Theory and Computation of Electromagnetic Fields

Part I: ELECTROMAGNETIC FIELD THEORY

CHAPTER 1

BASIC ELECTROMAGNETIC THEORY

The first half of this book covers the theory of electromagnetic fields and waves based on Maxwell’s equations from the engineering perspective. The coverage is focused on basic concepts and principles related to electromagnetic fields and theoretical approaches to analyzing electromagnetic radiation, propagation, and scattering problems. To start, this chapter presents basic electromagnetic theory, which includes a brief review of vector analysis that is essential for the mathematical treatment of electromagnetic fields, Maxwell’s equations in both integral and differential forms that govern all electromagnetic phenomena, the Lorentz force law that relates electric and magnetic fields to measurable forces, constitutive relations that characterize the electromagnetic properties of a medium, boundary conditions at interfaces between different media and at perfectly conducting surfaces, the concepts of electromagnetic energy and power, the energy conservation law as expressed by Poynting’s theorem, the concept of phasors for time-harmonic fields, and finally Maxwell’s equations and Poynting’s theorem in the complex form for time- harmonic fields. The presentation assumes that the reader has basic knowledge of vector calculus and electromagnetics at the undergraduate level [1–7].

1.1 REVIEW OF VECTOR ANALYSIS

We all know that both electric and magnetic fields are vectors since they have both a magnitude and a direction. Hence, the study of electromagnetic fields requires basic knowledge of vector analysis. The most useful concepts in vector analysis are those of divergence, curl, and gradient. In this section we present definitions and related integral theorems for these quantities. This is followed by the introduction of a new method that can easily deal with various vector identities and the description of the Helmholtz decomposition theorem, which will be very useful for the study of Maxwell’s equations.

1.1.1 Vector Operations and Integral Theorems

Assume that f is a vector function,¹ a quantity whose magnitude and direction vary as functions of its position in space. The divergence of the vector function f is defined by the limit

(1.1.1) c01e001001

where Δv denotes an infinitesimal volume and s denotes the closed surface of this volume. The differential surface ds is normal to s and points outward. By applying Equation 1.1.1 to the differential volume constructed in rectangular, cylindrical, and spherical coordinates, we obtain the expressions of the divergence as

(1.1.2) c01e001002

(1.1.3) c01e001003

(1.1.4)

in these three most important coordinate systems. It is important to remember that ∇ · f, a notation proposed by J. Willard Gibbs [8], is simply a mathematical notation for the divergence of f. It should not be interpreted as the dot product between the operator ∇ and the vector f; otherwise, mistakes can easily be made in the derivation of the expressions in cylindrical and spherical coordinates. Now, consider a finite volume denoted as V, which is enclosed by surface S. By dividing this volume into an infinite number of infinitesimal volumes, applying Equation 1.1.1 to each infinitesimal volume, and summing up the results, we obtain

(1.1.5) c01e001005

if the vector f and its first derivative are continuous in volume V as well as on its surface S. Equation 1.1.5 is known as the divergence theorem or Gauss’ theorem, which is very useful in electromagnetics.

In addition to the divergence, another operation that quantifies the variation of a vector function is called the curl. The curl of the vector function f is defined by the limit

(1.1.6) c01e001006

where Δv again denotes an infinitesimal volume enclosed by surface s. Again, we should remember that ∇ × f is simply a mathematical notation for the curl of f, and it should not be interpreted as the cross-product between the operator ∇ and the vector f. By applying Equation 1.1.6 to the differential volume constructed in rectangular, cylindrical, and spherical coordinates, we obtain the expressions of the curl as

(1.1.7)

(1.1.8)

(1.1.9)

Apparently, the curl itself is a vector that has a different magnitude and a different direction. Given a direction c01ue057 , the magnitude of the curl in this direction is given by

(1.1.10) c01e001010

where Δs is an infinitesimal surface normal to c01ue058 and c is a closed contour bounding Δs. The differential length dl is tangential to the contour c, and its direction is related to that of c01ue059 by the right-hand rule. Equation 1.1.10 can be derived by applying Equation 1.1.6 to an infinitesimal disk perpendicular to c01ue060 with a vanishing thickness. Now, consider an open surface S bounded by a closed contour C. We can divide S into an infinite number of infinitesimal surfaces, then apply Equation 1.1.10 to each of the infinitesimal surfaces, and finally sum up the results to find

(1.1.11) c01e001011

if the vector f and its first derivative are continuous on surface S as well as along C. Equation 1.1.11 is known as Stokes’ theorem, which is also very useful in the study of electromagnetics.

As we will see later, the divergence and curl are sufficient to characterize the variation of a vector function. The third useful operation in vector analysis is the gradient, which quantifies the variation of a scalar function. Let f be a scalar function of space. The gradient of this function is defined as

(1.1.12) c01e001012

which is a vector. Its magnitude along a given direction c01ue074 is given by

(1.1.13) c01e001013

which can be derived by applying Equation 1.1.12 to an infinitesimal circular disk perpendicular to c01ue061 with a vanishing radius and thickness. By applying Equation 1.1.12 to the differential volume constructed in rectangular, cylindrical, and spherical coordinates, we obtain the expressions of the gradient as

(1.1.14) c01e001014

(1.1.15) c01e001015

(1.1.16) c01e001016

In vector analysis, another important operation is to take the divergence on the gradient of a function such as c01ue088 . This operation is often referred to as the Laplacian, which is denoted as

(1.1.17) c01e001017

Its expressions in the three commonly used coordinates are given by

(1.1.18) c01e001018

(1.1.19) c01e001019

(1.1.20)

1.1.2 Symbolic Vector Method

In vector analysis, we often have to manipulate vector expressions into different and yet equivalent forms. A difficulty in such a manipulation is that the operator ∇ cannot be treated rigorously as a vector. This difficulty can be alleviated by the introduction of the symbolic vector method [8]. This symbolic vector, denoted as c01ue001 , is defined as

(1.1.21) c01e001021

where Δv denotes an infinitesimal volume, s denotes the closed surface of this volume, and c01ue062 denotes the unit vector normal to the surface s and pointing outward, which is related to ds by c01ue063 . The left-hand side of Equation 1.1.21, c01ue002 , represents an expression that contains the symbolic vector c01ue003 , such as c01ue004 , c01ue005 , c01ue006 , and c01ue007 . The integrand on the right-hand side, c01ue064 , represents the same expression with c01ue008 being replaced by c01ue065 , so the corresponding expressions for the four examples above are c01ue066 , c01ue096 , c01ue067 , and c01ue068 .

Based on the definition given in Equation 1.1.21, we can show easily that

(1.1.22)

and similarly, c01ue009 and c01ue010 . This indicates clearly that c01ue011 can be treated as a regular vector; hence, all valid vector manipulations and all algebraic identities are applicable to c01ue012 . However, by comparing Equation 1.1.21 with the definitions of the divergence, curl, and gradient, we also see that

(1.1.23) c01e001023

(1.1.24) c01e001024

(1.1.25) c01e001025

These equations establish a relation between the symbolic vector c01ue013 and the divergence, curl, and gradient operations. Given an expression that contains any of these operations, we can first convert it into an algebraic expression using Equations 1.1.23–1.1.25, then manipulate the algebraic expression using any of the valid algebraic identities, and finally convert the symbolic vector back to the divergence, curl, or gradient.

When a vector expression contains the symbolic vector c01ue014 and two arbitrary functions, since c01ue015 works on both functions, we can use the following chain rule to facilitate its manipulation:

(1.1.26) c01e001026

where a and b represent two functions that can either be scalar or vector, c01ue016 is the symbolic vector applying only to function a, and c01ue017 applies only to function b. Equation 1.1.26 should not come as a surprise to anyone who is familiar with the following well-known differentiation formula:

(1.1.27) c01e001027

To illustrate the application of Equation 1.1.26, we consider three examples. We first consider the expression ∇ · (ab). Using Equation 1.1.26, we find

(1.1.28)

Since c01ue018 , c01ue019 , and c01ue020 , we obtain the vector identity

(1.1.29) c01e001029

As the second example, we consider ∇ × (ab). Using Equation 1.1.26, we find

(1.1.30)

which yields the vector identity

(1.1.31) c01e001031

As the last example, we consider ∇ × (a × b). Using Equation 1.1.26 and the algebraic identity

(1.1.32) c01e001032

we find

(1.1.33)

which yields the vector identity

(1.1.34) c01e001034

These examples demonstrate the power of the symbolic vector in deriving various vector identities, which would otherwise be a rather tedious task.

Now, let us consider a finite volume V, which is enclosed by surface S. By dividing this volume into an infinite number of infinitesimal volumes, applying Equation 1.1.21 to each infinitesimal volume, and summing up the results, we obtain

(1.1.35) c01e001035

if the function involved in c01ue021 is continuous within volume V. Equation 1.1.35 is referred to as the generalized Gauss’ theorem, from which we can easily derive many integral theorems. For example, if we let c01ue022 , we obtain the standard Gauss’ theorem in Equation 1.1.5. If we let c01ue023 , we obtain the so-called curl theorem

(1.1.36) c01e001036

from which we can also derive Stokes’ theorem given in Equation 1.1.11 by applying it to a surface with a vanishing thickness.

1.1.3 Helmholtz Decomposition Theorem

In vector analysis, there are two special vectors. One is called the irrotational vector, whose curl vanishes. Denoting this vector as Fi, we have

(1.1.37) c01e001037

Another special vector is called the solenoidal vector, whose divergence is zero. Denoting this vector as Fs, we have

(1.1.38) c01e001038

Using the symbolic vector method, we can easily prove the following two very important vector identities:

(1.1.39) c01e001039

(1.1.40) c01e001040

These identities are valid for any continuous and differentiable scalar function φ and vector function A. Clearly, ∇φ is an irrotational vector and ∇ × A is a solenoidal vector.

Although a vector function can have a complicated variation, it can be shown that any smooth vector function F that vanishes at infinity can be decomposed into an irrotational and a solenoidal vector,

(1.1.41) c01e001041

By taking the divergence and curl of Equation 1.1.41, respectively, we obtain

(1.1.42) c01e001042

which clearly indicate that the solenoidal component is related to the curl of the function and the irrotational part is related to the divergence of the function. Therefore, once both the divergence and curl of a vector function are specified, the function is fully determined. This fact is known as the Helmholtz decomposition theorem.

1.1.4 Green’s Theorems

From Gauss’ theorem in Equation 1.1.5, we can derive some very useful integral theorems. If we substitute f = a∇b into Equation 1.1.5, where a and b are scalar functions, and apply a vector identity based on Equation 1.1.29, we obtain

(1.1.43) c01e001043

which is called the first scalar Green’s theorem. By exchanging the positions of a and b and subtracting the resulting equation from Equation 1.1.43, we obtain

(1.1.44) c01e001044

which is known as the second scalar Green’s theorem.

If we substitute f = a × ∇ × b into Equation 1.1.5, where both a and b are vector functions, and apply a vector identity, we obtain

(1.1.45)

which is called the first vector Green’s theorem. By switching the positions of a and b and subtracting the resulting equation from Equation 1.1.45, we obtain

(1.1.46)

which is known as the second vector Green’s theorem. Now, if we let c01ue069 , where c01ue070 is an arbitrary constant unit vector and b is a scalar function, and then substitute it into Equation 1.1.46, we can obtain after some vector manipulations

(1.1.47)

which can be called the scalar–vector Green’s theorem.

1.2 MAXWELL’S EQUATIONS IN TERMS OF TOTAL CHARGES AND CURRENTS

Maxwell’s equations are a set of four mathematical equations that relate precisely the electric and magnetic fields to their sources, which are electric charges and currents. They were established by James Clerk Maxwell (1831–1879) [9,10] based on the experimental discoveries of André-Marie Ampère (1775–1836) and Michael Faraday (1791–1867) and a law for electricity by Carl Friedrich Gauss (1777–1855), and were reformulated into the vector form by Heinrich Hertz (1857–1894) [11] and Oliver Heaviside (1850–1925) [12]. Maxwell’s equations can be expressed in both integral and differential forms. This section first presents Maxwell’s equations in integral form as the fundamental postulates of electromagnetic theory, and then derives Maxwell’s equations in differential form for fields in a continuous medium, which are subsequently used to derive the current continuity condition. This is followed by a brief description of the Lorentz force law that relates the electric and magnetic fields to measurable forces.

Picture credits

André-Marie Ampère: Engraved by Ambroise Tardieu, 1825, courtesy AIP Emilio Segre Visual Archives

Carl Friedrich Gauss: AIP Emilio Segre Visual Archives, Brittle Books Collection

Michael Faraday: Photo by John Watkins, courtesy AIP Emilio Segre Visual Archives

James Clerk Maxwell: AIP Emilio Segre Visual Archives

Heinrich Hertz: Deutsches Museum

Oliver Heaviside: AIP Emilio Segre Visual Archives, Brittle Books Collection

1.2.1 Maxwell’s Equations in Integral Form

Consider an open surface S bounded by a closed contour C. The first two Maxwell’s equations are given by

(1.2.1) c01e002001

(1.2.2)

where

x2130_SnellRoundhandLTStd-BlkScr_10n_000100 = electric field intensity (volts/meter)

x212C_SnellRoundhandLTStd-BlkScr_10n_000100 = magnetic flux density (webers/meter²)

x1D4A5_SnellRoundhandLTStd-BlkScr_10n_000100 total = electric current density (amperes/meter²)

ε0 = permittivity of free space (farads/meter)

μ0 = permeability of free space (henrys/meter).

The position vector r and time variable t are included explicitly to indicate that the associated quantities can be functions of position and time.² The subscript total in x1D4A5_SnellRoundhandLTStd-BlkScr_10n_000100 total is used to denote that this is the current density of total electric currents. In the MKS unit system, the numerical values for the free-space permittivity and permeability are

(1.2.3) c01e002003

(1.2.4) c01e002004

Equation 1.2.1 is called Faraday’s induction law, and Equation 1.2.2 is often called Ampère’s law or the Maxwell–Ampère law because Maxwell augmented the original Ampère’s law with the addition of the displacement current, the first term on the right-hand side. As we will see later, this term is very important because it predicts that electromagnetic fields can propagate as waves, which was experimentally verified by Hertz in 1887. Equations 1.2.1 and 1.2.2 indicate that a time-varying magnetic flux can generate an electric field, and an electric current and a time-varying electric field can generate a magnetic field.

In a folk song about Faraday’s law as expressed in Equation 1.2.1, the author, Dr. Walter Fox Smith of Haverford College, elaborated eloquently its physical meaning and practical importance in a humorous fashion. He wrote

Faraday’s law of induction

The law of all sea and all land—

No lies, no deceit, no corruption

In this law so complete and so grand!

Our children will sing it in chorus—

Circulation of vector cap E,

Yes they’ll sing as they march on before us,

"Equals negative d by dt

Of—

Magnetic flux through a surface,"

They’ll conclude as we strike up the band.

We’ll mark all our coins with our purpose—

On Maxwell’s equations we stand!

It’s Faraday’s law of induction

That allows us to generate pow’r.

It gives voltage increase or reduction—

We could sing on and on for an hour!

By denoting the total current and total electric flux passing through the surface S as

(1.2.5) c01e002005

(1.2.6) c01e002006

the Maxwell–Ampère law in Equation 1.2.2 can also be written as

(1.2.7) c01e002007

In a folk song titled Two great guys—one great law! Smith described the development history of this law and the contributions by Ampère and Maxwell:

Mr. Ampère’s magical, mystical, wonderful law!

Of Maxwell’s equations, it is the longest and strangest of all!

On the left side, he wrote circulation

Of magnetic field, ‘cause it was neat.

On the right hand side of his equation—

Mu-naught I—he thought it was complete.

Decades later, Maxwell saw disaster,

Although he thought of Ampère as a saint—

In between the plates of a capacitor

The right side’s zero, but the left side ain’t!

To fix this problem, he added to the right side

Displacement current, a brand new quantity!

It started mu-naught eps’lon-naught and ended by

The time derivative of phi-sub-E.

And so to Maxwell the myst’ry was revealed—

He saw how light could move through empty space.

The changing B-field made the changing E-field,

And vice-a-versa, all at the perfect pace.

Next, consider a volume V enclosed by a surface S. The other two Maxwell’s equations are given by

(1.2.8) c01e002008

(1.2.9) c01e002009

where ρe,total denotes the electric charge density (coulombs/meter³) in volume V. Again, the subscript total is used to denote that ρe,total represents the density of total charges. Equation 1.2.8 is called Gauss’ law, and Equation 1.2.9 is called Gauss’ law for the magnetic case. Clearly, Equation 1.2.9 indicates that the magnetic flux lines cannot be originated or terminated anywhere; they have to form closed loops. In contrast, the electric field lines, as indicated in Equation 1.2.8, can be originated from positive charges and terminated at negative charges.

By denoting the differential surface vector c01ue071 and the total charge enclosed inside V as

(1.2.10) c01e002010

Gauss’ law in Equation 1.2.8 can be rewritten as

(1.2.11) c01e002011

This equation is the subject of another folk song by Smith, which says

Inside, outside, count the lines to tell—

If the charge is inside, there will be net flux as well.

If the charge is outside, be careful and you’ll see

The goings in and goings out are equal perfectly.

If you wish to know the field precise,

And the charge is symmetric,

you will find this law is nice—

Q upon a constant—eps’lon naught they say—

Equals closed surface integral of E dot n dA.

Equations 1.2.1, 1.2.2, 1.2.8, and 1.2.9 are usually referred to as Maxwell’s equations in integral form. They are obtained directly from experiments and are valid everywhere for any case. They have been regarded as the fundamental postulates of electromagnetic theory ever since Maxwell formulated them over 140 years ago. The entire electromagnetic theory, valid from the static to the optical regimes and from subatomic to intergalactic length scales, is based on these four equations, as we will see repeatedly in this book.

1.2.2 Maxwell’s Equations in Differential Form

The integral-form Maxwell’s equations are valid everywhere. Now, consider a point in a continuous medium. The fields at such a point should be continuous; therefore, we can use Stokes’ and Gauss’ theorems to convert Maxwell’s equations in integral form into their counterparts in differential form. To be more specific, by applying Stokes’ theorem to Equations 1.2.1 and 1.2.2 and using the fact that these equations are valid for any surface S, we obtain

(1.2.12) c01e002012

(1.2.13)

respectively. Here, we omit the position vector and time variable for the sake of brevity. By applying Gauss’ theorem to Equations 1.2.8 and 1.2.9 and using the fact that these are valid for any volume V, we obtain

(1.2.14) c01e002014

(1.2.15) c01e002015

respectively. Equations 1.2.12 and 1.2.13 can also be obtained by shrinking the closed contour in Equations 1.2.1 and 1.2.2 to a point and then invoking the alternative definition of the curl given in Equation 1.1.10. Similarly, Equations 1.2.14 and 1.2.15 can also be obtained by shrinking the closed surface in Equations 1.2.8 and 1.2.9 to a point and then invoking the definition of the divergence in Equation 1.1.1. Therefore, Maxwell’s equations in differential form describe field behavior at a point in a continuous medium.

1.2.3 Current Continuity Equation

By taking the divergence of Equation 1.2.13 and applying the vector identity (Eq. 1.1.40) and Gauss’ law in Equation 1.2.14, we obtain

(1.2.16) c01e002016

To understand the implication of this equation, we can simply integrate it over a finite volume and apply Gauss’ theorem in Equation 1.1.5 to find

(1.2.17) c01e002017

It is evident that the left-hand side represents the net current leaving the volume and the right-hand side represents the reduction rate of the total charge in the volume. As a result, this equation represents the continuity of currents or conservation of charges. Because of this continuity equation, the four Maxwell’s equations are not independent for time-varying fields. This can be verified easily by taking the divergence of Equations 1.2.12 and 1.2.13 and then applying Equations 1.2.16 and 1.1.40, respectively, which would yield Equations 1.2.14 and 1.2.15. This, however, does not hold for the static fields because for such a case the currents and charges are no longer related and the electric and magnetic fields are completely decoupled; hence, all four equations have to be considered.

1.2.4 The Lorentz Force Law

When a particle carrying electric charge q is placed in an electric field, it experiences a force given by q x2130_SnellRoundhandLTStd-BlkScr_10n_000100 . When this charge is moving in a magnetic field, it experiences another force given by qv × x212C_SnellRoundhandLTStd-BlkScr_10n_000100 , where v represents the velocity vector of the charge. Combining the two forces, we obtain the total force exerted on a charged particle as

(1.2.18) c01e002018

which is known as the Lorentz force law. This law is useful for understanding the interaction between electromagnetic fields and matter, as we will discuss next. It is also the principle used in the design of many electrical devices such as electric motors, magnetrons, and particle accelerators.

1.3 CONSTITUTIVE RELATIONS

Maxwell’s equations, as presented above, are valid in any kind of media. Since a medium has a significant effect on electromagnetic fields, we have to consider this effect in the study of electromagnetic fields. A medium affects electromagnetic fields through three phenomena—electric polarization, magnetic polarization or simply magnetization, and electric conduction. This section discusses these three phenomena and formulates a set of equations, known as constitutive relations, to account for the effect of a medium on electromagnetic fields. These constitutive relations are then used to classify media into various categories.

1.3.1 Electric Polarization

We first consider the effect of electric charges in a medium on electromagnetic fields. It is well known that a matter that makes up a medium is made of molecules, which consist of atoms. In an atom, there is a nucleus consisting of neutrons and protons. The neutrons are not charged, but the protons are positively charged. Surrounding a nucleus are negatively charged electrons, whose number equals the number of protons. These electrons are bound to the nucleus by the electric force, so they normally cannot break free; instead, they orbit around the nucleus at high speed. The center of the orbit coincides with the center of the protons so that an entire atom is electrically neutral. A molecule is made up of one or more atoms. For some molecules, the atoms are arranged such that the center of positive charges coincides with that of negative charges. This type of molecule is called a non-polar molecule, and in such a case, the molecules and hence the matter that is made of non-polar molecules appear electrically neutral. For some other molecules, the interaction between atoms creates a small displacement between the effective centers of positive and negative charges, thus creating a tiny electric dipole and generating a weak electric field. This type of molecule is called a polar molecule. However, since all polar molecules are randomly oriented, the effects of tiny electric dipoles cancel each other and the matter that is made of polar molecules is also electrically neutral.

The scenario described above changes drastically when an electric field is applied to the medium. According to the Lorentz force law, the applied electric field exerts a force on positive charges in the direction of the field, whereas it exerts a force on negative charges in the opposite direction. As a result, in both atoms and non-polar molecules, the effective center of positive charges will be displaced from the effective center of negative charges, creating a tiny electric dipole in the direction of the electric field. (Here, we assume that the applied field is not strong enough to break the bound electrons loose from the nuclei. In such a case, the matter is often called a dielectric.) In the case of polar molecules, because of the Lorentz force, all the randomly oriented dipoles tend to line up with the applied electric field. When a large number of electric dipoles line in the same direction, the electric fields created by the dipoles add up and these electric fields are in the opposite direction to the applied field, resulting in a weaker total electric field in the medium. To quantify the effect of tiny dipoles, a vector quantity called the dipole moment is defined as

(1.3.1) c01e003001

where q denotes the charge and x1D4C1_SnellRoundhandLTStd-BlkScr_10n_000100 denotes the vector pointing from the effective center of the negative charge to that of the positive charge. The sum of dipole moments per unit volume is then

(1.3.2) c01e003002

where np denotes the number of dipoles contained in Δv. The dipole moment density x1D4AB_SnellRoundhandLTStd-BlkScr_10n_000100 is also called the polarization intensity or polarization vector.

When the dipole moment density is uniform, the positive charge of a dipole is completely canceled by the negative charge of the next dipole; hence, there is no net charge in the medium. However, when the dipole moment density is not uniform, the positive charge of a dipole cannot be completely canceled by the negative charge of the next dipole, resulting in a net charge at the point and hence a volume charge density. This volume charge density is given by

(1.3.3) c01e003003

where the subscript b is used to denote that this is the density of the bound charges. If the medium also contains free charges, the total charge density in the medium can then be expressed as

(1.3.4) c01e003004

where ρe,f denotes the density of free electric charges. Substituting this expression into Equation 1.2.14, we obtain

(1.3.5) c01e003005

By defining a new quantity, called the electric flux density, as

(1.3.6) c01e003006

which has a unit of coulombs/meter², Equation 1.3.5 can be written as

(1.3.7) c01e003007

This expression can be regarded as Gauss’ law expressed in terms of free electric charges. In addition to the volume charge density, the electric polarization also produces an electric current when it changes in time. In view of the current continuity equation (Eq. 1.2.16), the electric current density contributed by the electric polarization is

(1.3.8) c01e003008

When this current is separated from the total current, Equation 1.2.13 can also be expressed in terms of x1D49F_SnellRoundhandLTStd-BlkScr_10n_000100 defined in Equation 1.3.6.

In most dielectric materials, the polarization intensity is usually proportional to the electric field:

(1.3.9) c01e003009

where χe is called the electric susceptibility. Consequently, the electric flux density x1D49F_SnellRoundhandLTStd-BlkScr_10n_000100 is related to the electric field intensity x2130_SnellRoundhandLTStd-BlkScr_10n_000100 by

(1.3.10) c01e003010

where ε = ε0(1 + χe) is called the permittivity of the dielectric. In engineering practice, we often use the relative permittivity, defined as c01ue072 , to help us memorize the value. Since χe is usually a positive number, εr is usually greater than 1. Equation 1.3.10 is called the constitutive relation for the electric field. In free space such as vacuum and air, the polarization intensity x1D4AB_SnellRoundhandLTStd-BlkScr_10n_000100 either vanishes or is negligible; hence, the constitutive relation (Eq. 1.3.10) becomes

(1.3.11) c01e003011

1.3.2 Magnetization

Next, we consider what happens when a magnetic field is applied to a medium. As mentioned earlier, electrons orbit the nucleus continuously in an atom. Such orbiting creates a tiny current loop, which generates a very weak magnetic field. Such a current loop can be quantified by a vector called the magnetic dipole moment, which is defined as

(1.3.12) c01e003012

where I denotes the current and x1D4C8_SnellRoundhandLTStd-BlkScr_10n_000100 has a magnitude equal to the area of the current loop and a direction determined by the direction of the current flow via the right-hand rule. Quantum physics reveals that all electrons and protons rotate at high speed about their own axes, a motion called spin. Since electrons and protons are charged, such a rotation also creates current loops, which generate very weak magnetic fields and can be quantified by magnetic dipole moments as well. In the absence of any applied fields, the directions of all the magnetic dipoles are randomly oriented (except for those in a permanent magnet). As a result, the magnetic dipole moments cancel out macroscopically and the medium appears magnetically neutral. When a magnetic field is applied to the medium, the randomly oriented magnetic dipoles tend to align themselves either in the direction of the applied field or in the opposite direction. This produces an observable quantity called magnetization intensity or magnetization vector x2133_SnellRoundhandLTStd-BlkScr_10n_000100 , which is defined as the sum of the magnetic dipole moments per unit volume,

(1.3.13) c01e003013

where nm denotes the number of magnetic dipoles contained in Δv. This magnetization vector will either strengthen or weaken the total magnetic field.

When the magnetic dipole density is uniform, the electric current of a current loop is completely canceled by the current of the next current loop; hence, there is no net electric current in the medium. However, when the magnetic dipole density is not uniform, the electric current of a current loop cannot be completely canceled by the current of the next current loop, which then results in a net current at the point. The volume current density of this current is given by

(1.3.14) c01e003014

Adding this current to the current due to the electric polarization and the free current, we have the total current in the medium

(1.3.15) c01e003015

where x1D4A5_SnellRoundhandLTStd-BlkScr_10n_000100 f denotes the density of the free electric current. Substituting this into Equation 1.2.13, we obtain

(1.3.16) c01e003016

where we have also used Equation 1.3.6. By defining a new magnetic quantity, called the magnetic field intensity, as

(1.3.17) c01e003017

which has a unit of amperes/meter, Equation 1.3.16 can be written as

(1.3.18) c01e003018

This equation can be regarded as the Maxwell–Ampère law in terms of free electric currents. Note that there is no electric charge associated with x1D4A5_SnellRoundhandLTStd-BlkScr_10n_000100 m since ∇ · x1D4A5_SnellRoundhandLTStd-BlkScr_10n_000100 m = ∇ · (∇ × x2133_SnellRoundhandLTStd-BlkScr_10n_000100 ) ≡ 0.

Equation 1.3.17 can also be written as

(1.3.19) c01e003019

In most materials, the magnetization intensity is proportional to the magnetic field intensity:

(1.3.20) c01e003020

where χm is called the magnetic susceptibility. In such a case, Equation 1.3.19 becomes

(1.3.21) c01e003021

where μ = μ0(1 + χm) is called the permeability of the material. In engineering practice, we often use the relative permeability, defined as c01ue073 , to help us memorize the value. For most materials in reality, the magnetization is so small that μr ≈ 1 and such materials are called non-magnetic. Equation 1.3.21 is called the constitutive relation for the magnetic field. In free space such as vacuum and air, the magnetization intensity x2133_SnellRoundhandLTStd-BlkScr_10n_000100 either vanishes or is negligible; hence, the constitutive relation (Eq. 1.3.21) is reduced to

(1.3.22) c01e003022

1.3.3 Electric Conduction

In addition to the polarization and magnetization, a third phenomenon is called conduction, which happens in a medium containing free charges such as free electrons and ions. In the absence of any fields, these charges move in random directions so that they do not form electric currents macroscopically. However, when an electric field is applied to the medium, the free charges tend to flow either in the direction of the applied field or in the opposite direction depending on whether they are positively or negatively charged. As a result, they form electric currents, which are called conduction currents. In most materials, the current density of the conduction current is proportional to the electric field, which can be expressed as

(1.3.23) c01e003023

where σ is called the conductivity having a unit of siemens/meter. When the free charges such as electrons move in a medium, they collide with atomic lattices and their energy is dissipated and converted into heat. Hence, σ is also related to the dissipation of the energy. The conduction current can be regarded as a part of the free electric current.

1.3.4 Classification of Media

The preceding discussion indicates clearly that the electromagnetic properties of a medium are reflected in the following three constitutive relations:

(1.3.24) c01e003024

Therefore, the three parameters ε, μ, and σ fully characterize the electromagnetic properties of a medium. Consequently, we can classify media based on the forms and values of these parameters.

Classification Based on the Spatial Dependence

If any of ε, μ, or σ is a function of position in space, the medium is called inhomogeneous or heterogeneous. Otherwise, it is called a homogeneous medium, where ∇ε = ∇μ = ∇σ ≡ 0. A homogeneous medium affects electromagnetic fields through the polarization current x1D4A5_SnellRoundhandLTStd-BlkScr_10n_000100 p and the bound charges and currents on the surface of the medium.

Classification Based on the Time Dependence

If any of ε, μ, or σ is a function of time, the medium is called non-stationary; otherwise, it is called stationary. Note that even if a medium is physically stationary, it can still be electrically non-stationary if its electromagnetic properties change with time.

Classification Based on the Directions of x1D49F_SnellRoundhandLTStd-BlkScr_10n_000100 and x212C_SnellRoundhandLTStd-BlkScr_10n_000100

If the direction of x1D49F_SnellRoundhandLTStd-BlkScr_10n_000100 is parallel to that of x2130_SnellRoundhandLTStd-BlkScr_10n_000100 and the direction of x212C_SnellRoundhandLTStd-BlkScr_10n_000100 is parallel to that of x210B_SnellRoundhandLTStd-BlkScr_10n_000100 , the medium is called isotropic. Otherwise, it is called an anisotropic medium. For an anisotropic medium, the constitutive relations cannot be expressed in a simple form as in Equation 1.3.24. Instead, they have to be expressed as

(1.3.25)

which can be written compactly as

(1.3.26) c01e003026

where c01ue024 and c01ue025 are called permittivity and permeability tensors.³ When we discuss the reciprocity theorem, we will see that if these two tensors are symmetric, the medium is reciprocal; otherwise, it is non-reciprocal. A special case of general anisotropic media is crystals, which have a diagonal permittivity tensor,

(1.3.27) c01e003027

In this case, if all three diagonal elements are different, the medium is called biaxial. If any two of the three are the same, the medium is called uniaxial. Of course, if all three elements are the same, the medium is isotropic. A further generalization of the anisotropic medium is the so-called bianisotropic medium, whose constitutive relations are given by

(1.3.28) c01e003028

When c01ue026 , c01ue027 , c01ue028 , and c01ue029 reduce to scalars, the medium is called bi-isotropic. These kinds of materials are rare in nature, but they can be manufactured in laboratories.

Classification Based on the Field Dependence

If any value of ε, μ, or σ depends on the field intensities x2130_SnellRoundhandLTStd-BlkScr_10n_000100 and x210B_SnellRoundhandLTStd-BlkScr_10n_000100 , then the flux densities x1D49F_SnellRoundhandLTStd-BlkScr_10n_000100 and x212C_SnellRoundhandLTStd-BlkScr_10n_000100 and the conduction current density x1D4A5_SnellRoundhandLTStd-BlkScr_10n_000100 c are no longer linear functions of x2130_SnellRoundhandLTStd-BlkScr_10n_000100 and x210B_SnellRoundhandLTStd-BlkScr_10n_000100 . Such a medium is called non-linear; otherwise, it is called linear. Non-linear constitutive relations significantly complicate the study of the electromagnetic fields in the medium; nevertheless, non-linear media do exist in nature even though their applications are not widespread.

Classification Based on the Frequency Dependence

If any value of ε or μ depends on the frequency of the field such that ε = ε ( f ) or μ = μ( f ), where f denotes the frequency, the medium is called dispersive; otherwise, it is called non-dispersive. If a signal that contains multiple frequencies propagates in a dispersive medium, the shape of the signal will be distorted because different frequency components propagate at different speeds. Rigorously speaking, for a dispersive medium, the constitutive relations can no longer be written in the form of Equation 1.3.24. Because of the frequency dependence, they have to be written in terms of convolution:

(1.3.29)

(1.3.30)

where * denotes the temporal convolution. The convolution is due to the fact that the medium cannot polarize and magnetize instantaneously in response to the applied field and, therefore, the polarization and magnetization vectors are related to the fields at previous times.

Classification Based on the Value of Conductivity

If σ = 0, the medium is called a perfect dielectric or insulator. On the other hand, if σ → ∞, the medium is called a perfect electric conductor (PEC). In reality, there are no such things as perfect dielectrics or perfect conductors. But, in engineering practice, these are very useful concepts because the approximation of a very good conductor as a perfect conductor and the approximation of a good dielectric as a perfect dielectric can significantly simplify the analysis of electromagnetic problems. When σ has a non-negligible finite value, the medium is called lossy. It should be pointed out that the conduction characterized by σ represents only one of the loss mechanisms. When a medium is exposed to a time-varying electromagnetic field, the polarization and magnetization can also cause losses, especially when the frequency of the field is very high. This is because the directions of time-varying electric and magnetic fields change rapidly and, consequently, the electric and magnetic dipoles that follow the field directions change their directions as well. When these dipoles flip back and forth, the friction between the bound charges and dipoles causes energy dissipation. This phenomenon can be described mathematically in the time domain as a damping term in the motion equation for the dipoles [13]; however (and very fortunately) its description in the frequency domain is very simple. The Fourier transforms of the permittivity and permeability simply become two complex quantities with the imaginary parts representing the polarization and magnetization losses.

Classification Based on the Value of Permeability

As discussed earlier, when a magnetic field is applied to a medium, the randomly oriented magnetic dipoles tend to align themselves either in the direction of the applied field or in the opposite direction, producing a net magnetization intensity x2133_SnellRoundhandLTStd-BlkScr_10n_000100 . When this net magnetization intensity is very small and its direction is opposite to the direction of the applied field, the magnetic susceptibility χm is a very small negative number and the relative permeability μr is slightly less than 1. This type of medium is called diamagnetic. When the net magnetization intensity is again very small but its direction is in the direction of the applied field, the magnetic susceptibility χm is a very small positive number and the relative permeability μr is slightly greater than 1. The medium is called paramagnetic. For both diamagnetic and paramagnetic media, the value of μr differs from 1 by any amount on the order of 10−4. In most engineering applications, this difference can be neglected and μr can be practically approximated as μr ≈ 1.0; hence, the medium can be considered as non- magnetic. However, there is a type of medium in which the net magnetization intensity has a very large value and its direction is the same as that of the applied field, resulting in a large relative permeability μr. This type of medium is called ferromagnetic. Ferromagnetic materials usually have a high conductivity, and hence cannot sustain an appreciable electromagnetic field. There is yet another class of materials, called ferrites, which have a relatively large permeability and a very small conductivity at microwave frequencies. Because of this, ferrites find many applications in the design of microwave devices.

1.4 MAXWELL’S EQUATIONS IN TERMS OF FREE CHARGES AND CURRENTS

With the constitutive relations (Eq. 1.3.24), Maxwell’s equations in integral form can be written for x2130_SnellRoundhandLTStd-BlkScr_10n_000100 , x210B_SnellRoundhandLTStd-BlkScr_10n_000100 , x1D49F_SnellRoundhandLTStd-BlkScr_10n_000100 , and x212C_SnellRoundhandLTStd-BlkScr_10n_000100 in terms of free charges and currents as

(1.4.1) c01e004001

(1.4.2)

(1.4.3) c01e004003

(1.4.4) c01e004004

The free current x1D4A5_SnellRoundhandLTStd-BlkScr_10n_000100 f includes the conduction current x1D4A5_SnellRoundhandLTStd-BlkScr_10n_000100 c = σ x2130_SnellRoundhandLTStd-BlkScr_10n_000100 and the current supplied by impressed sources.

Equations 1.4.1–1.4.4 are asymmetric because of the lack of magnetic currents and charges. Although magnetic currents and charges do not exist or have not been found so far in reality, the concepts of such currents and charges are useful because sometimes we can introduce equivalent magnetic currents and charges to simplify the analysis of some electromagnetic problems. By incorporating magnetic currents and charges, Equations 1.4.1 and 1.4.4 become

(1.4.5)

(1.4.6)

where x2133_SnellRoundhandLTStd-BlkScr_10n_000100 f denotes the free magnetic current density (volts/meter²) and ρm,f denotes the free magnetic charge density (webers/meter³). With this modification, Maxwell’s equations become more symmetric. The reader is cautioned not to confuse the magnetic current density x2133_SnellRoundhandLTStd-BlkScr_10n_000100 f with the magnetization intensity x2133_SnellRoundhandLTStd-BlkScr_10n_000100 used previously.

The corresponding Maxwell’s equations in differential form for fields at a point in a continuous medium can be obtained by invoking Stokes’ and Gauss’ theorems. They can be written as

(1.4.7) c01e004007

(1.4.8)

(1.4.9) c01e004009

(1.4.10) c01e004010

The free charges and currents also satisfy the current continuity equations, which can be derived from Equations 1.4.7–1.4.10 by taking the divergence of Equations 1.4.7 and 1.4.8 and then applying the vector identity (Eq. 1.1.40) and Gauss’ laws in Equations 1.4.9 and 1.4.10. Their differential forms are given by

(1.4.11) c01e004011

(1.4.12) c01e004012

The corresponding integral forms can be obtained by integrating these two equations over a finite volume and then applying Gauss’ theorem in Equation 1.1.5, yielding

(1.4.13) c01e004013

(1.4.14) c01e004014

Because of these continuity conditions, the four Maxwell’s equations in Equations 1.4.7–1.4.10 are not independent for time-varying fields since Equations 1.4.9 and 1.4.10 can be derived from Equations 1.4.8 and 1.4.7, respectively.

Although Maxwell’s equations for free charges and currents appear quite different from those for total charges and currents, both can be written uniformly in the form presented in this section with the charge and current densities defined based on the constitutive relations used. This is the subject of Problem 1.15. In engineering, we often prefer Maxwell’s equations in terms of free charges and currents over the ones for total charges and currents because the total charges and currents are usually unknown before Maxwell’s equations are solved, whereas the constitutive parameters ε, μ, and σ can usually be measured experimentally.

1.5 BOUNDARY CONDITIONS

The differential-form Maxwell’s equations are valid at points in a continuous medium. They cannot be applied to discontinuous fields that may occur at interfaces between different media. Fortunately, we can employ Maxwell’s equations in integral form to find the relations between the fields on the two sides of an interface. Such relations are called boundary conditions. The relationship between the integral-form Maxwell’s equations and the differential-form Maxwell’s equations and the boundary conditions is illustrated in Figure 1.1. In this section we derive these boundary conditions using Maxwell’s equations for free charges and currents. Hence, all the charge and current quantities used in this section are pertinent to free charges and currents.

Figure 1.1 Relationship between Maxwell’s equations in integral and differential forms and boundary conditions.

Before deriving the boundary conditions, let us first introduce the concept of surface currents. So far, the current density x1D4A5_SnellRoundhandLTStd-BlkScr_10n_000100 f is actually a volume current density, which is often simply called current density. It represents the amount of current passing through a unit area normal to the direction of the current flow. Now, imagine a current flow confined in a thin layer. If the total current is kept constant while the thickness of the layer is reduced to zero, the volume current density approaches infinity, which can no longer describe the current sheet. In this case, the current distribution can be described by the surface current density, which is a vector denoted as x1D4A5_SnellRoundhandLTStd-BlkScr_10n_000100 s. Its value represents the amount of current passing through a unit width normal to the direction of the current flow and has a unit of amperes/meter. The surface magnetic current density x2133_SnellRoundhandLTStd-BlkScr_10n_000100 s is defined similarly, which has a unit of volts/meter.

Now, let us consider an interface between two different media, and for the sake of generality a free surface current with a density of x1D4A5_SnellRoundhandLTStd-BlkScr_10n_000100 s is assumed flowing on the interface. The normal unit vector c01ue075 on the interface is defined to point from medium 1 to medium 2. To apply Equation 1.4.2, we construct a small rectangular frame with one of its sides in medium 1 and the other in medium 2, as illustrated in Figure 1.2. The length of the frame is Δl and the width Δt is vanishingly small. Applying Equation 1.4.2 to this frame and letting Δt → 0, we have

(1.5.1) c01e005001

where c01ue076 is a tangential unit vector as shown in Figure 1.2. Since the direction of c01ue077 is not uniquely determined, it is desirable to remove it from Equation 1.5.1. For this, we rewrite c01ue078 as c01ue079 and employ the vector identity

(1.5.2) c01e005002

Figure 1.2 A rectangular frame across a discontinuous interface.

to find

(1.5.3)

Since the orientation of c01ue080 and, thus, c01ue081 , is arbitrary along the surface, we have

(1.5.4) c01e005004

which indicates that the tangential component of the magnetic field intensity is discontinuous across an interface carrying a free surface electric current. By applying the same approach to Equation 1.4.5, we obtain another boundary condition

(1.5.5) c01e005005

showing a discontinuity in the tangential component of the electric field intensity across an interface carrying a free surface magnetic current. Since the magnetic current does not exist in reality, the tangential component of the electric field intensity is always continuous across any interfaces.

Next, we consider an interface between two different media and we assume a free surface charge distribution over the interface. The surface charge density is defined as the amount of charge over a unit area on the surface. To apply Equation 1.4.3, we construct a small pillbox with one of its faces in medium 1 and the other in medium 2, as illustrated in Figure 1.3. Each face of the pillbox has an area Δs and its thickness Δt is vanishingly small. Applying Equation 1.4.3 to this pillbox and letting Δt → 0, we obtain

(1.5.6) c01e005006

or

(1.5.7) c01e005007

where ρe,s denotes the surface electric charge density having a unit of coulombs/meter². This reveals that the normal component of the electric flux density is discontinuous across an interface carrying a free surface electric charge. By applying the same procedure to Equation 1.4.6, we obtain

(1.5.8) c01e005008

which shows that the normal component of the magnetic flux density is discontinuous across an interface carrying a free surface magnetic charge. Here, ρm,s denotes the surface magnetic charge density and has a unit of webers/meter². However, since in reality the magnetic charges do not exist, the normal component of the magnetic flux density is always continuous across any interfaces.

Figure 1.3 A pillbox across a discontinuous interface.

Similar to the case for Maxwell’s equations, the four boundary conditions in Equations 1.5.4, 1.5.5, 1.5.7, and 1.5.8 are not independent. When the first two are satisfied, the latter two are usually satisfied as well. Also note that unless one of the media is a perfect conductor, the electromagnetic fields usually cannot induce free surface charges or currents at the interface. Hence, the tangential component of the magnetic field intensity and the normal component of the electric flux density are continuous across an interface between two different media. However, when one of the media is a perfect conductor, the situation is different. A perfect conductor is a medium full of free charges. When an electromagnetic field is applied to this medium, the free charges, being pushed by the applied field, move themselves such that they produce an opposing field that completely cancels the applied field. This causes the formation of the surface currents and charges on the surface of a perfect conductor. If it is a PEC, its surface can support a surface electric current and charge. If it is a perfect magnetic conductor (PMC), the surface can support a surface magnetic current and charge. Now, assuming that medium 1 is a PEC, the boundary conditions at the surface become

(1.5.9) c01e005009

(1.5.10) c01e005010

(1.5.11) c01e005011

(1.5.12) c01e005012

where the unit normal c01ue082 points away from the conductor. As mentioned earlier, it is unnecessary to enforce all these conditions when solving an electromagnetic problem. It is usually sufficient to enforce either Equation 1.5.9 or 1.5.12 since the other two conditions involve the induced surface current and charge densities, which are usually unknown. However, if the fields are known, Equations 1.5.10 and 1.5.11 provide a means to calculate the induced surface current and charge densities. The boundary conditions at the surface of a PMC can be deduced in a similar fashion.

We wish to point out that the boundary conditions are as important as Maxwell’s equations because they describe the field behavior across a discontinuous interface, whereas the differential-form Maxwell’s equations describe the field behavior in a continuous medium, as illustrated clearly in Figure 1.1. Without boundary conditions, an electromagnetic problem is usually not completely defined and cannot be solved. Furthermore, understanding these boundary conditions can allow us to have a general idea about the field distribution in a given electromagnetic problem and help us to deal with the problem more effectively.

1.6 ENERGY, POWER, AND POYNTING’S THEOREM

Energy and power are two of the most fundamental quantities in physics. They play very important roles in electromagnetics as well. In this section, we start from Maxwell’s equations and establish relations between electromagnetic fields and energy and power.

To start, we consider a medium characterized by permittivity ε, permeability μ, and conductivity σ. Maxwell’s equations (Eqs. 1.4.7 and 1.4.8) in such a medium can be written as

(1.6.1) c01e006001

(1.6.2) c01e006002

where x1D4A5_SnellRoundhandLTStd-BlkScr_10n_000100 i and x2133_SnellRoundhandLTStd-BlkScr_10n_000100 i represent the actual source of the field and are often referred to as the impressed currents. In Equation 1.6.2, the total current is separated into the conduction current and the impressed current. By taking the dot product of Equation 1.6.1 with x210B_SnellRoundhandLTStd-BlkScr_10n_000100 and the dot product of Equation 1.6.2 with