# 5.5 Magnetic Properties of Materials

Because of the similarity between the pattern of the magnetic field lines generated by a current loop and those exhibited by a permanent magnet, the loop can be regarded as a magnetic dipole with north and south poles (Section 5-2.2 and Fig. 5-13). The magnetic moment m of a loop of area A has magnitude m=I A and a direction normal to the plane of the loop (in accordance with the right-hand rule). Magnetization in a material is due to atomic scale current loops associated with (1) orbital motions of the electrons and protons around and inside the nucleus and (2) electron spin. The magnetic moment due to proton motion typically is three orders of magnitude smaller than that of the electrons; therefore, the total orbital and spin magnetic moment of an atom is dominated by the sum of the magnetic moments of its electrons.

The atoms of a diamagnetic material have no permanent magnetic moments. In contrast, both paramagnetic and ferromagnetic materials have atoms with permanent magnetic dipole moments, albeit with very different organizational structures.

# 5.5.1 Electron Orbital and Spin Magnetic Moments

This section presents a semiclassical, intuitive model of the atom, which provides quantitative insight into the origin of electron magnetic moments. An electron with charge of −e moving at constant speed u in a circular orbit of radius r (Fig. 5-20(a)) completes one revolution in time T=2 π r/u. This circular motion of the electron constitutes a tiny loop with current I given by

I=−e T=−e u 2 π r.(5.68)

# Figure 5-20 An electron generates (a) an orbital magnetic moment m o as it rotates around the nucleus and (b) a spin magnetic moment m s as it spins about its own axis.

Figure 5 20 Full Alternative Text

The magnitude of the associated orbital magnetic moment m o is

m o=I A=−e u 2 π r(π r 2)=−e u r 2=−e 2 m e L e,(5.69)

where L e=m e u r is the angular momentum of the electron and m e is its mass. According to quantum physics, the orbital angular momentum is quantized; specifically, L e is always some integer multiple of ℏ=h/2 π, where h is Planck’s constant. That is, L e=0,ℏ,2 ℏ,…. Consequently, the smallest nonzero magnitude of the orbital magnetic moment of an electron is

m o=−e ℏ 2 m e.(5.70)

Despite the fact that all materials contain electrons that exhibit magnetic dipole moments, most are effectively nonmagnetic. This is because, in the absence of an external magnetic field, the atoms of most materials are oriented randomly, as a result of which they exhibit a zero or very small net magnetic moment.

In addition to the magnetic moment due to its orbital motion, an electron has an intrinsic spin magnetic moment m s due to its spinning motion about its own axis (Fig. 5-20(b)). The magnitude of m s predicted by quantum theory is

m s=−e ℏ 2 m e,(5.71)

which is equal to the minimum orbital magnetic moment m o. The electrons of an atom with an even number of electrons usually exist in pairs with the members of a pair having opposite spin directions, thereby canceling each others’ spin magnetic moments. If the number of electrons is odd, the atom has a net nonzero spin magnetic moment due to its unpaired electron.

# 5.5.2 Magnetic Permeability

In Chapter 4, we learned that the relationship D=ε 0 E, between the electric flux and field in free space, is modified to D=ε 0 E+P in a dielectric material. Likewise, the relationship B=μ 0 H in free space is modified to

B=μ 0 H+μ 0 M=μ 0(H+M),(5.72)

where the magnetization vector M is defined as the vector sum of the magnetic dipole moments of the atoms contained in a unit volume of the material. Scale factors aside, the roles and interpretations of B, H, and M in Eq. (5.72) mirror those of D, E, and P in Eq. (4.93). Moreover, just as in most dielectrics P and E are linearly related, in most magnetic materials

M=χ m H,(5.73)

where χ m is a dimensionless quantity called the magnetic susceptibility of the material. For diamagnetic and paramagnetic materials, χ m is a (temperature-dependent) constant, resulting in a linear relationship between M and H at a given temperature. This is not the case for ferromagnetic substances; the relationship between M and H not only is nonlinear, but also depends on the “history” of the material, as explained in the next section.

Keeping this fact in mind, we can combine Eqs. (5.72) and (5.73) to get

B=μ 0(H+χ m H)=μ 0(1+χ m)H,(5.74)

or

B=μ H,(5.75)

where μ, the magnetic permeability of the material, relates to χ m as

Often it is convenient to define the magnetic properties of a material in terms of the relative permeability μ r:

A material usually is classified as diamagnetic, paramagnetic, or ferromagnetic on the basis of the value of its χ m (Table 5-2). Diamagnetic materials have negative susceptibilities, whereas paramagnetic materials have positive ones. However, the absolute magnitude of χ m is on the order of 10−5 for both classes of materials, which for most applications allows us to ignore χ m relative to 1 in Eq. (5.77).

# Table 5-2 Properties of magnetic materials.

Table 5 2 Full Alternative Text

Ferromagnetic materials are discussed next.

# 5.5.3 Magnetic Hysteresis of Ferromagnetic Materials

Ferromagnetic materials, which include iron, nickel, and cobalt, exhibit unique magnetic properties due to the fact that their magnetic moments tend to readily align along the direction of an external magnetic field. Moreover, such materials remain partially magnetized even after the external field is removed. Because of these peculiar properties, ferromagnetic materials are used in the fabrication of permanent magnets.

A key to understanding the properties of ferromagnetic materials is the notion of magnetized domains, that are microscopic regions (on the order of 10−10 m 3) within which the magnetic moments of all atoms (typically on the order of 10 19 atoms) are permanently aligned with each other. This alignment, which occurs in all ferromagnetic materials, is due to strong coupling forces between the magnetic dipole moments constituting an individual domain. In the absence of an external magnetic field, the domains take on random orientations relative to each other (Fig. 5-21(a)), resulting in zero net magnetization. The domain walls forming the boundaries between adjacent domains consist of thin transition regions. When an unmagnetized sample of a ferromagnetic material is placed in an external magnetic field, the domains partially align with the external field, as illustrated in Fig. 5-21(b). A quantitative understanding of how the domains form and how they behave under the influence of an external magnetic field requires a heavy dose of quantum mechanics and is outside the scope of the present treatment. Hence, we confine our discussion to a qualitative description of the magnetization process and its implications.

# Figure 5-21 Comparison of (a) unmagnetized and (b) magnetized domains in a ferromagnetic material.

(a) Unmagnetized domains

(b) Magnetized domains

The magnetization behavior of a ferromagnetic material is described in terms of its B−H magnetization curve, where B and H refer to the amplitudes of the B flux and H field in the material. Suppose that we start with an unmagnetized sample of iron, denoted by point O in Fig. 5-22. When we increase H continuously, for example, by increasing the current passing through a wire wound around the sample, B increases also along the B−H curve from point O to point A 1, at which nearly all the domains have become aligned with H. Point A 1 represents a saturation condition. If we then decrease H from its value at point A 1 back to zero (by reducing the current through the wire), the magnetization curve follows the path from A 1 to A 2. At point A 2, the external field H is zero (owing to the fact that the current through the wire is zero), but the flux density B in the material is not. The magnitude of B at A 2 is called the residual flux density B r. The iron material is now magnetized and ready to be used as a permanent magnet owing to the fact that a large fraction of its magnetized domains have remained aligned. Reversing the direction of H and increasing its intensity causes B to decrease from B r at point A 2 to zero at point A 3, and if the intensity of H is increased further while maintaining its direction, the magnetization moves to the saturation condition at point A 4. Finally, as H is made to return to zero and is then increased again in the positive direction, the curve follows the path from A 4 to A 1. This process is called magnetic hysteresis. Hysteresis means “lag behind.” The existence of a hysteresis loop implies that the magnetization process in ferromagnetic materials depends not only on the magnetic field H but also on the magnetic history of the material. The shape and extent of the hysteresis loop depend on the properties of the ferromagnetic material and the peak-to-peak range over which H is made to vary. Hard ferromagnetic materials are characterized by wide hysteresis loops (Fig. 5-23(a)). They cannot be easily demagnetized by an external magnetic field because they have a large residual magnetization B r. Hard ferromagnetic materials are used in the fabrication of permanent magnets for motors and generators. Soft ferromagnetic materials have narrow hysteresis loops (Fig. 5-23(b)); hence, they can be more easily magnetized and demagnetized. To demagnetize any ferromagnetic material, the material is subjected to several hysteresis cycles while gradually decreasing the peak-to-peak range of the applied field.

# Figure 5-22 Typical hysteresis curve for a ferromagnetic material.

Figure 5 22 Full Alternative Text

# Figure 5-23 Comparison of hysteresis curves for (a) a hard ferromagnetic material and (b) a soft ferromagnetic material.

(a) Hard material

Figure 5 23-a Full Alternative Text

(b) Soft material

Figure 5 23-b Full Alternative Text

Concept Question 5-12: What are the three types of magnetic materials, and what are typical values of their relative permeabilities?

Concept Question 5-13: What causes magnetic hysteresis in ferromagnetic materials?

Concept Question 5-14: What does a magnetization curve describe? What is the difference between the magnetization curves of hard and soft ferromagnetic materials?