in a uniform magnetic field \(\vec{B}\) experiences a torque \(\vec{\tau} = \vec{m} \times \vec{B}\), of magnitude \(\tau = mB\sin\theta\), where \(\theta\) is the angle between \(\vec{m}\) and \(\vec{B}\).
- The forces on the two poles are equal, opposite and non-collinear; they form a couple and produce pure rotation, with no translation.
- The torque is maximum (\(\tau = mB\)) at \(\theta = 90^\circ\) and zero at \(\theta = 0^\circ\) and \(\theta = 180^\circ\).
- The magnetic potential energy of the dipole is \(U_m = \int_0^\theta \tau\, d\theta = -mB\cos\theta\), with \(U_m\) taken as zero at \(\theta = 90^\circ\).
- At \(\theta = 0^\circ\), \(U_m = -mB\) (minimum energy, most stable); at \(\theta = 180^\circ\), \(U_m = +mB\) (maximum energy, most unstable); at \(\theta = 90^\circ\), \(U_m = 0\).
- For a suspended magnet disturbed through a small angle, \(\tau = I \dfrac{d^2\theta}{dt^2}\) and the restoring torque is \(\tau = -mB\sin\theta\). Using \(\sin\theta \approx \theta\) for small \(\theta\), \[\frac{d^2\theta}{dt^2} = -\left(\frac{mB}{I}\right)\theta\]
- This is angular S.H.M., with \(\omega = \sqrt{\dfrac{mB}{I}}\) and time period \[T = \frac{2\pi}{\omega} = 2\pi\sqrt{\frac{I}{mB}}\] where \(I\) is the moment of inertia of the magnet.
- A current-carrying loop behaves as a magnetic dipole. The face with anticlockwise current acts as the north pole and the face with clockwise current as the south pole.
Origin of Magnetism in Materials
- Magnetism in materials originates in the atom: revolving electrons behave as tiny current loops, and electrons also possess an intrinsic spin magnetic moment. The total magnetic moment of an atom is the vector sum of the orbital and spin contributions.
- For an electron of charge \(e\) revolving with speed \(v\) in a circular orbit of radius \(r\), the equivalent current is \[I = \frac{e}{T} = \frac{e\omega}{2\pi} = \frac{ev}{2\pi r}\]
- The orbital magnetic moment is \[m_{orb} = IA = \frac{ev}{2\pi r} \times \pi r^2 = \frac{1}{2}evr\]
- With orbital angular momentum \(L = m_e v r\), \[m_{orb} = \left(\frac{e}{2m_e}\right)L, \qquad \vec{m}_{orb} = -\frac{e}{2m_e}\vec{L}\] the minus sign arising because the electron is negatively charged.
- The ratio \(\dfrac{e}{2m_e}\) is called the gyromagnetic ratio; its value is \(8.8 \times 10^{10}\ \text{C kg}^{-1}\).
- From Bohr’s second postulate, \(L = \dfrac{nh}{2\pi}\), giving \(m_{orb} = \dfrac{enh}{4\pi m_e}\). For \(n = 1\), \(m_{orb} = \dfrac{eh}{4\pi m_e}\), called the Bohr magneton, of value \(9.274 \times 10^{-24}\ \text{A m}^2\).
- By Pauli’s exclusion principle, electrons paired in the same state (same \(n\), \(l\), \(m_l\)) have opposite spins, so their moments cancel. Atoms with unpaired electrons in the outermost orbit possess a net magnetic moment. Completely filled inner orbits do not contribute.
Magnetization, Intensity, Susceptibility and Permeability
- Magnetization is the net magnetic moment per unit volume: \(M = \dfrac{m_{net}}{\text{volume}}\), SI unit \(\text{A m}^{-1}\), dimensions \([\text{L}^{-1}\text{A}]\).
- For a rod inside a solenoid of \(n\) turns per unit length carrying current \(I\): \(B_0 = \mu_0 nI\), \(B_m = \mu_0 M\), and the net field is \(B = B_0 + B_m\).
- Magnetic intensity \(H = nI\); it does not depend on the material placed inside, SI unit \(\text{A m}^{-1}\). \[B = \mu_0 (H + M), \qquad H = \frac{B}{\mu_0} - M\]
- For fields that are not too strong, \(M = \chi H\), where \(\chi\) is the magnetic susceptibility, a dimensionless constant. \[B = \mu_0 H(1 + \chi) = \mu_0 \mu_r H = \mu H\] \[\mu_r = 1 + \chi, \qquad \mu = \mu_0 (1 + \chi)\]
- \(\mu\) is the permeability of the material and \(\mu_r\) its relative permeability (a pure number). For a superconductor \(\chi = -1\), giving \(\mu = 0\), so no lines of force pass through it (Meissner effect).
Classification of Magnetic Materials
- Diamagnetic: atoms have no permanent moment (paired electrons). A field induces, by Lenz’s law, a moment opposite to itself. Such materials are repelled by a magnet, move from stronger to weaker parts of a field, and a freely suspended rod sets itself perpendicular to the field. \(\chi\) is small and negative (\(-10^{-3}\) to \(-10^{-5}\)), \(\mu_r < 1\). Examples: copper, gold, bismuth, lead, silicon, glass, water, wood, plastics. Diamagnetism is present in all materials but is often masked.
- Paramagnetic: atoms have permanent moments, randomly oriented due to thermal agitation, so the net moment is zero without a field. In a field they align partly along it; they are attracted towards stronger parts of the field. \(\chi\) is small and positive (\(0\) to \(10^{-3}\)), \(\mu_r > 1\). Examples: magnesium, lithium, molybdenum, tantalum, \(\text{MnSO}_4 \cdot \text{H}_2\text{O}\), oxygen gas.
- Ferromagnetic: strong exchange interaction aligns neighbouring moments into domains (\(10^{-6}\) to \(10^{-4}\ \text{m}\) in size, containing \(10^{10}\) to \(10^{17}\) atoms), separated by domain walls. In an applied field the domains flip and merge into a giant domain. \(\chi\) is large and positive (\(10^2\) to \(10^3\)). Examples: iron, cobalt, nickel, some rare earths and transition metal alloys.
Effect of Temperature
- Curie’s law (paramagnetic materials): \(M = C\dfrac{B}{T}\), and with \(B = \mu_0 H\), \[\chi = \mu_r - 1 = \frac{C\mu_0}{T}, \qquad \chi \propto \frac{1}{T}\] Increasing the field and lowering the temperature both increase the magnetization. \(T\) must be in kelvin.
- Above the Curie temperature \(T_C\) the domain structure of a ferromagnetic material collapses and it behaves as a paramagnetic material, with \[\chi = \frac{C}{T - T_C} \quad \text{for } T > T_C\]
- Curie temperatures: metallic cobalt 1394 K, metallic iron 1043 K, \(\text{Fe}_2\text{O}_3\) 893 K, metallic nickel 631 K, metallic gadolinium 317 K.
Hysteresis and Its Applications
- The \(B\)–\(H\) curve of a ferromagnetic material is non-linear and irreversible, and records the magnetic history of the sample.
- Saturation: the state in which all domains are aligned and merged, beyond which increasing \(H\) produces no increase in \(B\).
- Retentivity (remanence): the value of \(B\) remaining when \(H\) is reduced to zero after saturation.
- Coercivity: the reverse value of \(H\) needed to reduce \(B\) to zero.
- One complete traversal of the closed \(B\)–\(H\) curve is a hysteresis cycle; the area inside the loop corresponds to the energy dissipated per unit volume per cycle, as in stress hysteresis in solids.
- Electromagnet: a soft ferromagnetic core (\(\mu_r > 1000\), low retentivity, narrow loop) inside a solenoid. The field rises a thousandfold with current on, and falls effectively to zero with current off. Used in electric bells, loudspeakers, circuit breakers, laboratory magnets and cranes; superconducting magnets give fields of a few tesla, used in NMR spectroscopy.
- Permanent magnet: a hard ferromagnetic rod, with high retentivity and high coercivity (wide loop), magnetised by a solenoid and remaining magnetised after the current is switched off.
- Magnetic shielding: a closed shell of soft ferromagnetic material draws the lines of force into itself, leaving the enclosed region almost free of field. The field is diverted, not destroyed. Used to protect instruments in spacecraft and in laboratories.
11.10 Objective Questions
The intensity of the magnetic field of the earth at a point inside a hollow iron box is (A) less than that outside (B) more than that outside (C) same as that outside (D) zero
Soft iron is used to make the core of a transformer because of its (A) low coercivity and low retentivity (B) low coercivity and high retentivity (C) high coercivity and high retentivity (D) high coercivity and low retentivity
Which of the following statements is correct for diamagnetic materials? (A) \(\mu_r < 1\) (B) \(\chi\) is negative and low (C) \(\chi\) does not depend on temperature (D) All of the above
A rectangular magnet suspended freely has a period of oscillation equal to \(T\). Now it is broken into two equal halves (each having half of the original length) and one piece is made to oscillate freely. If its period of oscillation is \(T'\), the ratio \(T'/T\) is (A) \(\dfrac{1}{2\sqrt{2}}\) (B) \(1/2\) (C) \(2\) (D) \(1/4\)
A magnetising field of \(360\ \text{A m}^{-1}\) produces a magnetic flux density \(B = 0.6\ \text{T}\) in a ferromagnetic material. What is its permeability in \(\text{T m A}^{-1}\)? (A) \(1/300\) (B) \(300\) (C) \(1/600\) (D) \(600\)
11.11 Check Your Concepts
A bar magnet placed in a uniform magnetic field turns but does not move bodily from its place. Explain why, in terms of the forces acting on its two poles.
Why is the position in which the magnetic moment is antiparallel to the field called an equilibrium position, even though it is described as the most unstable state?
In deriving the expression \(T = 2\pi\sqrt{I/mB}\) for a suspended magnet, the approximation \(\sin\theta \approx \theta\) is used. What physical assumption does this correspond to, and what would happen to the motion if the magnet were released from a very large angle?
A revolving electron constitutes a current although no source of e.m.f. is connected to it. Justify this statement.
Why are the vectors \(\vec{m}_{orb}\) and \(\vec{L}\) of an orbiting electron directed opposite to each other?
Two solenoids are identical and carry the same current, but one is empty and the other contains an iron rod. Compare the values of \(H\) and of \(B\) inside them, and explain the difference.
Diamagnetism is said to be present in every material, yet only some materials are called diamagnetic. Explain this apparent contradiction.
A freely suspended rod of a diamagnetic material sets itself perpendicular to an external magnetic field, while a paramagnetic rod sets itself along the field. Explain both observations using the idea of magnetic potential energy.
Why does lowering the temperature increase the magnetization of a paramagnetic sample kept in a fixed magnetic field?
What is the role of the exchange interaction in ferromagnetism? Why can a paramagnetic material never be made into a permanent magnet, however strong the field applied to it?
An unmagnetised iron nail shows no magnetism outside it, yet its interior is said to be strongly magnetised. Explain, using domain theory.
Above the Curie temperature the susceptibility of a ferromagnetic material follows \(\chi = C/(T - T_C)\) rather than \(\chi = C\mu_0/T\). What does the presence of \(T_C\) in the denominator indicate physically?
Why is the \(B\)–\(H\) curve of a ferromagnetic material said to carry information about the magnetic history of the sample? What happens to the retentivity of a specimen if it has never been taken to saturation?
Assertion: The core of an electromagnet should have a narrow hysteresis loop. Reason: The area of the hysteresis loop is a measure of the energy dissipated per cycle. State whether the assertion and the reason are correct, and whether the reason correctly explains the assertion.
Distinguish between retentivity and coercivity, and state the values each should ideally have for (a) a permanent magnet and (b) a transformer core.
A thin sheet of soft iron is held between a magnet and a compass needle, but the needle is still deflected. What is wrong with this arrangement as a shield, and how should it be corrected?
Is it correct to say that a soft ferromagnetic shell “blocks” or “absorbs” the magnetic field inside it? Explain what actually happens to the lines of force.
A copper box shields a sensitive instrument well against a rapidly alternating magnetic field but poorly against a steady one. Explain why.
Which property of soft iron makes it useful for preparing an electromagnet?
What happens to a ferromagnetic material when its temperature increases above the Curie temperature?
What should be the retentivity and coercivity of a permanent magnet?
Discuss Curie’s law for a paramagnetic material.
Obtain an expression for the orbital magnetic moment of an electron rotating about the nucleus in an atom.
What does the hysteresis loop represent?
Explain one application of an electromagnet.
11.12 Practice with Numericals
A bar magnet of magnetic moment \(0.8\ \text{A m}^2\) is placed in a uniform magnetic field of \(0.4\ \text{T}\). Calculate the torque acting on it when its axis makes angles of \(0^\circ\), \(30^\circ\) and \(90^\circ\) with the field.
A magnetic dipole of moment \(1.2\ \text{A m}^2\) is placed in a uniform field of \(0.25\ \text{T}\). Calculate its magnetic potential energy when it is (a) parallel to the field, (b) perpendicular to the field, and (c) antiparallel to the field. Hence find the work required to turn it from position (a) to position (c).
A bar magnet of moment of inertia \(9 \times 10^{-6}\ \text{kg m}^2\) and magnetic moment \(1.5 \times 10^{-2}\ \text{A m}^2\) performs 20 oscillations in 40 s in a horizontal plane. Calculate the magnitude of the magnetic field acting on it.
A magnetic needle has a period of oscillation of \(2\ \text{s}\) in a field of \(3.2 \times 10^{-5}\ \text{T}\). What will be its period of oscillation if it is taken to a place where the field is four times as strong, everything else remaining unchanged?
An electron revolves in a circular orbit of radius \(1.06 \times 10^{-10}\ \text{m}\) with a speed of \(1.1 \times 10^{6}\ \text{m s}^{-1}\). Calculate (a) the equivalent current, (b) the orbital magnetic moment, and (c) the orbital angular momentum. (\(e = 1.6 \times 10^{-19}\ \text{C}\), \(m_e = 9.1 \times 10^{-31}\ \text{kg}\).
The orbital angular momentum of an electron in the third Bohr orbit is \(3h/2\pi\). Taking \(h = 6.63 \times 10^{-34}\ \text{J s}\), \(e = 1.6 \times 10^{-19}\ \text{C}\) and \(m_e = 9.1 \times 10^{-31}\ \text{kg}\), calculate its orbital magnetic moment and express the answer in Bohr magnetons.
A solenoid of length \(0.4\ \text{m}\) has 1600 turns and carries a current of \(1.5\ \text{A}\). Calculate the magnetic intensity \(H\) inside it, and the magnetic field \(B\) when (a) the core is air and (b) the core is a material of relative permeability 900. (\(\mu_0 = 4\pi \times 10^{-7}\ \text{T m A}^{-1}\).
A magnetising field of \(1.5 \times 10^{3}\ \text{A m}^{-1}\) produces a magnetization of \(6.0 \times 10^{5}\ \text{A m}^{-1}\) in a specimen. Find its magnetic susceptibility, its relative permeability and its permeability.
A rod of a paramagnetic material of susceptibility \(2.4 \times 10^{-5}\) is placed inside a toroid winding. Calculate the percentage increase in the magnetic field inside the winding due to the rod.
A specimen of volume \(1.5 \times 10^{-5}\ \text{m}^3\) has a net magnetic moment of \(2.4\ \text{A m}^2\). Calculate its magnetization. If the magnetising field applied is \(4.0 \times 10^{3}\ \text{A m}^{-1}\), find the susceptibility of the specimen.
The susceptibility of a paramagnetic salt is \(6.0 \times 10^{-4}\) at \(200\ \text{K}\). Using Curie’s law, find its susceptibility at \(500\ \text{K}\), and the temperature at which the susceptibility would become \(2.0 \times 10^{-4}\).
A ferromagnetic material has a Curie temperature of \(600\ \text{K}\). Its susceptibility at \(900\ \text{K}\) is \(5.0 \times 10^{-3}\). Find its susceptibility at \(1100\ \text{K}\).
A domain in a ferromagnetic material is a cube of side \(2\ \mu\text{m}\). If the material contains \(8.5 \times 10^{28}\) atoms per cubic metre and each atom has a dipole moment of \(9.27 \times 10^{-24}\ \text{A m}^2\), estimate the number of atoms in the domain and its maximum possible magnetization.
A ferromagnetic specimen of volume \(5 \times 10^{-5}\ \text{m}^3\) is taken through hysteresis cycles at \(50\ \text{Hz}\). If the energy dissipated per unit volume per cycle is \(180\ \text{J m}^{-3}\), calculate the heat produced in the specimen in \(2\) minutes.
A short bar magnet of magnetic moment \(0.5\ \text{A m}^2\) is placed in a uniform magnetic field of 500 gauss. Calculate (a) the maximum torque it can experience and (b) the work done in rotating it from the position of minimum potential energy to a position at \(60^\circ\) to the field.
When a plate of magnetic material of size \(10\ \text{cm} \times 0.5\ \text{cm} \times 0.2\ \text{cm}\) (length, breadth and thickness respectively) is located in a magnetising field of \(0.5 \times 10^{4}\ \text{A m}^{-1}\), a magnetic moment of \(0.5\ \text{A m}^2\) is induced in it. Find out the magnetic induction in the plate.
A rod of magnetic material of cross section \(0.25\ \text{cm}^2\) is located in a \(4000\ \text{A m}^{-1}\) magnetising field. The magnetic flux passing through the rod is \(25 \times 10^{-6}\ \text{Wb}\). Find out (a) the relative permeability, (b) the magnetic susceptibility and (c) the magnetisation of the rod.
The work done in rotating a magnet of magnetic dipole moment \(m\) through \(90^\circ\) from its magnetic meridian is \(n\) times the work done to rotate it through \(60^\circ\). Find the value of \(n\).
An electron in an atom is revolving round the nucleus in a circular orbit of radius \(5.3 \times 10^{-11}\ \text{m}\), with a speed of \(2 \times 10^{6}\ \text{m s}^{-1}\). Find the resultant orbital magnetic moment and the angular momentum of the electron. (Charge on electron \(e = 1.6 \times 10^{-19}\ \text{C}\), mass of electron \(m = 9.1 \times 10^{-31}\ \text{kg}\).
A paramagnetic gas has \(2.0 \times 10^{26}\) atoms/m³ with an atomic magnetic dipole moment of \(1.5 \times 10^{-23}\ \text{A m}^2\) each. The gas is at \(27^\circ\text{C}\). (a) Find the maximum magnetization intensity of this sample. (b) If the gas in this problem is kept in a uniform magnetic field of \(3\ \text{T}\), is it possible to achieve saturation magnetization? Why? (\(k_B = 1.38 \times 10^{-23}\ \text{J K}^{-1}\).) (Hint: Find the ratio of the thermal energy of an atom of the gas, \(\tfrac{3}{2}k_B T\), to the maximum potential energy of the atom, \(mB\), and draw your conclusion.
A magnetic needle placed in a uniform magnetic field has a magnetic moment of \(2 \times 10^{-2}\ \text{A m}^2\) and a moment of inertia of \(7.2 \times 10^{-7}\ \text{kg m}^2\). It performs 10 complete oscillations in 6 s. What is the magnitude of the magnetic field?
A short bar magnet is placed in an external magnetic field of 700 gauss. When its axis makes an angle of \(30^\circ\) with the external magnetic field, it experiences a torque of \(0.014\ \text{N m}\). Find the magnetic moment of the magnet, and the work done in moving it from its most stable to its most unstable position.
A magnetic needle is suspended freely so that it can rotate freely in the magnetic meridian. In order to keep it in the horizontal position, a weight of \(0.1\ \text{g}\) is kept on one end of the needle. If the pole strength of this needle is \(20\ \text{A m}\), find the value of the vertical component of the earth’s magnetic field. (\(g = 9.8\ \text{m s}^{-2}\).
The susceptibility of a paramagnetic material is \(\chi\) at \(27^\circ\text{C}\). At what temperature will its susceptibility be \(\chi/3\)?