# 1.3 The Nature of Electromagnetism

Our physical universe is governed by four fundamental forces of nature:

- The nuclear force, which is the strongest of the four, but its range is limited to subatomic scales, such as nuclei.

- The electromagnetic force exists between all charged particles. It is the dominant force in microscopic systems, such as atoms and molecules, and its strength is on the order of 10−2 that of the nuclear force.

- The weak-interaction force, whose strength is only 10−14 that of the nuclear force. Its primary role is in interactions involving certain radioactive elementary particles.

- The gravitational force is the weakest of all four forces, having a strength on the order of 10−41 that of the nuclear force. However, it often is the dominant force in macroscopic systems, such as the solar system.

This book focuses on the electromagnetic force and its consequences. Even though the electromagnetic force operates at the atomic scale, its effects can be transmitted in the form of electromagnetic waves that can propagate through both free space and material media. The purpose of this section is to provide an overview of the basic framework of electromagnetism, which consists of certain fundamental laws governing the electric and magnetic fields induced by static and moving electric charges, the relations between the electric and magnetic fields, and how these fields interact with matter. As a precursor, however, we will take advantage of our familiarity with gravitational force by describing some of its properties because they provide a useful analogue to those of the electromagnetic force.

# 1.3.1 Gravitational Force: A Useful Analogue

According to Newton’s law of gravity, the gravitational force F g 21 acting on mass m 2 due to a mass m 1 at a distance R 12 from m 2 (Fig. 1-3) is given by

F g 21=−R^12 G m 1 m 2 R 12 2 (N),(1.2)

where G is the universal gravitational constant, R^12 is a unit vector that points from m 1 to m 2, and the unit for force is newton (N). The negative sign in Eq. (1.2) accounts for the fact that the gravitational force is attractive. Conversely, F g 12=−F g 21, where F g 12 is the force acting on mass m 1 due to the gravitational pull of mass m 2. Note that the first subscript of F g denotes the mass experiencing the force and the second subscript denotes the source of the force.

# Figure 1-3 Gravitational forces between two masses.

Figure 1 3 Full Alternative Text

The two objects do not have to be in direct contact for each to experience the pull by the other. This phenomenon of action at a distance has led to the concept of fields. An object of mass m 1 induces a gravitational field ψ 1 (Fig. 1-4) that does not physically emanate from the object, yet its influence exists at every point in space such that, if another object of mass m 2 were to exist at a distance R 12 from the object of mass m 1, the object of mass m 2 would experience a force acting on it equal to

F g 21=ψ 1 m 2,(1.3)

where

ψ 1=−R^G m 1 R 2 (N/kg).(1.4)

# Figure 1-4 Gravitational field ψ 1 induced by a mass m 1.

In Eq. (1.4), R^ is a unit vector that points in the radial direction away from object m 1; and therefore, −R^ points toward m 1. The force due to ψ 1 acting on a mass m 2, for example, is obtained from the combination of Eqs. (1.3) and (1.4) with R=R 12 and R^=R^12. The field concept may be generalized by defining the gravitational field ψ at any point in space such that, when a test mass m is placed at that point, the force F g acting on it is related to ψ by

ψ=F g m.(1.5)

The force F g may be due to a single mass or a collection of many masses.

# 1.3.2 Electric Fields

The electromagnetic force consists of an electrical component F e and a magnetic component F m.

We know from atomic physics that all matter contains a mixture of neutrons, positively charged protons, and negatively charged electrons with the fundamental quantity of charge being that of a single electron, usually denoted by the letter e. The unit by which electric charge is measured is the coulomb (C), named in honor of the eighteenth-century French scientist Charles Augustin de Coulomb (1736–1806). The magnitude of e is

The charge of a single electron is q e=−e and that of a proton is equal in magnitude but opposite in polarity: q p=e.

These properties constitute what today is called Coulomb’s law, which can be expressed mathematically as

where F e 21 is the electrical force acting on charge q 2 due to charge q 1 when both are in free space (vacuum), R 12 is the distance between the two charges, R^12 is a unit vector pointing from charge q 1 to charge q 2 (Fig. 1-5), and ε 0 is a universal constant called the electrical permittivity of free space [ε 0=8.854×10−12 farad per meter (F/m)]. The two charges are assumed to be isolated from all other charges. The force F e 12 acting on charge q 1 due to charge q 2 is equal to force F e 21 in magnitude, but opposite in direction: F e 12=−F e 21.

# Figure 1-5 Electric forces on two positive point charges in free space.

Figure 1 5 Full Alternative Text

The expression given by Eq. (1.7) for the electrical force is analogous to that given by Eq. (1.2) for the gravitational force, and we can extend the analogy further by defining the existence of an electric field intensity E due to any charge q as

where R is the distance between the charge and the observation point, and R^ is the radial unit vector pointing away from the charge. Figure 1-6 depicts the electric field lines due to a positive charge. For reasons that will become apparent in later chapters, the unit for E is volt per meter (V/m).

# Figure 1-6 Electric field E due to charge q.

Figure 1 6 Full Alternative Text

Electric charge exhibits two important properties.

If a volume contains n p protons and n e electrons, then its total charge is

q=n p e−n e e=(n p−n e)e(C).(1.9)

Even if some of the protons were to combine with an equal number of electrons to produce neutrons or other elementary particles, the net charge q remains unchanged. In matter, the quantum mechanical laws governing the behavior of the protons inside the atom’s nucleus and the electrons outside it do not allow them to combine.

This seemingly simple concept allows us in future chapters to compute the electric field due to complex distributions of charge without having to be concerned with the forces acting on each individual charge due to the fields by all of the other charges.

The expression given by Eq. (1.8) describes the field induced by an electric charge residing in free space. Let us now consider what happens when we place a positive point charge in a material composed of atoms. In the absence of the point charge, the material is electrically neutral with each atom having a positively charged nucleus surrounded by a cloud of electrons of equal but opposite polarity. Hence, at any point in the material not occupied by an atom, the electric field E is zero. Upon placing a point charge in the material, as shown in Fig. 1-7, the atoms experience forces that cause them to become distorted. The center of symmetry of the electron cloud is altered with respect to the nucleus with one pole of the atom becoming positively charged relative to the other pole. Such a polarized atom is called an electric dipole, and the distortion process is called polarization. The degree of polarization depends on the distance between the atom and the isolated point charge, and the orientation of the dipole is such that the axis connecting its two poles is directed toward the point charge, as illustrated schematically in Fig. 1-7. The net result of this polarization process is that the electric fields of the dipoles of the atoms (or molecules) tend to counteract the field due to the point charge. Consequently, the electric field at any point in the material is different from the field that would have been induced by the point charge in the absence of the material. To extend Eq. (1.8) from the free-space case to any medium, we replace the permittivity of free space ε 0 with ε, where ε is the permittivity of the material in which the electric field is measured and is therefore characteristic of that particular material. Thus,

# Figure 1-7 Polarization of the atoms of a dielectric material by a positive charge q.

Often, ε is expressed in the form

ε=ε r ε 0(F/m),(1.11)

where ε r is a dimensionless quantity called the relative permittivity or dielectric constant of the material. For a vacuum, ε r=1; for air near the Earth’s surface, ε r=1.0006; and the values of ε r for materials that we have occasion to use in this book are tabulated in Appendix B.

In addition to the electric field intensity E, we often find it convenient to also use a related quantity called the electric flux density D given by

with unit of coulomb per square meter (C/m 2).

# 1.3.3 Magnetic Fields

As early as 800 BC, the Greeks discovered that certain kinds of stones exhibit a force that attracts pieces of iron. These stones are now called magnetite (Fe 3 O 4), and the phenomenon they exhibit is known as magnetism. In the thirteenth century, French scientists discovered that, when a needle was placed on the surface of a spherical natural magnet, the needle oriented itself along different directions for different locations on the magnet. By mapping the directions indicated by the needle, it was determined that the magnetic force formed magnetic field lines that encircled the sphere and appeared to pass through two points diametrically opposite to each other. These points, called the north and south poles of the magnet, were found to exist for every magnet, regardless of its shape. The magnetic-field pattern of a bar magnet is displayed in Fig. 1-8. It was also observed that like poles of different magnets repel each other and unlike poles attract each other.

# Figure 1-8 Pattern of magnetic field lines around a bar magnet.

Figure 1 8 Full Alternative Text

If a permanent magnet is cut into small pieces, no matter how small each piece is, it will always have a north and a south pole.

The magnetic lines surrounding a magnet represent the magnetic flux density B. A magnetic field not only exists around permanent magnets but also can be created by electric current. This connection between electricity and magnetism was discovered in 1819 by the Danish scientist Hans Oersted (1777–1851), who observed that an electric current in a wire caused a compass needle placed in its vicinity to deflect and that the needle turned so that its direction was always perpendicular to the wire and to the radial line connecting the wire to the needle. From these observations, he deduced that the current-carrying wire induced a magnetic field that formed closed circular loops around the wire (Fig. 1-9). Shortly after Oersted’s discovery, French scientists Jean-Baptiste Biot and Felix Savart developed an expression that relates the magnetic flux density B at a point in space to the current I in the conductor. Application of their formulation, known today as the Biot–Savart law, to the situation depicted in Fig. 1-9 for a very long wire residing in free space leads to the result that the magnetic flux density B induced by a constant current I flowing in the z direction is given by

B=ϕ^μ 0 I 2 π r (T),(1.13)

where r is the radial distance from the current and ϕ^ is an azimuthal unit vector expressing the fact that the magnetic field direction is tangential to the circle surrounding the current. The magnetic field is measured in tesla (T), named in honor of Nikola Tesla (1856–1943). Tesla was an electrical engineer whose work on transformers made it possible to transport electricity over long wires without too much loss. The quantity μ 0 is called the magnetic permeability of free space [μ 0=4 π×10−7 henry per meter (H/m)], and it is analogous to the electric permittivity ε 0. In fact, as we will see in Chapter 2, the product of ε 0 and μ 0 specifies c, which is the velocity of light in free space:

# Figure 1-9 The magnetic field induced by a steady current flowing in the z direction.

Figure 1 9 Full Alternative Text

We noted in Section 1-3.2 that, when an electric charge q′ is subjected to an electric field E, it experiences an electric force F e=q′E. Similarly, if a charge q′ resides in the presence of a magnetic flux density B, it experiences a magnetic force F m, but only if the charge is in motion and its velocity u is in a direction not parallel (or anti-parallel) to B. In fact, as we learn in more detail in Chapter 5, F m points in a direction perpendicular to both B and u.

To extend Eq. (1.13) to a medium other than free space, μ 0 should be replaced with μ, which is the magnetic permeability of the material in which B is being observed. The majority of natural materials are nonmagnetic, meaning that they exhibit a magnetic permeability μ=μ 0. For ferromagnetic materials, such as iron and nickel, μ can be much larger than μ 0. The magnetic permeability μ accounts for magnetization properties of a material. In analogy with Eq. (1.11), μ of a particular material can be defined as

μ=μ r μ 0(H/m),(1.15)

where μ r is a dimensionless quantity called the relative magnetic permeability of the material. The values of μ r for commonly used ferromagnetic materials are given in Appendix B.

# 1.3.4 Static and Dynamic Fields

In EM, the time variable t, or more precisely if and how electric and magnetic quantities vary with time, is of crucial importance. Before we elaborate further on the significance of this statement, it will prove useful to define the following time-related adjectives unambiguously:

- Static—describes a quantity that does not change with time. The term dc (i.e., direct current) is often used as a synonym for static to describe not only currents, but other electromagnetic quantities as well.

- Dynamic—refers to a quantity that does vary with time, but conveys no specific information about the character of the variation.

- Waveform—refers to a plot of the magnitude profile of a quantity as a function of time.

- periodic—a quantity is periodic if its waveform repeats itself at a regular interval, namely its period T. Examples include the sinusoid and the square wave. By application of the Fourier series analysis technique, any periodic waveform can be expressed as the sum of an infinite series of sinusoids.

- Sinusoidal—also called ac (i.e., alternating current), describes a quantity that varies sinusoidally (or cosinusoidally) with time.

In view of these terms, let us now examine the relationship between the electric field E and the magnetic flux density B. Because E is governed by the charge q and B is governed by I=d q/d t, one might expect that E and B must be somehow related to each other. As we learn later, they may or may not be interrelated, depending on whether I is static or dynamic.

Let us start by examining the dc case in which I remains constant with time. Consider a small section of a beam of charged particles, all moving at a constant velocity. The moving charges constitute a dc current. The electric field due to that section of the beam is determined by the total charge q contained in it. The magnetic field does not depend on q, but rather on the rate of charge (current) flowing through that section. Few charges moving very fast can constitute the same current as many charges moving slowly. In these two cases, the induced magnetic field is the same because the current I is the same, but the induced electric field is quite different because the numbers of charges are not the same.

Electrostatics and magnetostatics refer to the study of EM under the specific, respective conditions of stationary charges and dc currents. They represent two independent branches, so characterized because the induced electric and magnetic fields do not couple to each other. Dynamics, the third and more general branch of electromagnetics, involves time-varying fields induced by time-varying sources, that is, currents and associated charge densities. If the current associated with the beam of moving charged particles varies with time, then the amount of charge present in a given section of the beam also varies with time, and vice versa. As we will see in Chapter 6, the electric and magnetic fields become coupled to each other in that case.

Table 1-3 provides a summary of the three branches of electromagnetics.

# Table 1-3 The three branches of electromagnetics.

Table 1 3 Full Alternative Text

The electric and magnetic properties of materials are characterized by the parameters ε and μ, respectively. A third fundamental parameter also needed, the conductivity of a material σ, which is measured in siemens per meter (S/m). The conductivity characterizes the ease with which charges (electrons) can move freely in a material. If σ=0, the charges do not move more than atomic distances, and the material is said to be a perfect dielectric. Conversely, if σ=∞, the charges can move very freely throughout the material, which is then called a perfect conductor.

# Table 1-4 Constitutive parameters of materials.

Table 1 4 Full Alternative Text

Concept Question 1-1: What are the four fundamental forces of nature, and what are their relative strengths?

Concept Question 1-2: What is Coulomb’s law? State its properties.

Concept Question 1-3: What are the two important properties of electric charge?

Concept Question 1-4: What do the electrical permittivity and magnetic permeability of a material account for?

Concept Question 1-5: What are the three branches and associated conditions of electromagnetics?