# Key Concepts of Magnetic Materials

## This article discusses the following basic concepts related to magnetic materials: the magnetic dipole moment, magnetization vectors, susceptibility, and permeability.

The properties of magnetic materials directly affect the performance of devices such as transformers, motors, generators, and electromagnets. This makes the science of magnetic materials an important field of study as well as a fascinating one—we use it both to understand the properties of these materials and to design and synthesize new materials that can meet the demands of different applications.

This article, the first in a series, introduces the concepts that underlie the behavior of magnetic materials. We’ll explore the world of magnetic materials further in subsequent articles, hopefully gaining a fuller appreciation of the complex role these substances play in our transformers and inductors.

### The Magnetic Dipole Moment

Different materials can have completely different responses to the same external magnetic field. To understand why, we first need to learn how magnetic dipoles determine magnetic behavior. We’ll do so by considering something called the *magnetic dipole moment*.

The magnetic dipole moment, or simply magnetic moment, is a useful concept in electromagnetics that allows us to understand and characterize the interaction between a current loop and a uniform magnetic field. The magnetic moment of a current loop with area *A* and current *I* is given by:

$$\overrightarrow{\mu} ~=~ I\overrightarrow{A}$$

**Equation 1.**

**Equation 1.**

Note that the area is defined as a vector, which makes the magnetic moment a vector quantity as well. Both vectors have the same direction.

The direction of the magnetic moment is perpendicular to the plane of the loop. It can be found by applying the right-hand rule—If you curl the fingers of your right hand in the direction of the current flow, your thumb shows the direction of the magnetic moment vector. This is illustrated in Figure 1.

**Figure 1.** The right-hand rule determines the direction of the magnetic moment.

**Figure 1.**The right-hand rule determines the direction of the magnetic moment.

The magnetic moment of a loop depends only on the current and the area. It is independent of the shape of the loop.

#### Torque and the Magnetic Moment

Consider Figure 2, which shows a current loop placed in a uniform magnetic field.

**Figure 2.** A current loop placed inside a magnetic field.

**Figure 2.**A current loop placed inside a magnetic field.

In the figure above:

*I*is the current.- \(\overrightarrow{B}\) is the magnetic field vector.
- \( \overrightarrow{\mu}\) is the magnetic moment.
- θ is the angle between the magnetic moment and the magnetic field vector.

Because the forces acting on the opposite sides of the loop cancel each other out, the total force acting on the loop is zero. However, the loop does experience a magnetic torque. The magnitude of the torque acting on the loop is given by:

$$\tau ~=~ \mu B \sin(\theta)$$

**Equation 2.**

**Equation 2.**

We can see from Equation 2 that τ is directly related to the magnetic moment. That’s because the magnetic moment behaves like a magnet—it experiences torque when placed in an external magnetic field. The torque always tends to rotate the loop toward the stable equilibrium position.

Stable equilibrium occurs when the magnetic field is perpendicular to the plane of the loop (θ = 0 degrees). If we rotate the loop slightly away from this position, the torque acts to force the loop back toward equilibrium.

The torque is also zero at θ = 180 degrees. However, in this position, the loop is in unstable equilibrium. A slight rotation away from θ = 180 degrees causes the torque to push the loop further from this point, toward θ = 0 degrees.

#### Why is the Magnetic Moment Important?

Many devices rely on the interaction of a current loop and a magnetic field. For example, the torque produced by an electric motor is based on the interaction between the magnetic field of the motor and the current-carrying conductors. In this interaction, the potential energy changes as the conductors rotate.

The interaction between the magnetic moment and the external field is what creates potential energy in our magnetic system. The angle between these two vectors determines the amount of energy stored in the system (*U*), as we see in the following equation:

$$U~=~- \mu B \cos{\theta}$$

**Equation 3.**

**Equation 3.**

Below are the stored energy values at a few different, important configurations:

- At θ = 0 degrees, we are at stable equilibrium and the minimum amount of energy is stored (
*U*= –μ*B*). - At θ = 90 degrees, the stored energy has increased to
*U*= 0. - At θ = 180 degrees, the stored energy reaches its maximum value,
*U*= μ*B*. This is the unstable equilibrium position.

#### Understanding the Net Magnetic Moment Through the Atomic Model

To fully understand how magnetic materials produce a magnetic field, we need to study quantum mechanics. That’s beyond the scope of this article, but we can still use the magnetic moment concept and the classical model of the atom to gain some insight into how materials interact with an external magnetic field. This model states that an electron orbits an atomic nucleus while also spinning around its own axis, as illustrated in Figure 3.

**Figure 3.** The orbital and spinning motions of electrons produce magnetic moments.

**Figure 3.**The orbital and spinning motions of electrons produce magnetic moments.

The orbital motion of the electron is similar to having a tiny current loop, which. As such, it produces a magnetic moment (\(\overrightarrow{\mu_{1}}\) in the figure above). The electron’s spin likewise produces a magnetic moment (\(\overrightarrow{\mu_{2}}\)). The electron’s net magnetic moment is the vector sum of these two magnetic moments, and the net magnetic moment of the atom is the vector sum of all its electrons’ magnetic moments. Though protons in an atom also have a magnetic dipole, their net effect is usually negligible in comparison to that of electrons.

An object’s net magnetic moment is the vector sum of the magnetic moments of all atoms within it.

### The Magnetization Vector

A material’s magnetic properties depend on the magnetic moments of its constituent particles. As we learned earlier in the article, the magnetic moments act as tiny magnets. When we place a material in an external magnetic field, the atomic magnetic moments of the material experience torque due to their interaction with the applied field. This tends to align the magnetic moments in the same direction.

The magnetic state of the substance depends on how many atomic magnetic moments are there in the material and how aligned they are. If the magnetic moments from microscopic current loops point in random directions, they tend to cancel each other out, so they can’t add up to much of a net magnetic field. To characterize the magnetic state of the substance, we use the magnetization vector, which is defined as the total magnetic moment per unit volume of the substance:

$$\overrightarrow{M}~=~ \frac{\overrightarrow{\mu}_{total}}{V}$$

**Equation 4.**

**Equation 4.**

where *V* is the volume of the material.

By placing the material in an external field, we cause its magnetic moments to align, creating a larger magnetization vector. The magnetization vector also depends on whether the material is classified as paramagnetic, ferromagnetic, or diamagnetic. Paramagnetic and ferromagnetic materials are made of atoms that have permanent magnetic moments. The atomic magnetic moments of diamagnetic materials aren’t permanent.

### Finding the Total Magnetic Field: Permeability and Susceptibility

Suppose that we place a material inside a magnetic field. The total magnetic field inside the material comes from two different sources:

- The magnetic field that was applied externally (
*B*_{0}). - The magnetization of the material in response to the external field (
*B*)._{m}

The total magnetic field inside the material is the sum of these two components:

$$B~=~B_0 ~+~ B_m$$

**Equation 5.**

**Equation 5.**

*B*_{0} is produced by a current-carrying conductor; *B _{m}* is produced by the magnetic substance. It can be shown that

*B*is proportional to the magnetization vector:

_{m}$$B_{m}~=~\mu_{0}M$$

**Equation 6.**

**Equation 6.**

where μ_{0} is a constant called permeability of free space. Therefore, we have:

$$B~=~B_0 ~+~ \mu_0 M$$

**Equation 7.**

**Equation 7.**

The magnetization vector is also related to external field by the following equation:

$$M ~=~ \frac{\chi}{\mu_0}B_0$$

**Equation 8.**

**Equation 8.**

where the Greek letter χ is a proportionality factor known as magnetic susceptibility. The value of χ depends on the type of material.

Combining the last two equations, we have:

$$B ~=~ (1 + \chi)B_0$$

**Equation 9.**

**Equation 9.**

This equation has a simple interpretation: it means that the total field inside the material is equal to the externally applied field times the factor (1 + χ). This factor, known as relative permeability, is a key parameter that describes how a material responds to a magnetic field. Relative permeability is typically denoted by μ_{r}.

#### Magnetic Susceptibility of Different Materials

Figure 4 illustrates the magnetic behavior of the three different material types when they’re placed in a uniform magnetic field. The area inside the material is represented by a yellow rectangle.

**Figure 4.** Behavior of a diamagnetic (a), paramagnetic (b), and ferromagnetic (c) material when placed in a uniform magnetic field.

**Figure 4.**Behavior of a diamagnetic (a), paramagnetic (b), and ferromagnetic (c) material when placed in a uniform magnetic field.

In Figure 4(a), the magnetic field lines inside the material are further away from each other compared to those on the outside. This represents the total magnetic field inside a diamagnetic material being slightly less than the externally applied field. For diamagnetic materials, χ is a small negative value—copper has a magnetic susceptibility of –9.8 × 10^{-6} at 300 K, for example—and so the material partly expels the magnetic field from its interior.

Figure 4(b) shows the response of a paramagnetic material. Here, the magnetic field lines inside the material are closer together compared to the lines of the outside field. We can conclude from this that the total magnetic field inside the material is slightly greater than the external field. For paramagnetic materials, χ is a small positive value. For example, the magnetic susceptibility of lithium is 2.1 × 10^{-5} at 300 K.

Finally, the ferromagnetic material in Figure 4(c) warps the magnetic field lines to pass them through the material. The material actually becomes magnetized, significantly enhancing the field inside the material. For ferromagnetic materials, χ has a positive value in the range of 1,000 to 100,000. Because of their high magnetic susceptibility, these materials produce a magnetic field much greater than the one that’s externally applied.

For ferromagnetic materials, χ isn’t a constant. Therefore, *M* is not a linear function of *B*_{0}.

### Wrapping Up

Magnetic materials play a key role in a wide array of applications, from transformers and motors to data storage devices. The magnetic state of a given substance depends on how many atomic magnetic moments are in the material and how well they align in the presence of an external field. As we briefly discussed, we can use these criteria to separate magnetic materials into three different types: paramagnetic, diamagnetic, and ferromagnetic. We’ll delve more deeply into these categories in a future article.

*Featured image used courtesy of Adobe Stock; all other images used courtesy of Steve Arar*

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