Zhang K., Li D. Electromagnetic Theory for Microwaves and Optoelectronics

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Chapter 1

Basic Electromagnetic

Theory

The foundation of this book is macroscopic electromagnetic theory. In the

beginning, a brief survey of the basic laws and theorems of electromagnetic

theory is necessary. These subjects are included in undergraduate texts on

electrodynamics or electromagnetic theory. Therefore, this chapter may be

read as a review. The contents of this chapter include:

Maxwell’s equations in vacuum and in continuous media;

Characteristics of material media;

Boundary conditions, Maxwell’s equations on the boundary of media;

Wave equations and Helmholtz’s equations;

Energy and power ﬂow in electromagnetic ﬁelds, Poynting’s theorem;

Potential functions and d’Alembert’s equations;

Hertzian vector potentials;

Duality and reciprocity.

1.1 Maxwell’s Equations

James Clark Maxwell (1831–1879) reviewed and grouped the most important

experimental laws on electric and magnetic phenomena, developed by previ-

ous scientists over more than one hundred years. He also developed Faraday’s

concept of ﬁeld, introduced the displacement current into Amp`ere’s circuital

law, and ﬁnally, in 1863, formulated a complete set of equations governing

the behavior of the macroscopic electromagnetic phenomenon [68]. In fact, it

was Oliver Heaviside (1850-1925) who ﬁrst expressed them in the form that

we know today, but J.C. Maxwell was the ﬁrst to state them clearly and to

recognize their signiﬁcance. This set of equations is usually and justly known

as Maxwell’s equations.

2 1. Basic Electromagnetic Theory

Figure 1.1: Position vector x and vector function A(x, t).

1.1.1 Basic Maxwell Equations

Instantaneous, or so-called time-domain, Maxwell equations in vacuum are

the basic Maxwell equations. The vectors and scalars in the equations are

functions of position x (vector) and time t (scalar). In rectangular coordi-

nates, x =

xx+

yy +

zz, where

y and

z are the unit vectors along the x, y

and z directions, respectively, refer to Fig. 1.1. An arbitrary vector function

A = A(x, t) is explained as follows:

A(x, t) =

(x, t) +

(x, t), (1.1)

where scalar functions A

, A

and A

are the comp onents of the vector

function A. This kind of vector functions and scalar functions with respect

to time t are known as instantaneous functions or instantaneous values.

The basic Maxwell equations in integral form and derivative form are

given as

E · dl = −

B · dS, ∇ × E = −

∂B

∂t

, (1.2)

· dl =

E · dS + I, ∇ ×

∂²

∂t

+ J , (1.3)

E · dS = q, ∇ · ²

E = %, (1.4)

B · dS = 0, ∇ · B = 0, (1.5)

where l is a closed contour surrounding an open surface S and a volume V

is bounded by the closed surface S. See Fig. 1.2.

1.1 Maxwell’s Equations 3

Figure 1.2: An open surface S surrounded by a closed contour l (a) and a

volume V bounded by a closed surface S (b).

Maxwell’s equations in derivative form are applicable in continuous

medium only. These constitute a set of simultaneous partial diﬀerential equa-

tions with respect to space and time. Equation (1.2) is the curl equation for

the electric ﬁeld, originating from Faraday’s law of induction; (1.3) is the curl

equation for the magnetic ﬁeld, originating from Amp`ere’s circuital law and

Maxwell’s hypothesis of displacement current; (1.4) is the divergence equa-

tion for the electric ﬁeld, originating from Gauss’s law; and ﬁnally, (1.5) is

the divergence equation for the magnetic ﬁeld, originating from the law of

continuity of magnetic ﬂux. The experimental foundation of Gauss’s law for

the electric ﬁeld is Coulomb’s law; the experimental foundation of Amp`ere’s

circuital law is the Biot–Savart law and Amp`ere’s law of force; and ﬁnally,

the experimental foundation of the law of continuity of magnetic ﬂux is that

there is no experimental evidence to prove the existence of magnetic charges

or so-called monopoles (until now).

In the above equations, % = %(x, t) is the electric charge density, a scalar

function in units of coulombs per cubic meter (C/m

); J = J (x, t) is

the electric current density, a vector function in amperes per square meter

(A/m

). The relation between them is the equation of continuity,

J · dS = −

, ∇ · J = −

∂%

∂t

, (1.6)

where q is the total electric charge in the volume V bounded by the closed

surface S, and we have

q =

% dV, (1.7)

and I is the total current ﬂowing across the cross section S surrounded by

4 1. Basic Electromagnetic Theory

closed contour l, and we have

I =

J · dS. (1.8)

The equation of continuity (1.6) is not independent, it can be derived from

the curl equation (1.3) and the divergence equation (1.4). Alternatively, we

may recognize the equation of continuity (1.6) as a basic law, and then the

divergence equation (1.4) can be derived from (1.6) and the curl equation

of the magnetic ﬁeld (1.3). Similarly, the divergence equation of magnetic

ﬁeld (1.5) is also not independent, it can be derived directly from the curl

equation of the electric ﬁeld (1.2).

In (1.2)–(1.5), E = E(x, t) is the vector function of electric ﬁeld strength,

the unit of E is volts per meter (V/m). The deﬁnition of electric ﬁeld strength

is the electric force exerted on a unit test point charge:

F = qE. (1.9)

The direction of the electric force is the same as that of the electric ﬁeld.

The vector function B = B (x, t) is the magnetic induction or magnetic

ﬂux density, in tesla (T) or weber per square meter (Wb/m

). The deﬁnition

of magnetic induction is based on the magnetic force exerted on a unit test

current element, i.e., the Amp`ere force, or the magnetic force exerted on a

moving point charge with velocity v, i.e., the Lorentz force. The direction

of the magnetic force is perpendicular to both the current element or the

velocity of the charge and the magnetic induction. The Amp`ere force and

the Lorentz force are given as

F = Idl × B, F = qv × B. (1.10)

In Maxwell’s equations, ²

and µ

are two constants, the permittivity and

the permeability of vacuum, respectively. The relation among ²

, µ

and the

speed of light in vacuum, c, is

= 1/µ

, c = 2.99792458 ×10

m/s ≈ 3 × 10

m/s.

In the international system of units (SI), the value of µ

has b een chosen to

= 4π × 10

−7

H/m,

which gives

= 8.85418782 × 10

−12

F/m ≈ (1/36π) × 10

−9

F/m.

The term

∂²

∂t

in equation (1.3) was originally introduced into the curl

equation for magnetic ﬁeld by Maxwell and is called the displacement current

density. Note that, displacement current density is not connected with the

1.1 Maxwell’s Equations 5

movement of electric charge but just the variation of electric ﬁeld with respect

to time, which has the same dimension of current density and has the same

eﬀect in the curl equation of magnetic ﬁeld. During the time of Maxwell, no

experimental veriﬁcation was made for this hypothesis.

Maxwell’s equations are the general description of macroscopic electro-

magnetic phenomenon. From this point of view, Coulomb’s law, the Biot–

Savart law, Amp`ere’s law and Faraday’s law are all particular examples. The

most important equations are the two curl equations including Faraday’s law

of induction and Maxwell’s hypothesis of displacement current. The interac-

tions between an electric ﬁeld and a magnetic ﬁeld in a time-varying state

were described by J.C. Maxwell, as were the wave behavior of electromag-

netic ﬁelds. Furthermore, the speed of an electromagnetic wave was obtained

theoretically from these equations and agrees with the experimental value for

the speed of light, within a small experimental error. J.C. Maxwell predicted

that time-varying electromagnetic ﬁelds exist in the form of waves and light

is an electromagnetic wave phenomenon in a special frequency band. 25 years

after this prediction, 9 years after the death of J.C. Maxwell, the electromag-

netic wave was generated and detected for the ﬁrst time in electromagnetic

experiments by Henrich R. Hertz in 1888, by which Maxwell’s hypothesis

of displacement current was experimentally proved. In 1895, the year after

the death of H.R. Hertz, experiments into the application of electromagnetic

waves in communications were realized by G. Marconi of Italy and A. Popov

of Russia independently and almost simultaneously. Unfortunately, the two

great prophets and forerunners, J.C. Maxwell and H.R. Hertz, didn’t see the

glorious result of their pioneering work.

In material media, Maxwell’s equations in vacuum (1.2)–(1.5) are still

correct, but all kinds of charges and currents must be included in % and J in

the equations. Both the free charge density %

and the bound charge density

produced by the polarization of media are included in the total electric

charge density %. All of the true (or free) current density, J

, including the

conduction current density and the convection current density, the polariza-

tion current density J

produced by the time-varying polarization of the

media and the molecular current density J

produced by the magnetization

of the media are included in the current density J . Thus the basic Maxwell

equations can be rewritten as

∇ × E = −

∂B

∂t

, (1.11)

∇ ×

∂²

∂t

+ J

, (1.12)

∇ · ²

E = %

+ %

, (1.13)

∇ · B = 0. (1.14)

6 1. Basic Electromagnetic Theory

1.1.2 Maxwell’s Equations in Material Media

When an electromagnetic ﬁeld is applied to material media, the phenomena

of conduction, polarization, and magnetization occur.

For conductive media, the conduction current ﬂowing as a result of the

existence of an electric ﬁeld is directly proportional to E,

J = σE, (1.15)

where σ is the conductivity of the material, in siemens per meter (S/m). This

is the diﬀerential form of Ohm’s law.

Polarization causes the appearance of aligned electric dipole moments or a

bound charge density %

. In addition, a time-varying polarization causes the

appearance of the polarization current density J

, which is the result of the

oscillation of bound charges. The result of magnetization is the appearance

of aligned magnetic dipole moments or a molecular current density J

. Two

new vector functions, the polarization vector P and the magnetization vector

M are introduced, where P is the vector sum of the electric dipole moments

per unit volume, i.e., the volume density of electric dipole moment and M

is the vector sum of the magnetic dipole moments per unit volume, i.e., the

volume density of magnetic dipole moment. The relation between P and %

∇ · P = −%

, (1.16)

and the relation between M and J

∇ × M = J

. (1.17)

From the equation of continuity (1.6), we have

∇ · J

= −

∂%

∂t

Substituting (1.16) into it, we obtain

∂P

∂t

. (1.18)

Substituting (1.16), (1.17) and (1.18) into Maxwell’s equations (1.11)–

(1.14) yields

∇ × E = −

∂B

∂t

, (1.19)

∇ ×

− ∇ × M =

∂²

∂t

∂P

∂t

+ J

, (1.20)

∇ · ²

E + ∇ · P = %

, (1.21)

∇ · B = 0. (1.22)

1.1 Maxwell’s Equations 7

The bound charge, the polarization current, and the molecular current dis-

appear in this set of equations, which are replaced by −∇ · P, ∂P/∂t and

∇ × M, respectively.

Introduce two new vector functions D and H. Their deﬁnitions are

D = ²

E + P, (1.23)

H =

− M, or B = µ

(H + M). (1.24)

D denotes the electric induction vector, and is also known as the electric

displacement vector or electric ﬂux density, the unit of D is coulombs per

square meter (C/m

). H denotes the magnetic strength vector, its unit is

amperes per meter (A/m).

For understanding the relations (1.16), (1.17), (1.18), equations (1.19) to

(1.22) and relations (1.23), (1.24), please refer to an undergraduate text on

electromagnetism.

Substituting (1.23) and (1.24) into equations (1.19) to (1.22) , and neglect-

ing the subscript ‘f ’ in %

and J

, we have the following Maxwell equations

and their integral form counterparts

E · dl = −

B · dS, ∇ × E = −

∂B

∂t

(1.25)

H · dl =

D · dS + I, ∇ × H =

∂D

∂t

+ J , (1.26)

D · dS = q, ∇·D = %, (1.27)

B · dS = 0, ∇ · B = 0. (1.28)

These equations are Maxwell’s equations in material media or, exactly,

Maxwell’s equations taking account of the eﬀect of polarization and magne-

tization of media. They are valid in material media as well as in vacuum.

In vacuum, we have P = 0 and M = 0, or D = ²

E and B = µ

H. Note

that only free charges and free (or true) currents are included in the charge

density and the current density terms in Maxwell’s equations in material me-

dia (1.25)–(1.28), but in the basic Maxwell’s equations or so called Maxwell’s

equations in vacuum (1.2)–(1.5), all kinds of charge and currents must be

included.

The term J

∂D

∂t

∂²

∂t

∂P

∂t

in equation (1.26) denotes the displace-

ment current density in material media, includes the variation of electric ﬁeld

and the variation of electric polarization with respect to time. The later rep-

resents the oscillation of bound charges. Equation of the curl of magnetic

ﬁeld (1.26) then becomes

∇ × H = J

+ J

8 1. Basic Electromagnetic Theory

It leads to the law of continuity of total current,

+ J

) · dS = 0.

We leave the proof of this law as an exercise, see Problem 1.4.

Equations (1.23)–(1.28) are universally applicable for any medium. But

when we deal with the relation between P and E, or the relation between M

and H, which are usually called the constitutive relations, the performances

of diﬀerent kinds of media are quite diﬀerent from each other.

(1) Simple Media

The non-dispersive, linear and isotropic media are called simple media. In

simple media, vector P is parallel to and proportional to vector E and vector

M is parallel to and proportional to vector B. The relationships can be

expressed as follows:

P = ²

E, (1.29)

M = χ

H, (1.30)

where χ

is the electric susceptibility and χ

is the magnetic susceptibility

of the medium, both of which are dimensionless quantities.

Substituting (1.29) and (1.30) into (1.23) and (1.24), respectively, we have

D = ²

(1 + χ

)E = ²

E, or D = ²E, (1.31)

B = µ

(1 + χ

)H = µ

H, or B = µH. (1.32)

In the above equations,

= 1 + χ

, ² = ²

= ²

(1 + χ

), (1.33)

= 1 + χ

, µ = µ

= µ

(1 + χ

), (1.34)

where ² is the permittivity, µ is the permeability of the medium, and ²

and

are the relative permittivity and relative permeability, respectively. The

unit of ² is the same as that of ²

and the unit of µ is the same as that of

. ²

and µ

are dimensionless quantities. Equations (1.31) and (1.32) are

constitutive equations or constitutive relations of the simple media, in which

² and µ are constitutive parameters.

In isotropic medium, the electric ﬁeld vector and the polarization vector

are in the same direction, and the susceptibilities in all directions are equal,

so are the magnetic ﬁeld and the magnetization vectors. In isotropic media,

² and µ are scalars. Hence in simple media, both the permittivity ² and the

permeability µ are constant scalars.

1.1 Maxwell’s Equations 9

Substituting (1.31) and (1.32) into (1.25)–(1.28), yields Maxwell’s equa-

tions for stable, uniform simple media:

∇ × E = −µ

∂H

∂t

, (1.35)

∇ × H = ²

∂E

∂t

+ σE + J , (1.36)

∇ · E =

, (1.37)

∇ · H = 0. (1.38)

True (or free) current J in equation (1.26) is divided into two terms in

(1.36). The one is conduction current, which comply with Ohm’s law and is

directly proportional to electric ﬁeld strength, J

= σE. It is considered as

a ﬁeld term. The second term is the current other than the conduction cur-

rent, and is not proportional to the electric ﬁeld, for example, the convection

current caused by the motion of charged particles in vacuum or plasma, and

the source current independent of the ﬁeld or so called impressed current. It

is considered as the source term J

or simply J in (1.36).

(2) Dispersive Media

In dispersive media, the responses of polarization and magnetization are not

instant. The D depends upon not only the present value of E but also

the time derivatives of all orders of E. So does B and H. For isotropic,

linear dispersive media, the constitutive relations become the following linear

diﬀerential equations [37]:

D = ²E + ²

∂E

∂t

+ ²

∂

∂t

+ ²

∂

∂t

+ ···, (1.39)

B = µH + µ

∂H

∂t

+ µ

∂

∂t

+ µ

∂

∂t

+ ···. (1.40)

Equations (1.39) and (1.40) reduce to (1.31) and (1.32), respectively, when

the coeﬃcients of the derivative terms become suﬃciently small, and the

medium can be treated as a non-dispersive medium.

Later, in section 1.1.4, we can see that, not only the dispersion but also

the polarization dissipation and magnetization dissipation are described in

the constitutive relations (1.39) and (1.40).

Detailed discussions of the dispersion of media and the electromagnetic

waves in dispersive media are given in Chapter 8.

10 1. Basic Electromagnetic Theory

(3) Nonlinear Media

In nonlinear media, the relation between D and E and the relation between

B and H are nonlinear. This means that the constitutive parameters ² and

µ depend upon the ﬁeld strengths. In this case, we must return to (1.23) and

(1.24). In practice, the experimental curves of D versus E and B versus H

are established in the following manner

D = f

(E), B = f

(H). (1.41)

An important phenomenon of nonlinear media is the hysteresis of ferro-

magnetic materials [82]. The investigation of nonlinear eﬀects of media in

the optical band, i.e., nonlinear optics, has been highly developed in recent

years [38, 116]. These subjects are not included in this book, refer to [16, 89].

(4) Anisotropic Media

If the electric ﬁeld induces polarization in a direction other than that of the

electric ﬁeld, and the susceptibilities are diﬀerent in diﬀerent directions, the

medium is known as an electric anisotropic medium. Similar behavior in

magnetization is called magnetic anisotropic medium. In anisotropic media,

the electric and/or magnetic susceptibilities are no longer scalars but tensors

of rank 2 or matrices [11, 53, 84].

For the electric anisotropic media, the electric susceptibility becomes a

tensor,

P = ²

· E, χ





exx

exy

exz

eyx

eyy

eyz

ezx

ezy

ezz





. (1.42)

For the magnetic anisotropic media, the magnetic susceptibility becomes a

tensor,

M = χ

· H, χ





mxx

mxy

mxz

myx

myy

myz

mzx

mzy

mzz





. (1.43)

Substituting (1.42) and (1.43) into (1.23) and (1.24), respectively, we have

D = ²

E + ²

· E = ²

(I + χ

) · E, (1.44)

B = µ

H + µ

· H = µ

(I + χ

) · H, (1.45)

where I is the unit tensor,

I =





1 0 0

0 1 0

0 0 1





. (1.46)