Chapter 1

Balanis c01.tex V3 - 11/22/2011 3:03 P.M. Page 1

CHAPTER 1Time-Varying and Time-Harmonic

Electromagnetic Fields

1.1 INTRODUCTION

Electromagnetic field theory is a discipline concerned with the study of charges, at rest and inmotion, that produce currents and electric-magnetic fields. It is, therefore, fundamental to thestudy of electrical engineering and physics and indispensable to the understanding, design, andoperation of many practical systems using antennas, scattering, microwave circuits and devices,radio-frequency and optical communications, wireless communications, broadcasting, geosciencesand remote sensing, radar, radio astronomy, quantum electronics, solid-state circuits and devices,electromechanical energy conversion, and even computers. Circuit theory, a required area in thestudy of electrical engineering, is a special case of electromagnetic theory, and it is valid whenthe physical dimensions of the circuit are small compared to the wavelength. Circuit concepts,which deal primarily with lumped elements, must be modified to include distributed elements andcoupling phenomena in studies of advanced systems. For example, signal propagation, distortion,and coupling in microstrip lines used in the design of sophisticated systems (such as computers andelectronic packages of integrated circuits) can be properly accounted for only by understandingthe electromagnetic field interactions associated with them.

The study of electromagnetics includes both theoretical and applied concepts. The theoreticalconcepts are described by a set of basic laws formulated primarily through experiments conductedduring the nineteenth century by many scientists—Faraday, Ampere, Gauss, Lenz, Coulomb,Volta, and others. They were then combined into a consistent set of vector equations by Maxwell.These are the widely acclaimed Maxwell’s equations . The applied concepts of electromagneticsare formulated by applying the theoretical concepts to the design and operation of practicalsystems.

In this chapter, we will review Maxwell’s equations (both in differential and integral forms),describe the relations between electromagnetic field and circuit theories, derive the boundaryconditions associated with electric and magnetic field behavior across interfaces, relate power andenergy concepts for electromagnetic field and circuit theories, and specialize all these equations,relations, conditions, concepts, and theories to the study of time-harmonic fields.

1.2 MAXWELL’S EQUATIONS

In general, electric and magnetic fields are vector quantities that have both magnitude anddirection. The relations and variations of the electric and magnetic fields, charges, and cur-rents associated with electromagnetic waves are governed by physical laws, which are known

1


2 TIME-VARYING AND TIME-HARMONIC ELECTROMAGNETIC FIELDS

as Maxwell’s equations. These equations, as we have indicated, were arrived at mostly throughvarious experiments carried out by different investigators, but they were put in their final formby James Clerk Maxwell, a Scottish physicist and mathematician. These equations can be writteneither in differential or in integral form.

1.2.1 Differential Form of Maxwell’s Equations

The differential form of Maxwell’s equations is the most widely used representation to solveboundary-value electromagnetic problems. It is used to describe and relate the field vectors, currentdensities, and charge densities at any point in space at any time. For these expressions to be valid,it is assumed that the field vectors are single-valued, bounded, continuous functions of positionand time and exhibit continuous derivatives . Field vectors associated with electromagnetic wavespossess these characteristics except where there exist abrupt changes in charge and current densi-ties. Discontinuous distributions of charges and currents usually occur at interfaces between mediawhere there are discrete changes in the electrical parameters across the interface. The variations ofthe field vectors across such boundaries (interfaces) are related to the discontinuous distributionsof charges and currents by what are usually referred to as the boundary conditions . Thus a com-plete description of the field vectors at any point (including discontinuities) at any time requiresnot only Maxwell’s equations in differential form but also the associated boundary conditions .

In differential form, Maxwell’s equations can be written as

∇ × � = −�i − ∂�

∂t= −�i − �d = −�t (1-1)

∇ × � = �i + �c + ∂�

∂t= �ic + ∂�

∂t= �ic + �d = �t (1-2)

∇ • � = qev

(1-3)

∇ • � = qmv

(1-4)

where

�ic = �i + �c (1-5a)

�d = ∂�

∂t(1-5b)

�d = ∂�

∂t(1-5c)

All these field quantities—�, �, �, �, �, �, and qv

—are assumed to be time-varying, andeach is a function of the space coordinates and time, that is, � = � (x , y , z ; t). The definitionsand units of the quantities are

� = electric field intensity (volts/meter)� = magnetic field intensity (amperes/meter)� = electric flux density (coulombs/square meter)� = magnetic flux density (webers/square meter)�i = impressed (source) electric current density (amperes/square meter)�c = conduction electric current density (amperes/square meter)�d = displacement electric current density (amperes/square meter)�i = impressed (source) magnetic current density (volts/square meter)�d = displacement magnetic current density (volts/square meter)q

ev= electric charge density (coulombs/cubic meter)

qmv

= magnetic charge density (webers/cubic meter)


MAXWELL’S EQUATIONS 3

The electric displacement current density �d = ∂�/∂t was introduced by Maxwell to completeAmpere’s law for statics, ∇ × � = �. For free space, �d was viewed as a motion of boundcharges moving in “ether,” an ideal weightless fluid pervading all space. Since ether proved to beundetectable and its concept was not totally reasonable with the special theory of relativity, it hassince been disregarded. Instead, for dielectrics, part of the displacement current density has beenviewed as a motion of bound charges creating a true current. Because of this, it is convenient toconsider, even in free space, the entire ∂�/∂t term as a displacement current density.

Because of the symmetry of Maxwell’s equations, the ∂�/∂t term in (1-1) has been des-ignated as a magnetic displacement current density. In addition, impressed (source) magneticcurrent density �i and magnetic charge density q

mvhave been introduced, respectively, in

(1-1) and (1-4) through the “generalized” current concept. Although we have been accustomed toviewing magnetic charges and impressed magnetic current densities as not being physically real-izable, they have been introduced to balance Maxwell’s equations. Equivalent magnetic chargesand currents will be introduced in later chapters to represent physical problems. In addition,impressed magnetic current densities, like impressed electric current densities, can be consideredenergy sources that generate fields whose expressions can be written in terms of these currentdensities. For some electromagnetic problems, their solution can often be aided by the introduc-tion of “equivalent” impressed electric and magnetic current densities. The importance of bothwill become more obvious to the reader as solutions to specific electromagnetic boundary-valueproblems are considered in later chapters. However, to give the reader an early glimpse of theimportance and interpretation of the electric and magnetic current densities, let us consider twofamiliar circuit examples.

In Figure 1-1a , an electric current source is connected in series to a resistor and a parallel-plate capacitor. The electric current density �i can be viewed as the current source that generatesthe conduction current density �c through the resistor and the displacement current density �dthrough the dielectric material of the capacitor. In Figure 1-1b, a voltage source is connected to awire that, in turn, is wrapped around a high-permeability magnetic core. The voltage source canbe viewed as the impressed magnetic current density that generates the displacement magneticcurrent density through the magnetic material of the core.

In addition to the four Maxwell’s equations, there is another equation that relates the variationsof the current density �ic and the charge density q

ev. Although not an independent relation, this

equation is referred to as the continuity equation because it relates the net flow of current out ofa small volume (in the limit, a point) to the rate of decrease of charge. It takes the form

∇ • �ic = −∂qev

∂t(1-6)

The continuity equation 1-6 can be derived from Maxwell’s equations as given by (1-1) through(1-5c).

1.2.2 Integral Form of Maxwell’s Equations

The integral form of Maxwell’s equations describes the relations of the field vectors, chargedensities, and current densities over an extended region of space. They have limited applications,and they are usually utilized only to solve electromagnetic boundary-value problems that possesscomplete symmetry (such as rectangular, cylindrical, spherical, etc., symmetries). However, thefields and their derivatives in question do not need to possess continuous distributions .

The integral form of Maxwell’s equations can be derived from its differential form by utilizingthe Stokes’ and divergence theorems . For any arbitrary vector A, Stokes’ theorem states that theline integral of the vector A along a closed path C is equal to the integral of the dot product ofthe curl of the vector A with the normal to the surface S that has the contour C as its boundary .



Dielectricslab

(a)

(b)

�i

�c

�d

�i

�d

Figure 1-1 Circuits with electric and magnetic current densities. (a) Electric current density. (b) Magneticcurrent density.

In equation form, Stokes’ theorem can be written as∮C

A • d� =∫∫

S(∇ × A) • ds (1-7)

The divergence theorem states that, for any arbitrary vector A, the closed surface integral of thenormal component of vector A over a surface S is equal to the volume integral of the divergence ofA over the volume V enclosed by S . In mathematical form, the divergence theorem is stated as

#SA • ds =

∫∫∫V

∇ • A dv (1-8)

Taking the surface integral of both sides of (1-1), we can write∫∫S(∇ × �) • ds = −

∫∫S

�i • ds −∫∫

S

∂�

∂t• ds = −

∫∫S

�i • ds − ∂

∂t

∫∫S

�i • ds (1-9)

Applying Stokes’ theorem, as given by (1-7), on the left side of (1-9) reduces it to∮C

� • d� = −∫∫

S�i • ds − ∂

∂t

∫∫S

� • ds (1-9a)


CONSTITUTIVE PARAMETERS AND RELATIONS 5

which is referred to as Maxwell’s equation in integral form as derived from Faraday’s law . In theabsence of an impressed magnetic current density, Faraday’s law states that the electromotiveforce (emf) appearing at the open-circuited terminals of a loop is equal to the time rate of decreaseof magnetic flux linking the loop.

Using a similar procedure, we can show that the corresponding integral form of (1-2) can bewritten as ∮

C� • d� =

∫∫S

�ic • ds + ∂

∂t

∫∫S

� • ds =∫∫

S�ic • ds +

∫∫S

�d • ds (1-10)

which is usually referred to as Maxwell’s equation in integral form as derived from Ampere’s law.Ampere’s law states that the line integral of the magnetic field over a closed path is equal to thecurrent enclosed.

The other two Maxwell equations in integral form can be obtained from the correspondingdifferential forms, using the following procedure. First take the volume integral of both sides of(1-3); that is, ∫∫∫

V∇ • � dv =

∫∫∫V

qev

dv = �e (1-11)

where �e is the total electric charge. Applying the divergence theorem, as given by (1-8), on theleft side of (1-11) reduces it to

#S� • ds =

∫∫∫V

qev

dv = �e (1-11a)

which is usually referred to as Maxwell’s electric field equation in integral form as derived fromGauss’s law. Gauss’s law for the electric field states that the total electric flux through a closedsurface is equal to the total charge enclosed.

In a similar manner, the integral form of (1-4) is given in terms of the total magnetic charge�m by

#S� • ds = �m (1-12)

which is usually referred to as Maxwell’s magnetic field equation in integral form as derived fromGauss’s law . Even though magnetic charge does not exist in nature, it is used as an equivalentto represent physical problems. The corresponding integral form of the continuity equation, asgiven by (1-6) in differential form, can be written as

#S�ic • ds = − ∂

∂t

∫∫∫V

qev

dv = −∂�e

∂t(1-13)

Maxwell’s equations in differential and integral form are summarized and listed in Table 1-1.

1.3 CONSTITUTIVE PARAMETERS AND RELATIONS

Materials contain charged particles, and when these materials are subjected to electromagneticfields, their charged particles interact with the electromagnetic field vectors, producing currentsand modifying the electromagnetic wave propagation in these media compared to that in freespace. A more complete discussion of this is in Chapter 2. To account on a macroscopic scale forthe presence and behavior of these charged particles, without introducing them in a microscopiclattice structure, we give a set of three expressions relating the electromagnetic field vectors.These expressions are referred to as the constitutive relations , and they will be developed inmore detail in Chapter 2.



TABLE 1-1 Maxwell’s equations and the continuity equation in differential and integral forms fortime-varying fields

Differential form Integral form

∇ × � = −�i − ∂�∂t

∮C

� · d� = −∫∫

S�i · ds − ∂

∂t

∫∫S

� · ds

∇ × � = �i + �c + ∂�∂t

∮C

� · d� =∫∫

S�i · ds +

∫∫S

�c · ds + ∂

∂t

∫∫S

� · ds

∇ · � = qev #S

� · ds = �e

∇ · � = qmv #S

� · ds = �m

∇ · �ic = −∂qev

∂t #S�ic · ds = − ∂

∂t

∫∫∫V

qev

dv = −∂�e

∂t

One of the constitutive relations relates in the time domain the electric flux density � to theelectric field intensity � by

� = ε ∗ � (1-14)

where ε is the time-varying permittivity of the medium (farads/meter) and ∗ indicates convolution.For free space

ε = ε0 = 8.854×10−12 � 10−9

36π(farads/meter) (1-14a)

and (1-14) reduces to a product.Another relation equates in the time domain the magnetic flux density � to the magnetic field

intensity � by� = μ ∗ � (1-15)

where μ is the time-varying permeability of the medium (henries/meter). For free space

μ = μ0 = 4π×10−7 (henries/meter) (1-15a)

and (1-15) reduces to a product.Finally, the conduction current density �c is related in the time domain to the electric field

intensity � by�c = σ ∗ � (1-16)

where σ is the time-varying conductivity of the medium (siemens/meter). For free space

σ = 0 (1-16a)

In the frequency domain or for frequency nonvarying constitutive parameters, the relations (1-14),(1-15) and (1-16) reduce to products. For simplicity of notation, they will be indicated everywherefrom now on as products, and the caret ( ) in the time-varying constitutive parameters will beomitted.

Whereas (1-14), (1-15), and (1-16) are referred to as the constitutive relations , ε, μ and σ arereferred to as the constitutive parameters , which are, in general, functions of the applied fieldstrength, the position within the medium, the direction of the applied field, and the frequency ofoperation.


CIRCUIT-FIELD RELATIONS 7

The constitutive parameters are used to characterize the electrical properties of a material.In general, materials are characterized as dielectrics (insulators), magnetics , and conductors ,depending on whether polarization (electric displacement current density), magnetization (mag-netic displacement current density), or conduction (conduction current density) is the predominantphenomenon. Another class of material is made up of semiconductors , which bridge the gapbetween dielectrics and conductors where neither displacement nor conduction currents are, ingeneral, predominant. In addition, materials are classified as linear versus nonlinear, homoge-neous versus nonhomogeneous (inhomogeneous), isotropic versus nonisotropic (anisotropic), anddispersive versus nondispersive, according to their lattice structure and behavior. All these typesof materials will be discussed in detail in Chapter 2.

If all the constitutive parameters of a given medium are not functions of the applied fieldstrength, the material is known as linear ; otherwise it is nonlinear . Media whose constitu-tive parameters are not functions of position are known as homogeneous; otherwise they arereferred to as nonhomogeneous (inhomogeneous). Isotropic materials are those whose constitu-tive parameters are not functions of direction of the applied field; otherwise they are designatedas nonisotropic (anisotropic). Crystals are one form of anisotropic material. Material whose con-stitutive parameters are functions of frequency are referred to as dispersive; otherwise they areknown as nondispersive. All materials used in our everyday life exhibit some degree of disper-sion, although the variations for some may be negligible and for others significant. More detailsconcerning the development of the constitutive parameters can be found in Chapter 2.

1.4 CIRCUIT-FIELD RELATIONS

The differential and integral forms of Maxwell’s equations were presented, respectively, inSections 1.2.1 and 1.2.2. These relations are usually referred to as field equations , since thequantities appearing in them are all field quantities . Maxwell’s equations can also be written interms of what are usually referred to as circuit quantities; the corresponding forms are denotedcircuit equations . The circuit equations are introduced in circuit theory texts, and they are specialcases of the more general field equations.

1.4.1 Kirchhoff’s Voltage Law

According to Maxwell’s equation 1-9a, the left side represents the sum voltage drops (use theconvention where positive voltage begins at the start of the path) along a closed path C , whichcan be written as ∑

v =∮

C� • d� (volts) (1-17)

The right side of (1-9a) must also have the same units (volts) as its left side. Thus, in the absenceof impressed magnetic current densities (�i = 0), the right side of (1-9a) can be written as

− ∂

∂t

∫∫S

� • ds = −∂ψm

∂t= − ∂

∂t(Ls i ) = −Ls

∂i

∂t(webers/second = volts) (1-17a)

because by definition ψm = Ls i where Ls is an inductance (assumed to be constant) and i is theassociated current. Using (1-17) and (1-17a), we can write (1-9a) with �i = 0 as∑

v = −∂ψm

∂t= − ∂

∂t(Ls i ) = −Ls

∂i

∂t(1-17b)

Equation 1-17b states that the voltage drops along a closed path of a circuit are equal to the timerate of change of the magnetic flux passing through the surface enclosed by the closed path, or



equal to the voltage drop across an inductor Ls that is used to represent the stray inductance ofthe circuit. This is the well-known Kirchhoff loop voltage law , which is used widely in circuittheory, and its form represents a circuit relation. Thus we can write the following field and circuitrelations:

Field Relation Circuit Relation∮C

� • d� = − ∂

∂t

∫∫S

� • ds = −∂ψm

∂t⇔

∑v = −∂ψm

∂t= −Ls

∂i

∂t(1-17c)

In lumped-element circuit analysis, where usually the wavelength is very large (or the dimen-sions of the total circuit are small compared to the wavelength) and the stray inductance of thecircuit is very small, the right side of (1-17b) is very small and it is usually set equal to zero. Inthese cases, (1-17b) states that the voltage drops (or rises) along a closed path are equal to zero,and it represents a widely used relation to electrical engineers and many physicists.

To demonstrate Kirchhoff’s loop voltage law, let us consider the circuit of Figure 1-2 wherea voltage source and three ideal lumped elements (a resistance R, an inductor L, and a capacitorC ) are connected in series to form a closed loop. According to (1-17b)

−vs + vR + vL + vC = −Ls∂i

∂t= −vsL (1-18)

where Ls , shown dashed in Figure 1-2, represents the total stray inductance associated with thecurrent and the magnetic flux generated by the loop that connects the ideal lumped elements (weassume that the wire resistance is negligible). If the stray inductance Ls of the circuit and thetime rate of change of the current is small (the case for low-frequency applications), the rightside of (1-18) is small and can be set equal to zero.

1.4.2 Kirchhoff’s Current Law

The left side of the integral form of the continuity equation, as given by (1-13), can be writtenin circuit form as ∑

i =#S

�ic • ds (1-19)

i

R

vR

vL

vC

LsvsL

vs

L

C

− +

+

−

+

+

−

−

−

+

Figure 1-2 RLC series network.


CIRCUIT-FIELD RELATIONS 9

where∑

i represents the sum of the currents passing through closed surface S . Using (1-19)reduces (1-13) to ∑

i = −∂�e

∂t= − ∂

∂t(Csv) = −Cs

∂v

∂t(1-19a)

since by definition �e = Csv where Cs is a capacitance (assumed to be constant) and v is theassociated voltage.

Equation 1-19a states that the sum of the currents crossing a surface that encloses a circuitis equal to the time rate of change of the total electric charge enclosed by the surface, or equalto the current flowing through a capacitor Cs that is used to represent the stray capacitance ofthe circuit. This is the well-known Kirchhoff node current law, which is widely used in circuittheory, and its form represents a circuit relation. Thus, we can write the following field and circuitrelations:

Field Relation Circuit Relation

#S�ic • ds = − ∂

∂t

∫∫∫V

qev

dv = −∂�e

∂t⇔

∑i = −∂�e

∂t= −Cs

∂v

∂t(1-19b)

In lumped-element circuit analysis, where the stray capacitance associated with the circuit is verysmall, the right side of (1-19a) is very small and it is usually set equal to zero. In these cases,(1-19a) states that the currents exiting (or entering) a surface enclosing a circuit are equal to zero.This represents a widely used relation to electrical engineers and many physicists.

To demonstrate Kirchhoff’s node current law, let us consider the circuit of Figure 1-3 wherea current source and three ideal lumped elements (a resistance R, an inductor L, and a capacitorC ) are connected in parallel to form a node. According to (1-19a)

−is + iR + iL + iC = −Cs∂v

∂t= −isC (1-20)

where Cs , shown dashed in Figure 1-3, represents the total stray capacitance associated with thecircuit of Figure 1-3. If the stray capacitance Cs of the circuit and the time rate of change ofthe total charge �e are small (the case for low-frequency applications), the right side of (1-20) issmall and can be set equal to zero. The current isC associated with the stray capacitance Cs also

vCLCsis

is

iR isC

iL

iC

S

R

+

−

Figure 1-3 RLC parallel network.



includes the displacement (leakage) current crossing the closed surface S of Figure 1-3 outside ofthe wires .

1.4.3 Element Laws

In addition to Kirchhoff’s loop voltage and node current laws as given, respectively, by (1-17b)and (1-19a), there are a number of current element laws that are widely used in circuit theory.One of the most popular is Ohm’s law for a resistor (or a conductance G), which states that thevoltage drop vR across a resistor R is equal to the product of the resistor R and the current iRflowing through it (vR = RiR or iR = vR/R = GvR). Ohm’s law of circuit theory is a special caseof the constitutive relition given by (1-16). Thus


�c = σ� ⇔ iR = 1

RvR = GvR

(1-21)

Another element law is associated with an inductor L and states that the voltage drop across aninductor is equal to the product of L and the time rate of change of the current through the inductor(vL = L diL/dt). Before proceeding to relate the inductor’s voltage drop to the corresponding fieldrelation, let us first define inductance. To do this we state that the magnetic flux ψm is equal to theproduct of the inductance L and the corresponding current i . That is ψm = Li . The correspondingfield equation of this relation is (1-15). Thus

Field Relation Circuit Relation� = μ� ⇔ ψm = LiL

(1-22)

Using (1-5c) and (1-15), we can write for a homogeneous and non-time-varying medium that

�d = ∂�

∂t= ∂

∂t(μ�) = μ

∂�

∂t(1-22a)

where �d is defined as the magnetic displacement current density [analogous to the electricdisplacement current density �d = ∂�/∂t = ∂(ε�)/∂t = ε∂�/∂t]. With the aid of the right sideof (1-9a) and the circuit relation of (1-22), we can write

∂

∂t

∫∫S

� • ds = ∂ψm

∂t= ∂

∂t(LiL) = L

∂iL∂t

= vL (1-22b)

Using (1-22a) and (1-22b), we can write the following relations:


�d = μ∂�

∂t⇔ vL = L

∂iL∂t

(1-22c)

Using a similar procedure for a capacitor C , we can write the field and circuit relationsanalogous to (1-22) and (1-22c):


� = ε� ⇔ �e = Cve (1-23)

�d = ε∂�

∂t⇔ iC = C

∂vC

∂t(1-24)

A summary of the field theory relations and their corresponding circuit concepts are listed inTable 1-2.


TA

BL

E1-

2R

elat

ions

betw

een

elec

trom

agne

tic

field

and

circ

uit

theo

ries

Fie

ldth

eory

Cir

cuit

theo

ry

1.�

(ele

ctri

cfie

ldin

tens

ity)

1.v

(vol

tage

)

2.�

(mag

netic

field

inte

nsity

)2.

i(c

urre

nt)

3.�

(ele

ctri

cflu

xde

nsity

)3.

q ev(e

lect

ric

char

gede

nsity

)

4.�

(mag

netic

flux

dens

ity)

4.q m

v(m

agne

ticch

arge

dens

ity)

5.�

(ele

ctri

ccu

rren

tde

nsity

)5.

i e(e

lect

ric

curr

ent)

6.�

(mag

netic

curr

ent

dens

ity)

6.i m

(mag

netic

curr

ent)

7.�

d=

ε∂� ∂t

(ele

ctri

cdi

spla

cem

ent

curr

ent

dens

ity)

7.i

=C

dv dt

(cur

rent

thro

ugh

aca

paci

tor)

8.�

d=

μ∂� ∂t

(mag

netic

disp

lace

men

tcu

rren

tde

nsity

)8.

v=

Ldi dt

(vol

tage

acro

ssan

indu

ctor

)

9.C

onst

itut

ive

rela

tion

s9.

Ele

men

tla

ws

(a)

�c

=σ

�(e

lect

ric

cond

uctio

ncu

rren

tde

nsity

)(a

)i

=G

v=

1 Rv

(Ohm

’sla

w)

(b)

�=

ε�

(die

lect

ric

mat

eria

l)(b

)�

e=

Cv

(cha

rge

ina

capa

cito

r)

(c)

�=

μ�

(mag

netic

mat

eria

l)(c

)ψ

=L

i(fl

uxof

anin

duct

or)

10.∮ C

�·d

�=

−∂ ∂t

∫∫ S�

·ds

(Max

wel

l–Fa

rada

yeq

uatio

n)10

.∑ v

=−L

s∂

i

∂t

�0

(Kir

chho

ff’s

volta

gela

w)

11.#

S�

ic·d

s=

−∂ ∂t

∫∫∫ Vq ev

dv

=−

∂�

e

∂t

11.∑ i

=−

∂�

e

∂t

=−C

s∂v

∂t

�0

(con

tinui

tyeq

uatio

n)(K

irch

hoff

’scu

rren

tla

w)

12.

Pow

eran

den

ergy

dens

itie

s12

.P

ower

and

ener

gy

(a)#

S(�

×�

)·d

s(i

nsta

ntan

eous

pow

er)

(a)

�=

vi

(pow

er–

volta

ge–

curr

ent

rela

tion)

(b)

∫∫∫ Vσ

�2dv

(dis

sipa

ted

pow

er)

(b)

�d

=G

v2

=1 R

v2

(pow

erdi

ssip

ated

ina

resi

stor

)

(c)

1 2

∫∫∫ Vε�

2dv

(ele

ctri

cst

ored

ener

gy)

(c)

1 2C

v2

(ene

rgy

stor

edin

aca

paci

tor)

(d)

1 2

∫∫∫ Vμ

�2dv

(mag

netic

stor

eden

ergy

)(d

)1 2

Li2

(ene

rgy

stor

edin

anin

duct

or)

11



1.5 BOUNDARY CONDITIONS

As previously stated, the differential form of Maxwell’s equations are used to solve for the fieldvectors provided the field quantities are single-valued, bounded, and possess (along with theirderivatives) continuous distributions. Along boundaries where the media involved exhibit discon-tinuities in electrical properties (or there exist sources along these boundaries), the field vectorsare also discontinuous and their behavior across the boundaries is governed by the boundaryconditions .

Maxwell’s equations in differential form represent derivatives, with respect to the space coor-dinates, of the field vectors. At points of discontinuity in the field vectors, the derivatives of thefield vectors have no meaning and cannot be properly used to define the behavior of the fieldvectors across these boundaries. Instead, the behavior of the field vectors across discontinuousboundaries must be handled by examining the field vectors themselves and not their derivatives.The dependence of the field vectors on the electrical properties of the media along boundaries ofdiscontinuity is manifested in our everyday life. It has been observed that cell phone, radio, ortelevision reception deteriorates or even ceases as we move from outside to inside an enclosure(such as a tunnel or a well-shielded building). The reduction or loss of the signal is governednot only by the attenuation as the signal/wave travels through the medium, but also by its behav-ior across the discontinuous interfaces. Maxwell’s equations in integral form provide the mostconvenient formulation for derivation of the boundary conditions.

1.5.1 Finite Conductivity Media

Initially, let us consider an interface between two media, as shown in Figure 1-4a , along whichthere are no charges or sources. These conditions are satisfied provided that neither of the twomedia is a perfect conductor or that actual sources are not placed there. Media 1 and 2 arecharacterized, respectively, by the constitutive parameters ε1, μ1, σ1 and ε2, μ2, σ2.

At a given point along the interface, let us choose a rectangular box whose boundary is denotedby C0 and its area by S0. The x, y, z coordinate system is chosen to represent the local geometryof the rectangle. Applying Maxwell’s equation 1-9a, with �i = 0, on the rectangle along C0 andon S0, we have ∮

C0

� • d� = − ∂

∂t

∫∫S0

� • ds (1-25)

As the height �y of the rectangle becomes progressively shorter, the area S0 also becomesvanishingly smaller so that the contributions of the surface integral in (1-25) are negligible. Inaddition, the contributions of the line integral in (1-25) along �y are also minimal, so that in thelimit (�y → 0), (1-25) reduces to

�1 • ax�x − �2 • ax�x = 0

�1t − �2t = 0 ⇒ �1t = �2t (1-26)

or

n × (�2 − �1) = 0 σ1, σ2 are finite (1-26a)

In (1-26), �1t and �2t represent, respectively, the tangential components of the electric field inmedia 1 and 2 along the interface. Both (1-26) and (1-26a) state that the tangential componentsof the electric field across an interface between two media, with no impressed magnetic currentdensities along the boundary of the interface, are continuous .


BOUNDARY CONDITIONS 13

e 2,m2, s2

e 1,m1, s1

e 2,m2, s2

e 1,m1, s1

(a)

(b)

z

x

y

Δy

zΔx

Δy x

y

C0

A0

A1

S0

1

1

2

2

n

n

n

n

�2, �2, �2, �2

�1, �1, �1, �1

�2, �2, �2, �2

�1, �1, �1, �1

Figure 1-4 Geometry for boundary conditions of tangential and normal components. (a) Tangential.(b) Normal.

Using a similar procedure on the same rectangle but for (1-10), assuming �i = 0, we can writethat

�1t − �2t = 0 ⇒ �1t = �2t (1-27)

orn × (�2 − �1) = 0 σ1, σ2 are finite (1-27a)

which state that the tangential components of the magnetic field across an interface between twomedia, neither of which is a perfect conductor, are continuous . This relation also holds if eitheror both media possess finite conductivity. Equations 1-26a and 1-27a must be modified if eitherof the two media is a perfect conductor or if there are impressed (source) current densities alongthe interface. This will be done in the pages that follow.

In addition to the boundary conditions on the tangential components of the electric and mag-netic fields across an interface, their normal components are also related. To derive these relations,



let us consider the geometry of Figure 1-4b where a cylindrical pillbox is chosen at a given pointalong the interface. If there are no charges along the interface, which is the case when there areno sources or either of the two media is not a perfect conductor, (1-11a) reduces to

#A0,A1

� • ds = 0 (1-28)

As the height �y of the pillbox becomes progressively shorter, the total circumferential area A1

also becomes vanishingly smaller, so that the contributions to the surface integral of (1-28) byA1 are negligible. Thus (1-28) can be written, in the limit (�y → 0), as

�2 • ay A0 − �1 • ay A0 = 0

�2n − �in = 0 ⇒ �2n = �1n (1-29)

orn • (�2 − �1) = 0 σ1, σ2 are finite (1-29a)

In (1-29), �1n and �2n represent, respectively, the normal components of the electric flux densityin media 1 and 2 along the interface. Both (1-29) and (1-29a) state that the normal componentsof the electric flux density across an interface between two media, both of which are imperfectelectric conductors and where there are no sources, are continuous . This relation also holds ifeither or both media possess finite conductivity. Equation 1-29a must be modified if either of themedia is a perfect conductor or if there are sources along the interface. This will be done in thepages that follow.

In terms of the electric field intensities, (1-29) and (1-29a) can be written as

ε2�2n = ε1�1n ⇒ �2n = ε1

ε2�1n ⇒ �1n = ε2

ε1�2n (1-30)

n • (ε2�2 − ε1�1) = 0 σ1, σ2 are finite (1-30a)

which state that the normal components of the electric field intensity across an interface arediscontinuous .

Using a similar procedure on the same pillbox, but for (1-12) with no charges along theinterface, we can write that

�2n − �1n = 0 ⇒ �2n = �1n (1-31)

n • (�2 − �1) = 0 (1-31a)

which state that the normal components of the magnetic flux density, across an interface betweentwo media where there are no sources, are continuous . In terms of the magnetic field intensities,(1-31) and (1-31a) can be written as

μ2�2n = μ1�1n ⇒ �2n = μ1

μ2�1n ⇒ �1n = μ2

μ1�2n (1-32)

n • (μ2�2 − μ1�1) = 0 (1-32a)

which state that the normal components of the magnetic field intensity across an interface arediscontinuous .



1.5.2 Infinite Conductivity Media

If actual electric sources and charges exist along the interface between the two media, or if eitherof the two media forming the interface displayed in Figure 1-4 is a perfect electric conductor(PEC), the boundary conditions on the tangential components of the magnetic field [stated by(1-27a)] and on the normal components of the electric flux density or normal components ofthe electric field intensity [stated by (1-29a) or (1-30a)] must be modified to include the sourcesand charges or the induced linear electric current density (�s) and surface electric charge density(q

es). Similar modifications must be made to (1-26a), (1-31a), and (1-32a) if magnetic sources

and charges exist along the interface between the two media, or if either of the two media is aperfect magnetic conductor (PMC).

To derive the appropriate boundary conditions for such cases, let us refer first to Figure 1-4a and assume that on a very thin layer along the interface there exists an electric surfacecharge density q

es(C/m2) and linear electric current density �s (A/m). Applying (1-10) along the

rectangle of Figure 1-4a , we can write that∮C0

� • d� =∫∫

S0

�ic • ds + ∂

∂t

∫∫S0

� • ds (1-33)

In the limit as the height of the rectangle is shrinking, the left side of (1-33) reduces to

lim�y→0

∮C0

� • d� = (�1 − �2) • ax�x (1-33a)

Since the electric current density �ic is confined on a very thin layer along the interface, the firstterm on the right side of (1-33) can be written as

lim�y→0

∫∫S0

�ic • ds

= lim�y→0

[�ic • az �x�y] = lim�y→0

[(�ic�y) • az �x ] = �s • az �x (1-33b)

Since S0 becomes vanishingly smaller as �y → 0, the last term on the right side of (1-33) reducesto

lim�y→0

∂

∂t

∫∫S0

� • ds = lim�y→0

∂

∂t

∫∫S0

� • az ds = 0 (1-33c)

Substituting (1-33a) through (1-33c) into (1-33), we can write it as

(�1 − �2) • ax�x = �s • az �x

or(�1 − �2) • ax − �s • az = 0 (1-33d)

Sinceax = ay × az (1-34)

(1-33d) can be written as(�1 − �2) • (ay × az ) − �s • az = 0 (1-35)

Using the vector identityA • B × C = C • A × B (1-36)



on the first term in (1-35), we can then write it as

az • [(�1 − �2) × ay ] − �s • az = 0 (1-37)

or{[ay × (�2 − �1)] − �s} • az = 0 (1-37a)

Equation 1-37a is satisfied provided

ay × (�2 − �1) − �s = 0 (1-38)

oray × (�2 − �1) = �s (1-38a)

Similar results are obtained if the rectangles chosen are positioned in other planes. Therefore,we can write an expression on the boundary conditions of the tangential components of themagnetic field, using the geometry of Figure 1-4a , as

n × (�2 − �1) = �s (1-39)

Equation 1-39 states that the tangential components of the magnetic field across an interface, alongwhich there exists a surface electric current density �s (A/m), are discontinuous by an amountequal to the electric current density .

If either of the two media is a perfect electric conductor (PEC), (1-39) must be reduced toaccount for the presence of the conductor. Let us assume that medium 1 in Figure 1-4a possessesan infinite conductivity (σ1 = ∞). With such conductivity �1 = 0, and (1-26a) reduces to

n × �2 = 0 ⇒ �2t = 0 (1-40)

Then (1-1) can be written as

∇ × �1 = 0 = −∂�1

∂t⇒ �1 = 0 ⇒ �1 = 0 (1-41)

provided μ1 is finite.In a perfect electric conductor, its free electric charges are confined to a very thin layer on the

surface of the conductor, forming a surface charge density qes

(with units of coulombs/squaremeter). This charge density does not include bound (polarization) charges (which contributeto the polarization surface charge density) that are usually found inside and on the surface ofdielectric media and form atomic dipoles having equal and opposite charges separated by anassumed infinitesimal distance. Here, instead, the surface charge density q

esrepresents actual

electric charges separated by finite dimensions from equal quantities of opposite charge.When the conducting surface is subjected to an applied electromagnetic field, the electric

surface charges are subjected to electric field Lorentz forces. These charges are set in motion andthus create a surface electric current density �s with units of amperes per meter. The surfacecurrent density �s also resides in a vanishingly thin layer on the surface of the conductor so thatin the limit, as �y → 0 in Figure 1-4a , the volume electric current density � (A/m2) reduces to

lim�y→0

(��y) = �s (1-42)

Then the boundary condition of (1-39) reduces, using (1-41) and (1-42), to

n × �2 = �s ⇒ �2t = �s (1-43)



which states that the tangential components of the magnetic field intensity are discontinuous nextto a perfect electric conductor by an amount equal to the induced linear electric current density .

The boundary conditions on the normal components of the electric field intensity, and theelectric flux density on an interface along which a surface charge density q

esresides on a very

thin layer, can be derived by applying the integrals of (1-11a) on a cylindrical pillbox as shownin Figure 1-4b. Then we can write (1-11a) as

lim�y→0#A0, A1

� • ds = lim�y→0

∫∫∫V

qeν

dν (1-44)

Since the cylindrical surface A1 of the pillbox diminishes as �y → 0, its contributions to thesurface integral vanish. Thus we can write (1-44) as

(�2 − �1) • n A0 = lim�y→0

[(qeν

�y)A0] = qes

A0 (1-45)

which reduces to

n • (�2 − �1) = qes

⇒ �2n − �1n = qes

(1-45a)

Equation 1-45a states that the normal components of the electric flux density on an interface,along which a surface charge density resides, are discontinuous by an amount equal to the surfacecharge density .

In terms of the normal components of the electric field intensity, (1-45a) can be written as

n • (ε2�2 − ε1�1) = qes

(1-46)

which also indicates that the normal components of the electric field are discontinuous across aboundary along which a surface charge density resides .

If either of the media is a perfect electric conductor (PEC) (assuming that medium 1 possessesinfinite conductivity σ1 = ∞), (1-45a) and (1-46) reduce, respectively, to

n • �2 = qes

⇒ �2n = qes

(1-47a)

n • �2 = qes

/ε2 ⇒ �2n = qes

/ε2 (1-47b)

Both (1-47a) and (1-47b) state that the normal components of the electric flux density, and corre-sponding electric field intensity, are discontinuous next to a perfect electric conductor .

1.5.3 Sources Along Boundaries

If electric and magnetic sources (charges and current densities) are present along the interfacebetween the two media with neither one being a perfect conductor, the boundary conditions onthe tangential and normal components of the fields can be written, in general form, as

−n × (�2 − �1) = �s (1-48a)

n × (�2 − �1) = �s (1-48b)

n • (�2 − �1) = qes

(1-48c)

n • (�2 − �1) = qms

(1-48d)



TABLE 1-3 Boundary conditions on instantaneous electromagnetic fields

Finiteconductivity media, Medium 1 of Medium 1 of

no sources or infinite electric infinitecharges conductivity magnetic

σ1, σ2 �= ∞ (�1 = �1 = 0) conductivity�s = 0; q

es= 0 σ1 = ∞; σ2 �= ∞ (�1 = �1 = 0)

General �s = 0; qms

= 0 �s = 0; qms

= 0 �s = 0; qes

= 0

Tangentialelectric fieldintensity

−n × (�2 − �1) = �s n × (�2 − �1) = 0 n × �2 = 0 −n × �2 = �s

Tangentialmagnetic fieldintensity

n × (�2 − �1) = �s n × (�2 − �1) = 0 n × �2 = �s n × �2 = 0

Normal electricflux density

n · (�2 − �1) = qes

n · (�2 − �1) = 0 n · �2 = qes

n · �2 = 0

Normal magneticflux density

n · (�2 − �1) = qms

n · (�2 − �1) = 0 n · �2 = 0 n · �2 = qms

where (�s , �s ) and (qms

, qes

) are the magnetic and electric linear (per meter) current and surface(per square meter) charge densities, respectively. The derivation of (1-48a) and (1-48d) proceedsalong the same lines, respectively, as the derivation of (1-48b) and (1-48c) in Section 1.5.2, butbegins with (1-9a) and (1-12).

A summary of the boundary conditions on all the field components is found in Table 1-3,which also includes the boundary conditions assuming that medium 1 is a perfect magneticconductor (PMC). In general, a magnetic conductor is defined as a material inside of which bothtime-varying electric and magnetic fields vanish when it is subjected to an electromagnetic field.The tangential components of the magnetic field also vanish next to its surface. In addition, themagnetic charge moves to the surface of the material and creates a magnetic current density thatresides on a very thin layer at the surface. Although such materials do not physically exist, theyare often used in electromagnetics to develop electrical equivalents that yield the same answersas the actual physical problems. PMCs can be synthesized approximately over a limited frequencyrange (band-gap); see Section 8.8 .

1.6 POWER AND ENERGY

In a wireless communication system, electromagnetic fields are used to transport information overlong distances. To accomplish this, energy must be associated with electromagnetic fields. Thistransport of energy is accomplished even in the absence of any intervening medium.

To derive the equations that indicate that energy (and forms of it) is associated with electro-magnetic waves, let us consider a region V characterized by ε, μ, σ and enclosed by the surface S ,as shown in Figure 1-5. Within that region there exist electric and magnetic sources represented,respectively, by the electric and magnetic current densities �i and �i . The fields generated by�i and �i that exist within S are represented by �, �. These fields obey Maxwell’s equations,and we can write using (1-1) and (1-2) that

∇ × � = −�i − ∂�

∂t= −�i − μ

∂�

∂t= −�i − �d (1-49a)


POWER AND ENERGY 19

S

V

e, m, s

n

�, �

�, ��i

�i

Figure 1-5 Electric and magnetic fields within S generated by �i and �i .

∇ × � = �i + �c + ∂�

∂t= �i + σ� + ε

∂�

∂t= �i + �c + �d (1-49b)

Scalar multiplying (1-49a) by � and (1-49b) by �, we can write that

� • (∇ × �) = −� • (�i + �d ) (1-50a)

� • (∇ × �) = � • (�i + �c + �d ) (1-50b)

Subtracting (1-50b) from (1-50a) reduces to

� • (∇ × �) − � • (∇ × �) = −� • (�i + �d ) − � • (�i + �c + �d ) (1-51)

Using the vector identity

∇ • (A × B) = B • (∇ × A) − A • (∇ × B) (1-52)

on the left side of (1-51), we can write that

∇ • (� × �) = −� • (�i + �d ) − � • (�i + �c + �d ) (1-53)

or∇ • (� × �) + � • (�i + �d ) + � • (�i + �c + �d ) = 0 (1-53a)

Integrating (1-53) over the volume V leads to∫∫∫V

∇ • (� × �) dv = −∫∫∫

V[� • (�i + �d ) + � • (�i + �c + �d )] dv (1-54)

Applying the divergence theorem (1-8) on the left side of (1-54) reduces it to

#S(� × �) • ds = −

∫∫∫V

[� • (�i + �d ) + � • (�i + �c + �d )] dv (1-55)

or

#S(� × �) • ds +

∫∫∫V

[� • (�i + �d ) + � • (�i + �c + �d )] dv = 0 (1-55a)

Equations 1-53a and 1-55a can be interpreted, respectively, as the differential and integral formsof the conservation of energy . To accomplish this, let us consider each of the terms included in(1-55a).



The integrand, in the first term of (1-55a), has the form

= � × � (1-56)

where is known as the Poynting vector. It has the units of power density (watts/square meter),since � has units of volts/meter and � has units of ampere/meter, so that the units of arevolts · ampere/meter2 = watts/meter2. Thus the first term of (1-55a), written as

�e =#S

(� × �) • ds =#S

• ds (1-57)

represents the total power �e exiting the volume V bounded by the surface S .The other terms in (1-55a), which represent the integrand of the volume integral, can be written

as

ps = −(� • �i + � • �i ) (1-58a)

� • �d = � •∂�

∂t= μ� •

∂�

∂t= 1

2μ

∂�2

∂t= ∂

∂t

(1

2μ�2

)= ∂

∂twm (1-58b)

pd = � • �c = � • (σ�) = σ�2 (1-58c)

� • �d = � •∂�

∂t= ε� •

∂�

∂t= 1

2ε∂�2

∂t= ∂

∂t

(1

2ε�2

)= ∂

∂twe (1-58d)

where

wm = 1

2μ�2 = magnetic energy density(J/m3) (1-58e)

we = 1

2ε�2 = electric energy density(J/m3) (1-58f)

ps = −(� • �i + � • �i ) = supplied power density(W/m3) (1-58g)

pd = σ�2 = dissipated power density(W/m3) (1-58h)

Integrating each of the terms in (1-58a) through (1-58d), we can write the corresponding formsas

�s = −∫∫∫

V(� • �i + � • �i ) dv =

∫∫∫V

ps dv (1-59a)

∫∫∫V(� • �d ) dv = ∂

∂t

∫∫∫V

1

2μ�2 dv = ∂

∂t

∫∫∫V

wm dv = ∂

∂tm (1-59b)

�d =∫∫∫

V(� • �c) dv =

∫∫∫V

σ�2 dv =∫∫∫

Vpd dv (1-59c)

∫∫∫V(� • �d ) dv = ∂

∂t

∫∫∫V

1

2ε�2 dv = ∂

∂t

∫∫∫V

we dv = ∂

∂te (1-59d)


TIME-HARMONIC ELECTROMAGNETIC FIELDS 21

where m = magnetic energy (J)e = electric energy (J)�s = supplied power (W)�e = exiting power (W)�d = dissipated power (W)

Using (1-57) and (1-59a) through (1-59d), we can write (1-55a) as

�e − �s + �d + ∂

∂t(e + m) = 0 (1-60)

or�s = �e + �d + ∂

∂t(e + m) (1-60a)

which is the conservation of power law . This law states that within a volume V , bounded by S ,the supplied power �s is equal to the power �e exiting S plus the power �d dissipated withinthat volume plus the rate of change (increase if positive) of the electric (e) and magnetic (m)

energies stored within that same volume.A summary of the field theory relations and their corresponding circuit concepts is found listed

in Table 1-2.

1.7 TIME-HARMONIC ELECTROMAGNETIC FIELDS

Maxwell’s equations in differential and integral forms, for general time-varying electromagneticfields, were presented in Sections 1.2.1 and 1.2.2. In addition, various expressions involvingand relating the electromagnetic fields (such as the constitutive parameters and relations, circuitrelations, boundary conditions, and power and energy) were also introduced in the precedingsections. However, in many practical systems involving electromagnetic waves, the time variationsare of cosinusoidal form and are referred to as time-harmonic. In general, such time variationscan be represented by1 ejωt , and the instantaneous electromagnetic field vectors can be related totheir complex forms in a very simple manner. Thus for time-harmonic fields, we can relate theinstantaneous fields, current density and charge (represented by script letters) to their complexforms (represented by roman letters) by

�(x , y , z ; t) = Re[E(x , y , z )ejωt ] (1-61a)

�(x , y , z ; t) = Re[H(x , y , z )ejωt ] (1-61b)

�(x , y , z ; t) = Re[D(x , y , z )ejωt ] (1-61c)

�(x , y , z ; t) = Re[B(x , y , z )ejωt ] (1-61d)

�(x , y , z ; t) = Re[J(x , y , z )ejωt ] (1-61e)

q(x , y , z ; t) = Re[q(x , y , z )ejωt ] (1-61f)

where �, �, �, �, �, and q represent the instantaneous field vectors, current density and charge,while E, H, D, B, J, and q represent the corresponding complex spatial forms which are onlya function of position. In this book we have chosen to represent the instantaneous quantities bythe real part of the product of the corresponding complex spatial quantities with ejωt . Another

1 Another representation form of time-harmonic variations is e−jωt (most scientists prefer eiωt or e−iωt where i = √−1).Throughout this book, we will use the ejωt form, which when it is not stated will be assumed. The e−jωt fields are relatedto those of the ejωt form by the complex conjugate.



option would be to represent the instantaneous quantities by the imaginary part of the products.It should be stated that throughout this book the magnitudes of the instantaneous fields representpeak values that are related to their corresponding root-mean-square (rms) values by the squareroot of 2 (peak = √

2 rms). If the complex spatial quantities can be found, it is then a very simpleprocedure to find their corresponding instantaneous forms by using (1-61a) through (1-61f). Inwhat follows, it will be shown that Maxwell’s equations in differential and integral forms fortime-harmonic electromagnetic fields can be written in much simpler forms using the complexfield vectors.

1.7.1 Maxwell’s Equations in Differential and Integral Forms

It is a very simple exercise to show that, by substituting (1-61a) through (1-61f) into (1-1) through(1-4) and (1-6), Maxwell’s equations and the continuity equation in differential form for time-harmonic fields can be written in terms of the complex field vectors as shown in Table 1-4. Usinga similar procedure, we can write the corresponding integral forms of Maxwell’s equations andthe continuity equation listed in Table 1-1 in terms of the complex spatial field vectors as shownin Table 1-4. Both of these derivations have been assigned as exercises to the reader at the endof the chapter.

By examining the two forms in Table 1-4, we see that one form can be obtained from theother by doing the following:

1. Replace the instantaneous field vectors by the corresponding complex spatial forms, or viceversa.

2. Replace ∂/∂t by jω(∂/∂t = jω), or vice versa.

The second step is very similar to that followed in circuit analysis when Laplace transforms areused to analyze RLC a.c. circuits. In these analyses ∂/∂t is replaced by s (∂/∂t ≡ s). For steady-state conditions ∂/∂t is replaced by jω(∂/∂t ≡ s ≡ jω). The reason for using Laplace transformsis to transform differential equations to algebraic equations, which are simpler to solve. The sameintent is used here to write Maxwell’s equations in forms that are easier to solve. Thus, if it isdesired to solve for the instantaneous field vectors of time-harmonic fields, it is easier to usethe following two-step procedure, instead of attempting to do it in one step using the generalinstantaneous forms of Maxwell’s equations:

1. Solve for the complex spatial field vectors, current densities and charges (E, H, D, B, J,M, q), using Maxwell’s equations from Table 1-4 that are written in terms of the complexspatial field vectors, current densities and charges.

2. Determine the corresponding instantaneous field vectors, current densities and charges using(1-61a) through (1-61f).

Step 1 is obviously the most difficult, and it is often the only step needed. Step 2 is straight-forward, and it is often omitted. In practice, the time variations of ejωt are stated at the outset,but then are suppressed.

1.7.2 Boundary Conditions

The boundary conditions for time-harmonic fields are identical to those of general time-varyingfields, as derived in Section 1.5, and they can be expressed simply by replacing the instantaneousfield vectors, current densities and charges in Table 1-3 with their corresponding complex spatialfield vectors, current densities and charges. A summary of all the boundary conditions for time-harmonic fields, referring to Figure 1-4, is found in Table 1-5.

In addition to the boundary conditions found in Table 1-5, an additional boundary conditionon the tangential components of the electric field is often used along an interface when one of


TA

BL

E1-

4In

stan

tane

ous

and

tim

e-ha

rmon

icfo

rms

ofM

axw

ell’

seq

uati

ons

and

cont

inui

tyeq

uati

onin

diff

eren

tial

and

inte

gral

form

s

Inst

anta

neou

sT

ime

harm

onic

Dif

fere

ntia

lfo

rm

∇×

�=

−�i−

∂� ∂t

∇×

E=

−Mi−

jωB

∇×

�=

�i+

�c+

∂� ∂t

∇×

H=

J i+

J c+

jωD

∇·�

=q ev

∇·D

=q e

v

∇·�

=q m

v∇

·B=

q mv

∇·�

ic=

−∂q ev

∂t

∇·J

ic=

−jω

q ev

Inte

gral

form

∮ C�

·d�

=−

∫∫ S�

i·d

s−

∂ ∂t

∫∫ S�

·ds

∮ CE

·d�

=−

∫∫ SM

i·d

s−

jω∫∫ S

B·d

s

∮ C�

·d�

=∫∫ S

�i·d

s+

∫∫ S�

c·d

s+

∂ ∂t

∫∫ S�

·ds

∮ CH

·d�

=∫∫ S

J i·d

s+

∫∫ SJ c

·ds+

jω∫∫ S

D·d

s

#S

�·d

s=

�e

#S

D·d

s=

Qe

#S

�·d

s=

�m

#S

B·d

s=

Qm

#S

�ic

·ds=

−∂�

e

∂t

#S

J ic·d

s=

−jω

Qe

23



TABLE 1-5 Boundary conditions on time-harmonic electromagnetic fields

Finiteconductivity media, Medium 1 of Medium 1 of

no sources or infinite electric infinitecharges conductivity magnetic

σ1, σ2 �= ∞ (E1 = H1 = 0) conductivityJs = Ms = 0 σ1 = ∞; σ2 �= ∞ (E1 = H1 = 0)

General qes = qms = 0 Ms = 0; qms = 0 Js = 0; qes = 0

Tangentialelectric fieldintensity

−n × (E2 − E1) = Ms n × (E2 − E1) = 0 n × E2 = 0 −n × E2 = Ms

Tangentialmagnetic fieldintensity

n × (H2 − H1) = Js n × (H2 − H1) = 0 n × H2 = Js n × H2 = 0

Normal electricflux density

n · (D2 − D1) = qes n · (D2 − D1) = 0 n · D2 = qes n · D2 = 0

Normal magneticflux density

n · (B2 − B1) = qms n · (B2 − B1) = 0 n · B2 = 0 n · B2 = qms

the two media is a very good conductor (material that possesses large but finite conductivity).This is illustrated in Figure 1-6 where it is assumed that medium 1 is a very good conductorwhose surface, as will be shown in Section 4.3.1, exhibits a surface impedance Zs (ohms) given,approximately, by (4-42) or

Zs = Rs + jXs = (1 + j )

√ωμ1

2σ1(1-62)

with equal real and imaginary (inductive) parts (σ1 is the conductivity of the conductor). At thesurface there exists a linear current density Js (A/m) related to the tangential magnetic field inmedium 2 by

Js � n × H2 (1-63)

Since the conductivity is finite (although large), the most intense current density resides at thesurface, and it diminishes (in an exponential form) as the observations are made deeper into theconductor. This is demonstrated in Example 5.7 of Section 5.4.1. In addition, the electric fieldintensity along the interface cannot be zero (although it may be small). Thus, we can write that

E2, H

2, D2, B

2E

1, H1, D

1, B1

e 2,m2, s2

e 1,m1, s1

1

2

y

xZs = Rs + jXs

large)(s1

z

n

Figure 1-6 Surface impedance along the surface of a very good conductor.



the tangential component of the electric field in medium 2, along the interface, is related to theelectric current density Js and tangential component of the magnetic field by

Et2 = ZsJs = Zs n × H2 = n × H2

√ωμ1

2σ1(1 + j ) (1-64)

For time-harmonic fields, the boundary conditions on the normal components are not inde-pendent of those on the tangential components, and vice-versa, since they are related throughMaxwell’s equations. In fact, if the tangential components of the electric and magnetic fields sat-isfy the boundary conditions, then the normal components of the same fields necessarily satisfythe appropriate boundary conditions. For example, if the tangential components of the electricfield are continuous across a boundary, their derivatives (with respect to the coordinates on theboundary surface) are also continuous. This, in turn, ensures continuity of the normal componentof the magnetic field.

To demonstrate that, let us refer to the geometry of Figure 1-6 where the local surface alongthe interface is described by the x, z coordinates with y being normal to the surface. Let usassume that Ex and Ez are continuous, which ensures that their derivatives with respect to x andz (∂Ex/∂x , ∂Ex/∂z , ∂Ez /∂x , ∂Ez /∂z ) are also continuous. Therefore, according to Maxwell’s curlequation of the electric field

∇ × E = ∇ × (ax Ex + az Ez ) =

∣∣∣∣∣∣∣∣ax ay az

∂

∂x0

∂

∂zEx 0 Ez

∣∣∣∣∣∣∣∣= ax (0) + ay

(∂Ex

∂z− ∂Ez

∂x

)+ az (0)

∇ × E = ay

(∂Ex

∂z− ∂Ez

∂x

)= −jωμH (1-65)

or

By = μHy = − 1

jω

(∂Ex

∂z− ∂Ez

∂x

)(1-65a)

According to (1-65a), By , the normal component of the magnetic flux density along the interface,is continuous across the boundary if ∂Ex/∂z and ∂Ez /∂x are also continuous across the boundary.

In a similar manner, it can be shown that continuity of the tangential components of themagnetic field ensures continuity of the normal component of the electric flux density (D).

1.7.3 Power and Energy

In Section 1.6, it was shown that power and energy are associated with time-varying electromag-netic fields. The conservation-of-energy equation, in differential and integral forms, was statedrespectively by (1-53a) and (1-55a). Similar equations can be derived for time-harmonic electro-magnetic fields using the complex spatial forms of the field vectors. Before we attempt this, letus first rewrite the instantaneous Poynting vector in terms of the complex field vectors.

The instantaneous Poynting vector was defined by (1-56) and is repeated here as

= � × � (1-66)

The electric and magnetic fields of (1-61a) and (1-61b) can also be written as

�(x , y , z ; t) = Re[E(x , y , z )ejωt ] = 12 [Eejωt + (Eejωt )∗] (1-67a)

�(x , y , z ; t) = Re[H(x , y , z )ejωt ] = 12 [Hejωt + (Hejωt )∗] (1-67b)



where the asterisk (*) indicates complex conjugate. Substituting (1-67a) and (1-67b) into (1-66),we have that

= � × � = 12 (Eejωt + E∗e−jωt ) × 1

2 (Hejωt + H∗e−jωt )

= 12

{12 [E × H∗ + E∗ × H] + 1

2 [E × Hej 2ωt + E∗ × H∗e−j 2ωt ]}

= 12

{12 [E × H∗ + (E × H∗)∗] + 1

2 [E × Hej 2ωt + (E × Hej 2ωt )∗]}

(1-68)

Using the equalities (1-67a) or (1-67b) in reverse order, we can write (1-68) as

= 12 [Re(E × H∗) + Re(E × Hej 2ωt )] (1-69)

Since both E and H are not functions of time and the time variations of the second term are twicethe frequency of the field vectors, the time-average Poynting vector (average power density) overone period is equal to

av = S = 12 Re[E × H∗] (1-70)

Since E × H∗ is, in general, complex and the real part of E × H∗ represents the real part ofthe power density, what does the imaginary part represent? As will be seen in what follows, theimaginary part represents the reactive power. With (1-69) and (1-70) in mind, let us now derivethe conservation-of-energy equation in differential and integral forms using the complex formsof the field vector.

From Table 1-4, the first two of Maxwell’s equations can be written as

∇ × E = −Mi − jωμH (1-71a)

∇ × H = Ji + Jc + jωεE = Ji + σE + jωεE (1-71b)

Dot multiplying (1-71a) by H∗ and the conjugate of (1-71b) by E, we have that

H∗ • (∇ × E) = −H∗ • Mi − jωμH • H∗ (1-72a)

E • (∇ × H∗) = E • J∗i + σE • E∗ − jωεE • E∗ (1-72b)

Subtracting (1-72a) from (1-72b), we can write that

E • (∇ × H∗) − H∗ • (∇ × E)

= H∗ • Mi + E • J∗i + σE • E∗ − jωεE • E∗ + jωμH • H∗ (1-73)

Using the vector identity (1-52) reduces (1-73) to

∇ • (H∗ × E) = H∗ • Mi + E • J∗i + σ |E|2 + jωμ|H|2 − jωε|E|2 (1-74)

or−∇ • (E × H∗) = H∗ • Mi + E • J∗

i + σ |E|2 + jω(μ|H|2 − ε|E|2) (1-74a)

Dividing both sides by 2, we can write that

−∇ • ( 12 E × H∗) = 1

2 H∗ • Mi + 12 E • J∗

i + 12σ |E|2 + j 2ω( 1

4μ|H|2 − 14ε|E|2) (1-75)

For time-harmonic fields, (1-75) represents the conservation-of-energy equation in differentialform.

To verify that (1-75) represents the conservation-of-energy equation in differential form, it iseasier to examine its integral form. To accomplish this, let us first take the volume integral of



both sides of (1-75) and then apply the divergence theorem (1-8) to the left side. Doing both ofthese steps reduces (1-75) to

−∫∫∫

V∇ • ( 1

2 E × H∗) dv = −#S

( 12 E × H∗) • ds

= 1

2

∫∫∫V(H∗ • Mi + E • J∗

i ) dv

+ 1

2

∫∫∫V

σ |E|2 dv + j 2ω

∫∫∫V( 1

4μ|H|2 − 14ε|E|2) dv

or

−1

2

∫∫∫V(H∗ • Mi + E • J∗

i ) dv =#S

( 12 E × H∗) • ds + 1

2

∫∫∫V

σ |E|2 dv

+ j 2ω

∫∫∫V( 1

4μ|H|2 − 14ε|E|2) dv (1-76)

which can be written asPs = Pe + Pd + j 2ω(W m − W e) (1-76a)

where

Ps = −1

2

∫∫∫V(H∗ • Mi + E • J∗

i ) dv = supplied complex power (W) (1-76b)

Pe =#S

(1

2E × H∗

)• ds = exiting complex power (W) (1-76c)

Pd = 1

2

∫∫∫V

σ |E|2 dv = dissipated real power (W) (1-76d)

W m =∫∫∫

V

14μ|H|2 dv = time-average magnetic energy (J) (1-76e)

W e =∫∫∫

V

14ε|E|2 dv = time-average electric energy (J) (1-76f)

For an electromagnetic source (represented in Figure 1-5 by electric and magnetic current densitiesJi and Mi , respectively) supplying power in a region within S , (1-76) and (1-76a) represent theconservation-of-energy equation in integral form. Now, it is also much easier to accept that (1-75), from which (1-76) was derived, represents the conservation-of-energy equation in differentialform. In (1-76a), Ps and Pe are in general complex and Pd is always real, but the last two termsare always imaginary and represent the reactive power associated, respectively, with magneticand electric fields. It should be stated that for complex permeabilities and permittivities thecontributions from their imaginary parts to the integrals of (1-76e) and (1-76f) should both becombined with (1-76d), since they both represent losses associated with the imaginary parts ofthe permeabilities and permittivities.

It should be stated that the imaginary term of the right side of (1-76), including its signs, whichrepresents the complex stored power (inductive and capacitive), does conform to the notation ofconventional circuit theory. For example, defining the complex power P , assuming V and I arepeak values, as

P = 1

2(VI ∗) (1-77)



the complex power of a series circuit consisting of a resistor R in series with an inductor L, witha current I through both R and L and total voltage V across both the resistor and inductor, canbe written, based on (1-77), as

P = 1

2(VI ∗) = 1

2(ZI )I ∗ = 1

2Z |I |2 = 1

2(R + jωL)|I |2 (1-77a)

The imaginary part of (1-77a) is positive. Similarly, for a parallel circuit consisting of a conductorG in parallel with a capacitor C , with a voltage V across G and C and a total current I (I =IG + IC , where IG is the current through the conductor and IC is the current through the capacitor),its complex power P , based on (1-77), can be expressed as

P = 1

2(VI ∗) = 1

2V (YV )∗ = 1

2Y ∗|V |2 = 1

2(G + jωC )∗|V |2 = 1

2(G − jωC )|V |2 (1-77b)

TABLE 1-6 Relations between time-harmonic electromagnetic field and steady-state a.c. circuittheories

Field theory Circuit theory

1. E (electric field intensity) 1. v (voltage)

2. H (magnetic field intensity) 2. i (current)

3. D (electric flux density) 3. qev (electric charge density)

4. B (magnetic flux density) 4. qmv (magnetic charge density)

5. J (electric current density) 5. ie (electric current)

6. M (magnetic current density) 6. im (magnetic current)

7. Jd = jωεE(electric displacementcurrent density)

7. i = jωCv(current through acapacitor)

8. Md = jωμH(magnetic displacementcurrent density)

8. v = jωLi(voltage across aninductor)

9. Constitutive relations 9. Element laws

(a) Jc = σE(electric conductioncurrent density)

(a) i = Gv = 1

Rv (Ohm’s law)

(b) D = εE (dielectric material) (b) Qe = Cv (charge in a capacitor)

(c) B = μH (magnetic material) (c) ψ = Li (flux of an inductor)

10.∮

CE · d� = −jω

∫∫S

B · ds(Maxwell–Faradayequation)

10.∑

v = −jωLs i � 0(Kirchhoff’svoltage law)

11.#S

Jic · ds = −jω∫∫∫

Vqevdv = −∂Qe

∂t(continuity equation)

11.∑

i = −jωQe = −jωCsv � 0(Kirchoff’s current law)

12. Power and energy densities 12. Power and energy(v and i represent peak values)

(a)1

2#S(E × H∗) · ds (complex power) (a) P = 1

2vi

(power-voltage-currentrelation)

(b)1

2

∫∫∫V

σ |E|2dv (dissipated real power) (b) Pd = 1

2Gv2 = 1

2

v2

R(power dissipatedin a resistor)

(c)1

4

∫∫∫V

ε|E|2dv(time-average electricstored energy)

(c)1

4Cv2 (energy stored in a

capacitor)

(d)1

4

∫∫∫V

μ|H|2dv(time-average magneticstored energy)

(d)1

4Li 2 (energy stored in an

inductor)


MULTIMEDIA 29

The imaginary part of (1-77b) is negative. Therefore the imaginary parts of (1-77a) and (1-77b) conform, respectively, to the notation (positive and negative) of the imaginary parts of thecomplex power in (1-76) due to the H and E fields.

The field and circuit theory relations for time-harmonic electromagnetic fields are similar tothose found in Table 1-2 for the general time-varying electromagnetic fields, but with the instan-taneous field quantities (represented by script letters) replaced by their corresponding complexfield quantities (represented by roman letters) and with ∂/∂t replaced by jω (∂/∂t ≡ jω). Theseare shown listed in Table 1-6.

Over the years many excellent introductory books on electromagnetics, [1] through [28], andadvanced books, [29] through [40], have been published. Some of them can serve both purposes,and a few may not now be in print. Each is contributing to the general knowledge of electro-magnetic theory and its applications. The reader is encouraged to consult them for an even betterunderstanding of the subject.

1.8 MULTIMEDIA

On the website that accompanies this book, the following multimedia resources are included forthe review, understanding and presentation of the material of this chapter.

• Power Point (PPT) viewgraphs, in multicolor.

REFERENCES

1. F. T. Ulaby, E. Michielssen and U. Ravaioli, Fundamentals of Applied Electromagnetics , Sixth Edition,Pearson Education, Inc., Upper Saddle River, NJ, 2010.

2. M. N. Sadiku, Elements of Electromagnetics , Fifth Edition, Oxford University Press, Inc., New York,2010.

3. S. M. Wenthworth, Fundamentals of Electromagnetics with Engineering Applications , John Wiley &Sons, 2005.

4. C. R. Paul, Electromagnetics for Engineers: With Applications to Digital Systems and ElectromagneticInterference, John Wiley & Sons, Inc., 2004.

5. N. Ida, Engineering Electromagnetics , Second Edition, Springer, NY, 2004.

6. U. S. Inan and A. S. Inan, Electromagnetic Waves , Prentice-Hall, Inc., Upper Saddle River, NJ, 2000.

7. M. F. Iskander, Electromagnetic Fields & Waves , Waveland Press, Inc., Long Grove, IL, 2000.

8. K. R. Demarest, Engineering Electromagnetics , Prentice-Hall, Inc., Upper Saddle River, NJ, 1998.

9. G. F. Miner, Lines and Electromagnetic Fields for Engineers , Oxford University Press, Inc., New York,1996.

10. D. H. Staelin, A. W. Morgenthaler and J. A. Kong, Electromagnetic Waves , Prentice-Hall, Inc., Engle-wood Cliffs, NJ, 1994.

11. J. D. Kraus, Electromagnetics , Third Edition, McGraw-Hill, New York, 1992.

12. S. Ramo, J. R. Whinnery and T. Van Duzer, Fields and Waves in Communication Electronics , SecondEdition, John Wiley & Sons, New York, 1984.

13. W. H. Hayt, Jr. and John A. Buck, Engineering Electromagnetics , Sixth Edition, McGraw-Hill, NewYork, 2001.

14. D. T. Paris and F. K. Hurd, Basic Electromagnetic Theory , McGraw-Hill, New York, 1969.

15. C. T. A. Johnk, Engineering Electromagnetic Fields and Waves , Second Edition, John Wiley & Sons,New York, 1988.

16. H. P. Neff, Jr., Basic Electromagnetic Fields , Second Edition, John Wiley & Sons, New York, 1987.



17. S. V. Marshall, R. E. DuBroff and G. G. Skitek, Electromagnetic Concepts and Applications , FourthEdition, Prentice-Hall, Englewood Cliffs, NJ, 1996.

18. D. K. Cheng, Field and Wave Electromagnetics , Addison-Wesley, Reading, MA, 1983.

19. C. R. Paul and S. A. Nasar, Introduction to Electromagnetic Fields , Third Edition, McGraw-Hill, NewYork, 1998.

20. L. C. Shen and J. A. Kong, Applied Electromagnetism , Third Edition, PWS Publishing Co., Boston,MA, 1987

21. N. N. Rao, Elements of Engineering Electromagnetics , Fourth Edition, Prentice-Hall, Englewood Cliffs,NJ, 1994.

22. M. A. Plonus, Applied Electromagnetics , McGraw-Hill, New York, 1978.

23. A. T. Adams, Electromagnetics for Engineers , Ronald Press, New York, 1971.

24. M. Zahn, Electromagnetic Field Theory , John Wiley & Sons, New York, 1979.

25. L. M. Magid, Electromagnetic Fields, Energy, and Waves , John Wiley & Sons, New York, 1972.

26. S. Seely and A. D. Poularikas, Electromagnetics: Classical and Modern Theory and Applications ,Dekker, New York, 1979.

27. D. M. Cook, The Theory of the Electromagnetic Field , Prentice-Hall, Englewood Cliffs, NJ, 1975.

28. R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics: Mainly Electromag-netism and Matter , Volume II, Addison-Wesley, Reading, MA, 1964.

29. W. R. Smythe, Static and Dynamic Electricity , McGraw-Hill, New York, 1939.

30. J. A. Stratton, Electromagnetic Theory , Wiley-Interscience, New York, 2007.

31. R. E. Collin, Field Theory of Guided Waves , IEEE Press, New York, 1991.

32. E. C. Jordan and K. G. Balmain, Electromagnetic Waves and Radiating Systems , Second Edition,Prentice-Hall, Englewood Cliffs, NJ, 1968.

33. R. F. Harrington, Time-Harmonic Electromagnetic Fields , McGraw-Hill, New York, 1961.

34. J. R. Wait, Electromagnetic Wave Theory , Harper & Row, New York, 1985.

35. J. A. Kong, Theory of Electromagnetic Waves , John Wiley & Sons, New York, 1975.

36. C. C. Johnson, Field and Wave Electrodynamics , McGraw-Hill, New York, 1965.

37. J. D. Jackson, Classical Electrodynamics , Third Edition, John Wiley & Sons, New York, 1999.

38. D. S. Jones, Methods in Electromagnetic Wave Propagation , Oxford Univ. Press (Clarendon), Lon-don/New York, 1979.

39. M. Kline (Ed.), The Theory of Electromagnetic Waves , Interscience, New York, 1951.

40. J. Van Bladel, Electromagnetic Fields , Second Edition, Wiley-Interscience, New York, 2007.

PROBLEMS

1.1. Derive the differential form of the continuityequation, as given by (1-6), from Maxwell’sequations 1-1 through 1-4.

1.2. Derive the integral forms of Maxwell’sequations and the continuity equation, aslisted in Table 1-1, from the correspondingones in differential form.

1.3. The electric flux density inside a cube isgiven by:(a) D = ax (3 + x)

(b) D = ay (4 + y2)

Find the total electric charge enclosed insidethe cubical volume when the cube is in the

first octant with three edges coincident withthe x , y , z axes and one corner at the origin.Each side of the cube is 1 m long.

1.4. An infinite planar interface between media,as shown in the figure, is formed by havingair (medium #1) on the left of the inter-face and lossless polystyrene (medium #2)(with a dielectric constant of 2.56) to theright of the interface. An electric surfacecharge density qes = 0.2 C/m2 exists alongthe entire interface.The static electric flux density inside thepolystyrene is given by

D2 = 6ax + 3az C/m2


PROBLEMS 31

Determine the corresponding vector:(a) Electric field intensity inside the

polystyrene.(b) Electric polarization vector inside the

polystyrene.(c) Electric flux density inside the air

medium.(d) Electric field intensity inside the air

medium.(e) Electric polarization vector inside the air

medium.Leave your answers in terms of εo , μo .

Aireo, mo

Polystyrene2.56eo, mo

z

x

y

qes

#1 #2n

Figure P1-4

1.5. An infinite planar interface between media,as shown in the figure, is formed by havingair (medium #1) on the left of the interfaceand lossless magnetic material (medium #2)(with a relative permeability of 4 and rela-tive permittivity of 2.56) to the right of theinterface.The static magnetic field intensity inside theair is given by

H1 = 3ax + 9az A/m

Magnetic material2.46eo, 4mo

Aireo, mo

z

x

y

#1 #2

Figure P1-5

Determine the corresponding vector:(a) Magnetic flux density in the air medium.(b) Magnetic polarization in the air

medium.(c) Magnetic field intensity in the magnetic

material.(d) Magnetic flux density in the magnetic

material.(e) Magnetic polarization in the magnetic

material.Leave your answers in terms of ε0, μo .

1.6. A static electric field of intensity/strength Eo

is established inside a free-space medium asshown below. The static electric field inten-sity is oriented at an angle of 30◦ relative tothe principal z axis. A semi-infinite dielec-tric slab of relative permittivity of 4 andrelative permeability of unity is immersedinto the initially established static electricfield, as shown below.Determine the:(a) Total electric field intensity E1 and

total electric flux density D1 within thedielectric slab. Leave your answers interms of Eo, εo , μo , and any constants.

(b) Angle θ (in degrees).

z

Eo, Do

Eo, Do

E1, D1

z

30°

30°

q

eo, mo

eo, mo

4eo, mo

Figure P1-6

1.7. A static magnetic field of field inten-sity/strength Ho is established inside a free-space medium as shown on the next page. Thestatic magnetic field intensity is oriented at anangle of 30◦ relative to the principal z axis.A semi-infinite magnetic slab of relative per-meability of 4 and relative permittivity of 9 isimmersed into the initially established staticmagnetic field, as shown on the next page.Determine the:(a) Total static magnetic field intensity H1

and total static magnetic flux density B1

within the magnetic slab. Leave youranswers in terms of Ho, εo , μo and anyconstants.

(b) Angle θ (in degrees).



z

Ho, Bo

Ho, Bo

H1, B1

z

30°

30°

q

eo, mo

eo, mo 9eo, 4mo

Figure P1-7

1.8. A dielectric slab, with a thickness of 6 cmand dielectric constant of 4, is sandwichedbetween two different media; free space tothe left and another dielectric, with a dielec-tric constant of 9, to the right. If the electricfield in the free-space medium is at an angleof 30◦ at a height of 3 cm at the lead-ing interface, as shown in the figure below,determine the:(a) Angle α (in degrees, as measured from

the normal to the interface) the electricfield will make in the dielectric mediumto the right of the center slab.

(b) Height h (in cm) the electric field willhave at the trailing interface.

h = ?

Freespace

er = 4 er = 9

3 cm

a = ?

6 cm30°

Figure P1-8

1.9. The electric field inside a circular cylinderof radius a and height h is given by

x

y

z

a

h

Figure P1-9

E = az

[− c

h+ b

6ε0(3z 2 − h2)

]

where c and b are constants. Assuming themedium within the cylinder is free space,find the total charge enclosed within thecylinder.

1.10. The static magnetic field on the inside partof the surface of an infinite length dielec-tric cylinder of circular cross section ofradius a = 4 cm and of magnetic materialwith a relative permittivity and permeabilityof εr = 4, μr = 9 is given by

H= aρ3+aφ6 + az 8 A/m at ρ = 4− cm

The cylinder is surrounded on the outsidewith air. Refer to Figure 3-4 for the cylindri-cal coordinate system and its units vectors.Determine the:(a) Magnetic flux density on the inside part

of the surface of the cylinder (ρ = 4−cm; magnetic material).

(b) Magnetic field on the outside part of thecylinder surface (ρ = 4+ cm; air).

(c) Magnetic flux density on the outside partof the cylinder surface (ρ = 4+ cm; air).

x

y

z

er, mr

4 cm

Figure P1-10

1.11. The instantaneous electric field inside asource-free, homogeneous, isotropic, andlinear medium is given by

� = [ax A(x + y) + ay B(x − y)] cos(ωt)

Determine the relations between A and B .

1.12. The magnetic flux density produced on itsplane by a current-carrying circular loop of


PROBLEMS 33

radius a = 0.1 m, placed on the xy plane atz = 0, is given by

� = az10−12

1 + 25ρcos(1500π t) Wb/m2

where ρ is the radial distance in cylindricalcoordinates. Find the:(a) Total flux in the z direction passing

through the loop.(b) Electric field at any point ρ within

the loop. Check your answer by usingMaxwell’s equation 1-1.

1.13. The instantaneous magnetic flux density infree space is given by

� = ax Bx cos(2y) sin(ωt − πz )

+ ay By cos(2x) cos(ωt − πz )

where Bx and By are constants. Assum-ing there are no sources at the observationpoints x , y , determine the electric displace-ment current density.

1.14. The displacement current density within asource-free (�i = 0) cube centered aboutthe origin is given by

�d = ax yz + ay y2 + az xyz

Each side of the cube is 1 m long and themedium within it is free space. Find the dis-placement current leaving, in the outwarddirection, through the surface of the cube.

1.15. The electric flux density in free spaceproduced by an oscillating electric chargeplaced at the origin is given by

� = ar10−9

4π

1

r2cos(ωt − βr)

where β = ω√

μ0ε0. Find the time-averagecharge that produces this electric flux den-sity.

1.16. The electric field radiated at large distancesin free space by a current-carrying small cir-cular loop of radius a , placed on the xy planeat z = 0, is given by

� = aφE0 sin θcos(ωt − β0r)

r, r � a

where E0 is a constant, β0 = ω√

μ0ε0, ris the radial distance in spherical coordi-nates, and θ is the spherical angle measuredfrom the z axis that is perpendicular to the

plane of the loop. Determine the correspond-ing radiated magnetic field at large distancesfrom the loop (r � a).

1.17. A time-varying voltage source of v(t) =10 cos(ωt) is connected across a paral-lel plate capacitor with polystyrene (ε =2.56ε0, σ = 3.7×10−4 S/m) between theplates. Assuming a small plate separation of2 cm and no field fringing, determine at:(a) f = 1 MHz(b) f = 100 MHz

the maximum values of the conduction anddisplacement current densities within thepolystyrene and compare them.

1.18. A dielectric slab of polystyrene (ε =2.56ε0, μ = μ0) of height 2h is boundedabove and below by free space, as shownin Figure P1-18. Assuming the electric fieldwithin the slab is given by

� = (ay 5 + az 10) cos(ωt − βx)

where β = ω√

μ0ε, determine the:(a) Corresponding magnetic field within the

slab.(b) Electric and magnetic fields in free

space right above and below the slab.

x

e0, m0

e0, m0

2.56 e0, m0

2.56 e0, m0

y

z

h

h

Figure P1-18

1.19. A finite conductivity rectangular strip,shown in Figure P1-19, is used to carryelectric current. Because of the strip’s lossynature, the current is nonuniformly dis-tributed over the cross section of the strip.The current density on the upper and lowersides is given by

� = az 104 cos(2π×109t) A/m2

and it rapidly decays in an exponential fash-ion from the lower side toward the center bythe factor e−106y , or

� = az 104e−106y cos(2π×109t) A/m2



A similar decay is experienced by the cur-rent density from the upper side toward thecenter. Assuming no variations of the cur-rent density with respect to x , determine thetotal current flowing through the wire.

x

y

z5 mm

0.5 mm

Figure P1-19

1.20. The instantaneous electric field inside aconducting rectangular pipe (waveguide) ofwidth a is given by

� = ay E0 sin(π

ax)

cos(ωt − βz z )

where βz is the waveguide’s phase constant.Assuming there are no sources within thefree-space-filled pipe determine the:(a) Corresponding instantaneous magnetic

field components inside the conductingpipe.

(b) Phase constant βz .The height of the waveguide is b.

1.21. The instantaneous electric field intensityinside a source-free coaxial line with innerand outer radii of a and b, respectively, thatis filled with a homogeneous dielectric ofε = 2.25ε0, μ = μ0, and σ = 0, is given by

� = aρ

(100

ρ

)cos(108t − βz )

where β is the phase constant and ρ is thecylindrical radial distance from the center ofthe coaxial line. Determine the:(a) Corresponding instantaneous magnetic

field �.(b) Phase constant β.(c) Displacement current density �d .

1.22. A coaxial line resonator with inner andouter conductors at a = 5 mm and b = 20mm, and with conducting plates at z = 0and z = �, is filled with a dielectric withεr = 2.56, μr = 1, and σ = 0. The instan-taneous magnetic field intensity inside the

source-free dielectric medium is given by

� = aφ

(2

ρ

)cos

(π

�z)

cos(4π×108t)

Find the following:(a) Electric field intensity within the dielec-

tric.(b) Surface current density �s at the con-

ductor surfaces at ρ = a and ρ = b.(c) Displacement current density �d at any

point within the dielectric.(d) Total displacement current flowing

through the circumferential surface ofthe resonator.

1.23. Using the instantaneous forms of Maxwell’sequation and the continuity equations listedin Tables 1-1 and 1-4, derive the correspond-ing time-harmonic forms (in differential andintegral forms) listed in Table 1-4. Use def-initions (1-61a) through (1-61f).

1.24. Show that the electric and magnetic fields(1-61a) and (1-61b) can be written, respec-tively, as in (1-67a) and (1-67b).

1.25. The time-harmonic instantaneous electricfield traveling along the z -axis, in a free-space medium, is given by

�(z , t) = ax Eo sin[(ωt − βoz ) +

(π

2

)]where Eo is a real constant and βo =ω

√μoεo .

(a) Write an expression for the complexspatial electric field intensity E(z ).

(b) Find the corresponding complex spatialmagnetic field intensity H(z ).

(c) Determine the time-average Poyntingvector (average power density) Save.

1.26. An electric line source of infinite length andconstant current, placed along the z axis,radiates in free space at large distances fromthe source (ρ � 0) a time-harmonic com-plex magnetic field given by

H = aφH0e−jβ0ρ

√ρ

, ρ � 0

where H0 is a constant, β0 = ω√

μ0ε0, and ρ

is the radial cylindrical distance. Determinethe corresponding electric field for ρ � 0.

1.27. The time-harmonic complex electric fieldradiated in free space by a linear radiatingelement is given by

E = ar Er + aθ Eθ


PROBLEMS 35

Er = E0cos θ

r2

[1 + 1

jβ0r

]e−jβ0r

Eθ = jE0β sin θ

2r

[1 + 1

jβ0r− 1

(β0r)2

]e−jβ0r

where ar and aθ are unit vectors in thespherical directions r and θ , E0 is a con-stant, and β0 = ω

√μ0ε0. Determine the cor-

responding spherical magnetic field compo-nents.

1.28. The time-harmonic complex electric fieldradiated by a current-carrying small circularloop in free space is given by

E = aφE0sin θ

r

[1 + 1

jβ0r

]e−jβ0r

where aφ is the spherical unit vector inthe φ direction, E0 is a constant, andβ0 = ω

√μ0ε0. Determine the correspond-

ing spherical magnetic field components.

1.29. The complex electric field inside aninfinitely long rectangular pipe, with all fourvertical walls perfectly electric conducting,as shown in Figure P1-29, is given by

E = az (1 + j ) sin(π

ax)

sin(π

by)

Assuming that there are no sources withinthe box and a = λ0, b = 0.5λ0, and μ = μ0,where λ0 = free space, infinite medium

x

y

z

a

b

Figure P1-29

wavelength, find the:(a) Conductivity.(b) Dielectric constant.of the medium within the box.

1.30. A time-harmonic electromagnetic field infree space is perpendicularly incident upona perfectly conducting semi-infinite planarsurface, as shown in Figure P1-30. Assum-ing the incident Ei and reflected Er complexelectric fields on the free-space side of theinterface are given by

Ei = ax e−jβ0z

Er = −ax e+jβ0z

whereβ0 = ω

√μ0ε0

determine the current density Js induced onthe surface of the conducting surface. Eval-uate all the constants.

zy

x

s = ∞Incident

e0, m0

Reflected

Figure P1-30

1.31. The free-space incident Ei and reflected Er

fields of a time-harmonic electromagneticfield obliquely incident upon a perfectlyconducting semi-infinite planar surface ofFigure P1-31 are given by

Ei = ay E0e−jβ0(x sin θi +z cos θi )

Er = ay E0�h e−jβ0(x sin θi −z cos θi )

zy

x

s = ∞

Incident

e0, m0Reflected

qi

qi

Figure P1-31



where E0 is a constant and β0 = ω√

μ0ε0.Determine the coefficient �h .

1.32. For Problem 1.31, determine the:(a) Corresponding incident and reflected

magnetic fields.(b) Electric current density along the inter-

face between the two media.

1.33. Repeat Problem 1.31 when the incident andreflected electric fields are given by

Ei = (ax cos θi − az sin θi )

× E0e−jβ0(x sin θi +z cos θi )

Er = (ax cos θi + az sin θi )

× �eE0e−jβ0(x sin θi −z cos θi )


μ0ε0.Determine the coefficient �e by applying theboundary conditions on the tangential com-ponents.

1.34. Repeat Problem 1.33 except that �e shouldbe determined using the boundary condi-tions on the normal components. Comparethe answer with that obtained in Problem1.33. Explain.

1.35. For Problem 1.33 determine the:(a) Corresponding incident and reflected

magnetic fields.(b) Electric current density along the inter-

face between the two media.

1.36. A time-harmonic electromagnetic field trav-eling in free space and perpendicularly inci-dent upon a flat surface of distilled water(ε = 81ε0, μ = μ0), as shown in Figure P1-36, creates a reflected field on the free-spaceside of the interface and a transmitted fieldon the water side of the interface. Assum-ing the incident (Ei ), reflected (Er ), and

x

y

z

Incident

Transmitted

e0, m0

m = m0

e = 81e0

Reflected

Figure P1-36

transmitted (Et ) electric fields are given,respectively, by

Ei = ax E0e−jβ0z

Er = ax �0E0e+jβ0z

Et = ax T0E0e−jβz

determine the coefficients �0 and T0. E0 isa constant, β0 = ω

√μ0ε0, β = ω

√μ0ε.

1.37. When a time-harmonic electromagnetic fieldis traveling in free space and is obliquelyincident upon a flat surface of distilled water(ε = 81ε0, μ = μ0), it creates a reflectedfield on the free-space side of the interfaceand a transmitted field on the water side ofthe interface. Assume the incident, reflected,and transmitted electric and magnetic fieldsare given by

Ei = ay E0e−jβ0(x sin θi +z cos θi )

Hi = (−ax cos θi + az sin θi )

×√

ε0

μ0E0e−jβ0(x sin θi +z cos θi )

Er = ay�h E0e−jβ0(x sin θi −z cos θi )

Hr = (ax cos θi + az sin θi )

×√

ε0

μ0�h E0e−jβ0(x sin θi −z cos θi )

Et = ay Th E0e−jβ0

(x sin θi +z

√εε0

− sin2 θi

)

Ht =(

−ax

√1 − ε0

εsin2 θi + az

√ε0

εsin θi

)

×√

ε

μ0Th E0e

−jβ0

(x sin θi +z

√εε0

− sin2 θi)

xy

z

Incident

Transmitted

e0, m0

m = m0

qt

qiqi

e = 81e0

Reflected

Figure P1-37


PROBLEMS 37


μ0ε0.Determine the coefficients �h and Th byapplying the boundary conditions on the tan-gential components. Evaluate all the con-stants.

1.38. Repeat Problem 1.37 except that �h andTh should be determined using the bound-ary conditions on the normal components.Compare the answers to those obtained inProblem 1.37. Explain.

1.39. Repeat Problem 1.37 when the incident,reflected, and transmitted electric and mag-netic fields are given by

Ei =(ax cos θi −az sin θi )

× E0e−jβ0(x sin θi +z cos θi )

Hi = ay

√ε0

μ0E0e−jβ0(x sin θi +z cos θi )

Er =(ax cos θi +az sin θi )

× �eE0e−jβ0(x sin θi −z cos θi )

Hr =−ay

√ε0

μ0�eE0e−jβ0(x sin θi −z cos θi )

Et =[

ax

√1 − ε0

εsin2 θi − az

√ε0

εsin θi

]

× TeE0e−jβ0

(x sin θi +z

√εε0

− sin2 θi)

Ht = ay

√ε

μ0TeE0e

−jβ0

(x sin θi +z

√εε0

− sin2 θi)

�e and Te should be determined using theboundary conditions on the tangential com-ponents.

1.40. Repeat Problem 1.39 except that �e andTe should be determined using the bound-ary conditions on the normal components.Compare the answers to those obtained inProblem 1.39. Explain.

1.41. For Problem 1.16 find the:(a) Average power density at large dis-

tances.(b) Total power exiting through the surface

of a large sphere of radius r(r � a).

1.42. A uniform plane wave traveling in a freespace medium is incident at an oblique angleθi upon an infinite and flat perfect electricconductor (PEC, σ = ∞). The normalizedincident and reflected magnetic fields at the

surface of the PEC (y = 0, on the free spacepart of the PEC), are given by

Hincident(on surface of PEC)

= 1

377(−ax cos θi + az sin θi )

Hreflected(on surface of PEC)

= 1

377(−ax cos θi − az sin θi )

Find the total electric current density Js

induced on the surface of the PEC.

PEC (s = ∞)

x

yqi

free space

Reflectedfield

Incidentfield

Figure P1-42

1.43. The time-harmonic complex field inside asource-free conducting pipe of rectangu-lar cross section (waveguide), shown inFigure P1-43 filled with free space, is givenby

E = ay E0 sin(π

ax)

e−jβz z ,

0 ≤ x ≤ a , 0 ≤ y ≤ b

xz

y

a

b e0, m0

Figure P1-43

where

βz = β0

√1 −

(λ0

2a

)2

E0 is a constant, and β0 = 2π/λ0 =ω

√μ0ε0. For a section of waveguide of

length l along the z axis, determine the:



(a) Corresponding complex magnetic field.(b) Supplied complex power.(c) Exiting complex power.(d) Dissipated real power.(e) Time-average magnetic energy.(f) Time-average electric energy.Ultimately verify that the conservation-of-energy equation in integral form is satisfiedfor this set of fields inside this section of thewaveguide.

1.44. For the waveguide and its set of fields ofProblem 1.43, verify the conservation-of-energy equation in differential form for anyobservation point within the waveguide.

1.45. The normalized time-harmonic electric fieldinside an air-filled, source-free rectangu-lar pipe/waveguide of infinite length andwith cross-sectional dimensions of a and b,whose four walls (left-right, top-bottom) areperfect electric conductors (PEC, σ = ∞), isgiven by

Ex = cos(βx x) sin(βy y)

Ey = sin(βx x) cos(βy y)

where βx and βy are real constants. For non-trivial(nonzero) fields, determine all possiblevalues of βx in terms of a , and βy in termsof b.

a

b

z

x

y

s = ∞

Figure P1-45

1.46. At microwave frequencies, high Q resonantcavities are usually constructed of enclosed

conducting pipes (waveguides) of differentcross sections. One such cavity is that ofrectangular cross section that is enclosed onall six sides, as shown in Figure P1-46. Oneset of complex fields that can exist insidesuch a source-free cavity filled with freespace is given by

E = ay E0 sin(π

ax)

sin(π

cz)

such that

ω = ωr = 1√μ0ε0

√(π

a

)2 +(π

c

)2

where E0 is a constant and ωr is referred toas the resonant radian frequency. Within thecavity, determine the:(a) Corresponding magnetic field.(b) Supplied complex power.(c) Dissipated real power.(d) Time-average magnetic energy.(e) Time-average electric energy.Ultimately verify that the conservation ofenergy equation in integral form is sat-isfied for this set of fields inside thisresonant cavity.

x

c

z

y

a

b

e0, m0

Figure P1-46

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