NAME OF STUDENT: EMERIBE CHIKE IKWUDIMMA

STUDENT NUMBER: UAD14303SEL21755

DEPARTMENT: ELECTRICAL ENGINEERING DEPARTMENT

DEGREE: MASTERS/DOCTORATE

COURSE TITLE: ELECTROMAGNETIC FIELDS & WAVES THEORY

ASSIGNMENT PAPER

1.0. INTRODUCTION

The field of electrical engineering can be broadly classified into two namely the

electric circuit theory and electromagnetic fields and wave theory. While circuit

theory deals with electrical analysis, synthesis and design of a wired networks,

electromagnetic theory focuses on the propagation of electrical signals through a

wireless medium.

This assignment takes an in depth look at some concepts in electromagnetic theory

as well as its applications.

In the field of electromagnetic theory, one name stands very tall, the man whom

many have described as the wizard of electromagnetic theory. He is James Clerk

Maxwell. In 1863, James Clerk Maxwell before the Royal Academy of science

proved that electrical signals can be propagated through a wireless medium in form

of electric and magnetic fields. This he proved possible through the introduction

of a displacement current concept. Years latter, this theory of Maxwell was

practically verified and proved correct by the likes of Guglielmo Marconi and

Hertz. James Clerk Maxwell developed four basic equations that characterized

both electric and magnetic fields, these equations till date form the basis for the

study and analysis of electromagnetic fields and wave theory.

1.1. THE CONCEPT OF ELECTRODYNAMICS

Electrodynamics deals with electric and magnetic fields, its inter-

relationships under various charge arrangements. Electrodynamics can be

classified into two areas viz: Electrostatics and Magneto statics

ELECTROSTATICS: This deals with theory that describes physical phenomena

related to the interaction between stationary electric charges or charge distributions

in space.

Recall from coulomb’s law, the force on a charge q located at a distance r

due to the existence of another charge located at a distance is given by the

relation

The interpretation of the above equation(1) is that the electrical force between two

charges separated by distance R is directly proportional to the product of the

magnitude of the charges and inversely proportional to the distances separating the

charge quantities. The distance R is obtained by the vectorial differences of the

position coordinates of the two charge quantities (ie R= r - ). The other

components of the equation (ie 4 ) is a constant associated with the geometry of

space.

Hence the electrostatic field produced by the charge on the test charge q is given

as force per unit charge.

………………(2)

In equation (2) above, the limit taken as q tends to zero implies that the test charge

q is so small to the extent that it has no effect on the field produced by the charge

under study ( )

Substituting equ (1) into equ (2)

……………………………(3)

Here equation (3) is similar to equation (1) above except that the electrostatic field

is produced by the charge under study due to the earlier assumption that the

quantity of the test charge tends to zero and hence the electric field produced by

the test charge is negligible.

Suppose that the magnitude of the discrete electric charges are so minute and the

number of charges so massive, then the concept of electric charge density will be

introduced. Hence,

..........................(4)

The above equation(4) is exactly the same as equation(3) but while equation (3)

considers a point charge, equation (4) considers charge density within a volume

(It is worthy of note here that in reality, equation (4) is what is obtainable as it is

difficult to study point charges).

For the purpose of analysis and study of the property of the electrostatic field E,

the Curl and divergence of the field is taken as below.

Prior to this analysis, let’s look at what divergence ( and Curl ( means.

First of all, divergence and curl are both operators used to study vector and scalar

fields. Now, the divergence of a vector is the outflow of flux( a flux is nothing

other than lines of electric and magnetic fields) from a small closed surface per

unit volume as the volume shrinks to zero. Divergence ordinarily tells us how

much flux is leaving a small area on a per unit volume basis without any direction.

Similarly, the Curl of a vector is the measure of the strength of rotation or vorticity

of the vector field around the corresponding coordinate direction. In order to obtain

the characteristics of a vector or scalar field, divergence and curl operators are

applied to such quantities.

Recall, “from vector field identities, the curl of any well behaved scalar field

is equivalent to zero”[2]

i.e . Applying this in the electrostatic field E.

…………(5)

The physical interpretation of equ (5) is that the electrostatic field is irrotational.

This is so because the electrostatic field is a scalar quantity(ie it has magnitude

without direction). In all of the equations used here, the subscripts indicate the

position coordinate of the vector quantity (for instance, implies the charge

density at the position coordinate of ( ). Similarly the integral sign( in any

of the equations implies the summation of the total charge density within the

enclosed volume.

Similarly, taking the divergence of the electrostatic field E, we have.

………(6)

The above equ(6) is a differential form of Gauss’s law. That is to say that equation

(6) conforms to the earlier statement of Gauss that “ the total amount of fields

radiating from a closed surface is equal to the total number of charges within the

same closed volume”.

1.2 ANALYSIS AND DERIVATIONS OF MATHEMATICAL MODELS

FOR ELECTRIC FIELD OF DIFFERENT CHARGE

DISTRIBUTIONS AND APPLICATION OF GAUSS’S LAW

FIELD OF N-POINT CHARGES

Since coulomb forces are linear, the electric field intensity due to two point

charges Q1 and Q2 is the sum of the forces on Q1 caused by Q1 and Q2 acting alone.

This also implies that electric fields produced by different point charges are also

linear.

Mathematically this can be expressed thus:

E = a + a + - - - + a

OR

FIELD DUE TO A CONTINOUS VOLUME CHARGE DISTRIBUTION

The small amount of charges in a small volume is given by relation

= …………. (8)

Hence the total charge within some finite volume is given thus:

Q =

E =

E =

FIELD OF A LINE CHARGE ( )

In the determination of electric field E at any point resulting from a uniform line

charge density , we use symmetry to determine two factors:

(a) With which coordinate does the field not vary and

(b) which components of the field are not present.

The line – field varies only with r – coordinate.

No element of charge produces a component

Hence = . Each element does produce an Er and EZ component, but the

contributions of charge due to + EZ and - EZ which are equal distances above

and below the point at which the field is being determined will cancel out

leaving only the Er component, which varies only with r.

Hence, OR

=

………….(11)

Equation (11) looks at the differential electric field within the length of the sheet, when integrated within the entire length of the wire, equation (12) is obtained.

Replacing by and summing the contribution from every element of

charge.

………(12)

FIELD OF A SHEET OF CHARGE (SURFACE CHARGE DENSITY )

Another basic charge configuration is the infinite sheet of charge having a uniform

density of such a charge distribution may be used to approximate that

found on the conductors of a strip transmission here that static charges reside on the conductor surfaces and not in their interiors

The line – charge density or charge per unit length is ..... (13)

Where

This is the position vector of the charge density(that is the vectorial distance)

= -

Hence, ……………..(14)

The first line of the above equation looks at the differential field dE in the x direction while the second line takes the integral of differential field so as to obtain the total field of the charge distribution within the sheet.

1.3 GAUSS’S LAW

This states that the electric Flux passing through any closed surface is equal to the total charge enclosed by that surface.

The total flux ( ) passing through the closed surface is obtained by adding the

differential contributions crossing each surface element s

That is ……….. (15)

Expressing Gauss’s statement mathematically, we have

…………(16)

Equation (16) implies that total the amount of electric or magnetic flux emanating from an enclosed surface filled with charges is equal to the total number of

charges( ) within that volume. This is true because each charge quantity radiates

its own field so the total fields from a volume is the same as the sum of the fields from each of the charges within the volume.

Where = Electric Flux Density

One the applications of Gauss’s Law that it is used to convert surface integral to volume integral.

1.4 APPLICATION OF GAUSS LAW TO SOME SYMETRICAL CHARGE DISTRIBUTIONS

The objective here is to use Gauss law to determine (ie the electric flux density)

if the charge distribution is known. To be able to do this a closed surface which

satisfies two conditions has to be chosen. The conditions are (a) where is

everywhere either normal or tangential to the closed surface so that becomes

or zero respectively. (b) on that portion of the closed surface for which

is not zero, = constant. This allows the replacement of the dot product

with the products of the scalars and and then to bring outside the integral

sign(ie the above two conditions if met, makes the integration of the resultant

integral possible). The remaining integral then becomes over that portion of

the closed surface which crosses normally and this is simply the area of this

section of that surface. The Gaussian surface for an infinite line charge is a right

circular cylinder of length L and radius . Also worthy of mention is the fact that

Gauss law cannot be used to obtain solution for electric flux density D if symmetry does not exist.

Using Gaussian law, the electric flux density for spherical surface is given thus

……………….(17)

Recall

Hence, , ie Electric flux density is charge per unit area.

, This is the electric field relation for spherical surface charge

distribution.

The electric flux density ( ) of a uniform line charge for a circular cylinder can be

obtained thus

, This equation takes a look at the flux density round the entire cylinder, based on the divergence theorem which states that “the integral of the normal component of any vector field over a closed surface is equal to the integral of the divergence of this vector field throughout the volume enclosed by the closed surface”. From the foregoing, the circular cylinder has the normal component of the electric flux density by the sides. Hence the field at any other part of the cylinder amounts to zero as indicated in the above integral equation.

………………………(19)

In terms of the charge density the total charge enclosed is

Hence,

1.5 APPLICATION OF GAUSS’S LAW TO A DIFFRENTIAL VOLUME ELEMENT

The aim here is to apply Gauss’s law to obtain electric flux density ( ) for

problems that do not possess any symmetry at all. Without symmetry, a simple

Gaussian surface cannot be chosen such that the normal component of is

constant or zero everywhere on the surface and without such a surface the integral cannot be evaluated.

The approach to this problem is to choose such a very small closed surface that

is almost constant over the surface and the small change in may be adequately

represented using the first two terms of the Taylor’s series expansions of . The

result approaches the correct value as the volume enclosed by the Gaussian surface decreases and approaches zero.

Using this approach will not give the exact value of as obtained in the

symmetrical case above, rather it will give a useful information about the way varies in the

Fig1.1

region of our small surface

Consider any point P in a given volume as shown below

Hence, ……………(20)

…………………………..(21)

Equations (20) and (21) implies that the charge enclosed in the volume is

approximately equal to. Here the approximate result is obtained due to lack of symmetry by the chosen surface thereby making the integral equation unintegrable.

1.6 MAGNETOSTATICS

This is the second part of electrodynamics which deals with stationary electric currents, that is, electric charges moving with constant speeds, and the interactions between these currents.

While coulomb looks at the force produced when a charge imparts on another

charge q called the test charge) Ampere shows that electric current interact in much the same way. Let F denotes such a force acting on a small loop C carrying a

current J located at r, due to the presence of a small loop C carrying a current

located at

According to Ampere’s law, this force in vacuum is given by

Hence ………………………(22)

dL

j

r-

r

O Fig 1.2

The diagram above illustrates Ampere’s law.

“Hence, the differential element dB (r) of the static magnetic field set up, at the

field point r by a small line element d of stationary current at the source point

ri is shown thus”[2]:

………………………………(23

Taking the integral of equation (23) above:

= …………………(24)

In order to assess the properties of B, its divergence and curl are determined thus:

Taking the divergence of both rides of equation (24)

………………(25)

Similarly, taking the curl of equation (24)

……………(26)

Thus the uncoupled equations of electrodynamics for E & B fields are as follows.

1.

2.

3.

4.

1.7 UNIFICATION OF ELECTRODYNAMIC THEORY

Here, the objective is to derive expressions that will couple the E and B fields. However, the unification of the theories of electrostatics and magneto statics is based on these two empirically established facts:

i. “Electric charge is a conserved quantity and electric current is a transport of electric charge. This is expressed in the equation of continuity and as a consequence, in max well’s displacement current”[2].

ii. “A charge in the magnetic flux through a loop will induce an electromotive force (emf) in the loop. This is the celebrated Faraday’s Law of induction”[2].

Hence, the unified theories of electrodynamics is expressed by the four Maxwell’s equations given thus:

1) ……………………………(27)

2) …………………………(28)

3) ……………………………(29)

4) ………………(30)

1.8 PHYSICAL INTERPRETATION OF MAXWELL’S EQUATION

1. Maxwell’s first equation states that the electric flux per unit volume leaving a vanishingly small volume unit is exactly equal to the volume

charge density within that same volume (ie = ) this is differential form

of Gauss Law.

2. The second equation of Maxwell states that whenever there is a rate of charge of magnetic flux (B) an emf is induced. This statement stems from

faraday’s law (ie )

3. The third equation states that the total magnetic flux density within an

enclosed surface is equal to zero (ie )

4. Maxwell’s fourth equation implies that the magneto motive force around a closed path (loop) is equal to the total current enclosed by the loop plus

the displacement current (ie. ). This stems

from Ampere’s experiment (Law).

1.9 ELECTROMAGNETIC DUALITY

Electromagnetic duality takes Maxwell’s equations a step ahead by making the following assumptions

i. “That there exist magnetic monopoles represented by a

magnetic charge density, denoted as ”[2].

ii. “That there exist a magnetic current density denoted as ”[2]

With these new quantities included in the theory, we have a new form of Maxwell’s equation as below

1.

2.

3.

4.

The above equations are called Dirac symmetrised form of Maxwell’s equations or the electro magneto dynamic equations.

It is important to state here that magnetic charge ( ) and magnetic current ( )

exist only in the imaginative sense.

2.0 ELECTRIC POLARIZATION AND DIPOLE

ELECTRIC DIPOLE

An electric dipole is the name given to two point charges of equal magnitude and opposite sign, separated by a distance which is small compared to the distance to the point P at which electric and potential fields (E&V) are to be determined

“Dielectrics (insulators) are materials whose dominant charges in atoms and molecules are bound negative and positive charges that are held in place by atomic and molecular forces, and they are not free to travel. However, when

external fields are applied these bound negative and positive charges would have their respective centroids shifted slightly in positions relative to each other, thus creating numerous electric dipoles”[2]. This is illustrated in the figure below

L

Fig1.3a Fig1.3b

Absence of applied under applied field Under applied Field

Mathematically, the dipole moment is given thus:

d = Q (31)

“where Q is the magnitude (in coulombs) of each of the negative and

positive charges whose centroids are displaced vectorially by distance ”[2]

Z

+Q

r

y

d

-Q

x

Fig 1.4

From the figure above, the electric field of a dipole is given thus.

…………………………..(32)

Hence the dipole potential V = ………………………(33)

ELECTRIC POLARIZATION

This is the dipole moment per unit volume. The total dipole moment of a

material is obtained by summing the dipole moments of all the orientational

polarization dipoles for a volume where there are Ne electric dipoles,

we have mathematically:

………………(34)

Electric polarization for dielectrics can be produced by any of the following three mechanisms Viz:

(a) DIPOLE OR ORIENTATIONAL POLARIZATION

“This Polarization is evident in material that in the absence of an applied field and owing to their structure permanent dipole moments that are randomly oriented. However when an electric field is applied the dipoles tend to align with the applied field such materials are known as polar materials e.g water”[1].

(b) IONIC OR MOLECULAR POLARISTION

“This polarization is evident in materials such as sodium chloride (NaCl), that posses positive and negative ions and that tend to displace themselves when an electric field is applied”[1].

(c) ELECTRONIC POLARISATION

“This polarization is evident in most material and it exists when an applied electric field displaces the electric cloud of an atom relative to the center of the nucleus”[1].

P

- - - - - - - - - -

Fig1.5

The setup in fig 1.5 above has two sections, one contains free space while the

second section contains a dielectric slab. Whereas the applied electric field

maintains its value, the electric flux density (D) inside the dielectric materials differ from what would exist where the dielectric material is replaced by free space. In the free space part of the parallel plate capacitor of fig 1.5, the electric

flux density D is given by = ……..(36)

In the dielectric portion, the electric flux density D is related to that in free space

by the relation.

D = + P ……………(37)

The mathematical relationship of polarization to applied electric field is

P = …………..(38)

= ………………(39)

Where represents the electric susceptibility which is the degree of displacement

of dielectric charges wherever electric potential is applied.

Hence equation (37) now becomes

=

= …………….(40)

2.1 ELECTROMAGNETIC WAVE EQUATIONS

The first part of this assignment x-rayed in details electric and magnetic fields. These fields during propagation is space, at certain frequency propagate as waves. Hence this section of the assignment is concerned with derivation and understanding mathematical wave equations used to calculate the electric and magnetic field components (E & B) from prescribed charge and current

distributions ( & J). the implications of these wave equations is also covered in

this section.

THE WAVE EQUATION FOR ELECTRIC FIELD (E) (TIME VARYING FIELD)

The wave equation for both E and B field vectors are derived here based on the following assumptions.

(1)“That in a volume, the net charges is equal to zero (i.e = 0 )”[2]

(2)“That there is no electromotive force induced (i.e EMF = 0)”[2]

WAVE EQUATION FOR E

Taking the curl of equation (28)

)…………..(41)

Recall, from vector field identities

……………(42)

Also based on the first assumption above = 0

Hence,

Also from the second assumption

EMF = 0, hence from ohm’s law total current

=> …………………….(43)

Hence equation (41) now becomes:

…………….(44)

Recalls, = 1/C2, where C is the speed of light, opening the bracket of equation

(44)

………………….(45)

……………………..(46)

Equation (46) is the homogeneous equation for the determination of the electric field E.

2.2 THE WAVE EQUATION FOR MAGNETIC FIELD (B) (TIME VARYING FIELDS)

The wave equation for magnetic field (B) is derived in much the same way as the wave equation for electric field (E).

Taking the curl of equation (30) and applying ohm’s law of J =

= ………………(47)

Recall …………..(48)

Recall from equation (29)

Hence, …………………(49)

Equation (50) represents the homogeneous wave equation for magnetic field

2.3 TIME HARMONIC WAVE EQUATION FOR E – FIELD

Time harmonic fields are fields that can be expressed using Fourier component

For the E – field, the time harmonic wave equation can be obtained by replacing E

with

Substituting this into equation (46)

………………(52)

………………(53)

………………….(54)

Equation (54) is the time harmonic wave equation for the electric field (E)

2.4 TIME HARMONIC WAVE EQUATION FOR B – FIELD

Similarly, replacing the B – field with the Fourier component (i.e B =

)

From equation (50)

……………………(55)

……………(56)

…………………(57)

…………………(58)

Equation (58) is the time harmonic equation for the magnetic field B.

2. 5 LORENTZ TRANSFORMATION

This is the property of an inertia system whereby all geometrical objects (vectors, tensors) in an equation describing the physical, process must transform in the same way

Consider the figure below

y

P(t,x,y,z)

P(

X

Z

Fig1.6

“According to Einstein, the two postulates of special relativity require that the spatial coordinates and times as measured by an observer in

Taking the difference between the square of (a) and the square of (b) we have

From equation (c) and (d) above, it is observed that Y and Z coordinates are unaffected by the transformation motion of the inertial system

………………..(60)

Equation (60) means that if a light wave is transmitted from the coinciding origins

O and O’ at time

Observer concludes that the speed of light in vacuum is C. Hence the speed of light

in

A linear coordinate transformation which has this property is called a (HOMOGENEOUS LORENTZ TRANSFORMATION).

2.6 GREENS FUNCTION

In electromagnetic, solutions to many problems are obtained using a second order

uncoupled partial differential equation derived from Maxwell’s equation as shown

in the other part of this assignment. The form of most of these types of solutions is

an infinite series, provided the partial differential equation and the boundary

conditions representing the problems are separable in the coordinate system chosen

The difficulty in using these type of solutions to obtain an insight into the

behaviour of the function is that they are usually slowly convergent especially at

regions where rapid changes occur. Greens function is used to obtain a closed form

solutions for some problems and their associated regions. With the Green’s

function technique, a solution to the partial differential equation is obtained using a

unit source (impulse, Dirac delta) as the driving function. This driving function is

known as the GREEN’S FUNCTION. The solution to the actual driving function is

written as a super position of the impulse response solutions with the Green’s

function is nothing else but the impulse response of a system. “For a given

problem, the Green’s function can take various forms. One form of its solution can

be expressed in terms of finite explicit functions. Another form of the Green’s

function is to construct its solution by an infinite series of suitably chosen

orthonormal functions.

The form of the Green’s Function that is most appropriate depends on the problem

in question”[1]

PROPERTIES OF GREEN’S FUNCTIONS

(1) G (x, x1) satisfies the homogenous differential equation except at x = x1

(2) G (x, x1) is symmetrical with respect to x and x

(3) G (x, x1) satisfies certain homogenous boundary conditions

(4) G (x, x1) is continuous at x = x

(5) [ d G (x, x1)] / dx has a discontinuity of 1/ [ p(x1)] at x = x1

DERIVATION OF A CLOSED FORM GREEN’S FUNCTION FOR A

ONE DIMENSIONAL DIFFERENTIAL EQUATION

This section of the assignment considers derivation of Green’s functions for the

one dimensional differential equation of the STURM LIOUVILLE form thus:

d [p (x) ] - = F(x)...........(61)

Equation (61) above is called STURM-LIOURILLE form of one dimensional

differential equation subject to homogeneous boundary conditions.

Also equation (61) can be re-written as thus

dxdd

dx

Ly =

Where L is the STURM-LOUVILLE operator

L = __ q ---- (63)

Hence, every general one – dimensional, source excited, second order differential

equation of the form

………..(64)

Can be converted to Sturm – Liouville form following the procedure below

Expanding equation

……………..(65)

Dividing equation (64) by A(x) and

Dividing equation (65) by P(x) we obtain.

…………………….(66)

………….(67)

Comparing equation (66) and (67) it will be observed that

…………………………………(68)

………………………………………(69)

dx d P

dx d

………………………………………….(70)

From equation (68)

……………………………………(71)

Equation (71) is linear first order differential equation with a particular solution

………………………………(72)

From equ (69) ……………………(73)

With a particular solution

………………………(74)

From equ (70)

With particular solution

…………………………..(75)

Summarily, from the foregoing analysis a one dimensional, source excited, second

order differential equation of the form in equ (64) is converted to a STURM –

LIOUVILLE form of equ (61) by letting

be that of equ (72 ) be that of equ (74), and f(x) be that of equ(75).

Now, having shown that each general second – order, source – excited differential

equation can be converted to a STURM – LIOUVILLE form, Green’s function can

be derived by using a more general form of STURN – LIOUVILLE equation of the

form

………………….(76)

OR

………………………….(77)

Where L is the STURM – LIOUVILLE operator. represent, eiyeenvalues,

determined by the nature and boundary of the region of interest.

For the homogenous equation (76)

(a) Let y, (x) be in nontrivial solution in the internal a x’ satisfying the

boundary conditions at x = a

Hence, y,(x) and G(x, x’) are related thus:

G(x, ) = (x) a ………………….(78)

(b) Let y2(x) represent a nontrivial solution of equation (76) in the interval x

Satisfying the boundary condition at x = b

Hence y2(x) and G(x, x’) are related thus:

………..(79)

Recall, must be continuous at

………(80)

…………………………(81)

Also recall, the derivation of the Green’s function must be discontinuous at x =

by an amount of

Hence …………………..(82)

Solving (81) and (82) simultaneously

……………………..(83)

…………….............(84)

Where w (x1) is the wrong kin of Y1 and Y2 at x = x1 defined as

………………….(85)

Hence, the chosed form of Green’s function for the differential equation of (76)

can be written as

………………….(86)

Where Y1 (x) and Y2 (x) are two independent solution of the homogeneous form of

the differential equation (76) each satisfying respectively, the boundary condition

at x = a and x = b.

The above procedure used to derive the closed form of Green’s function is

predicated on the following premises.

(1) The solution to the homogenous differential equation (76) is known

(2) The Green’s function is desired in closed form, instead of an infinite series

of orthogonal functions.

2.7 THE INTEGRAL EQUATION AND MOMENT METHOD

Because of the limitations of some known methods of solution to radiation and

scattering characteristic of electromagnetic fields and waves by some medium.

Scientists and engineers over the years have embarked on painstaking research to

overcome this trend.

The methods of solutions such as Geometrical optics, physical optics and modal

solutions have been observed to generate approximate (inaccurate) results of

induced current on the surface of the finite size target/medium (strip and rectangle

plate). These inaccurate results obtained form the above named methods normally

appear inform of infinite series with little or no convergence especially if the

dimensions of the medium (i.e the target) exceed about one wavelength.

However, integral equation (IE) was developed to overcome the above stated

limitations of the earlier methods of solutions. The integral equation are used to

model/describe radiation and scattering characteristic (i.e. determine the accurate

current induced) of some radiating and scattering system/medium with finite size.

The induced current is modeled using an integral equation containing the induced

current density which forms the actual variable to be determined.

To solve this integral equation, moment method of solution (i.e. numerical

technique)is employed. The moment method is used to decompose the integral

equation into matrix form which is then computed. With the integral equations,

many practical scattering and radiating systems / medium can be analyzed.

y

x

h

P

t

w/2 w/2

Fig 1.7 Geometry of a line source with a two dimensional finite width strip.

From fig 1.7 above, the field radiated by a line source of steady current Iz without

the strip width is given thus:

consider the situation where the field from source is directed at the strip width,

once this is done, a linear current density Jz is induced on the strip. This induced

current now re-radiates and produces an electric field know as scattered field or

reflected field.

The scattered field if given as:

= 8)

Where Jz ( )

Hence, ………(90)

is know as the Hankel function of the second kind of order zero.

From the foregoing, the ELECTRIC FILED INTEGRAL EQUATION (EFIE)

without the scattered filed is given as

………(91)

However, to any observer at any point, the total field is the sum of the direct field

and scattered field.

i.e

This implies that,

Equations (91) and (93) can be resolved into a matrix of the form:

Where is called the basis function.

=

The values of …… aN are the solutions of the induced current density as

required. The solution of the matrix can be obtained using moment method (ie. Numerical computation). It is important to emphasize here that equation (91) is based on the boundary conditions of vanishing total tangential electric field on the conducting strip surface.

2.8 SCATTERING PHENOMENA OF ELECTROMAGNETIC WAVES

Scattering and diffraction are both properties of electromagnetic field and waves. This section of the assignment examined the scattering phenomena of electromagnetic waves by scattering targets +that are plan, cylindrical and spherical in geometry. The nature of the scattered field produced by these geometrical shapes/target is also examined.

Scattering phenomena in electromagnetic ware propagation is a very useful

concept in antenna engineering and RAD The scattered fields are used to

identify objects at a distance.

This application is very useful in military and aviation.

SCATTERING BY PLANAR SURFACES

In a target with planar surface, electromagnetic wave propagated along the part of

such target is scattered by what is known as the echo area or radar cross section (

).

The echo area is the area that intercepts the amount of power that when scattered in equal directions, produces at the receiver a density that is equal to the density scattered by the actual target [ 1 ]. One of the important parameters for a two-dimensional target is known as scattering width (SW).

y

x

Z w

Fig1.8

y

x

Fig1.9

Figures 1.8 and 1.9 shows uniform plane wave incident on a finite width strip. Fig. 1.8 is Transverse magnetic polarization ™, fig 1.9 is Transverse Electric polarization (TE).

From the figures above, the incident electric and magnetic fields can be expressed thus:

……………(95)

+ + ysin .....................(96)

Eo = magnitude of the incident electric field

= wave impedance.

For the reflected electric and magnetic fields are expressed as follows:-

……(97)

…………………….(98)

From the equations, the current density induced on the surface of the target is obtained thus

……………(99)

………………(100)

SCATTERING BY CIRCULAR CYCLINDERS

The study of the scattering characteristics of cylindrical surfaces is very important considering the fact that most practical target targets have such geometry.

Consider a Transverse magnetic polarized electromagnetic wave by a conducting circular cylinder. The electric and magnetic field functions are given mathematically thus:

………………………..(101)

Applying Bessel transformation,

Where = the unknown Amplitude, Given as:

Hence the electrical scattered field of equation (104) is now

Hence,

Corresponding, the magnetic field component for both and coordinate is given

thus

For the scattered magnetic field component we have

Hence,

Which implies

SCATTERING BY SPHERICAL SURFACE

Spherical geometrical structures are one of the most popular geometries encountered in the study and practical application of electromagnetic wave propagation and scattering characteristics. To this end its analysis and study of

field equations associated with scatterers with such geometries becomes highly imperative.

“Electromagnetic propagation on spherical geometry has angular variations , this

variation within the interval 0 ≤ ≥ is expressed by the Legendre polynomials

P(cos ) as well as the Legendre function (cos )”[1].

“The series of solutions to this polynomials are known as TESSERAL HARMONICS which are always orthogonal to the spherical surfaces. Hence, wave functions defined over a spherical structure can be represented using TESSERAL HARMONICS”[1].

The total electric field expression for the r, and coordinate components of the

spherical surface are as follows.

=

Similarly, the total magnetic field expression for the r, coordinate

components of the spherical surface are as follows:

=

In the above mathematical expressions, and

From equations (121) and (122), , and are constants while ( ) and

are spherical Bessel and Hankel functions respectively.

Similarly, equation (115) to (120) show the expression of the total fields a receiver is expected to capture whenever an electromagnetic wave encounters a scatterer (target) with spherical surface structure. The nature of these fields captured by the

receiver gives a description of the type or target to an observer at a distance. This application is today common in RADAR, SONAR etc.

2.9 CONCLUSION

This assignment x-rayed into the field of electromagnetic fields and waves. It started with analysis and derivation of electric and magnetic fields of various charge distributions, electrostatics, magneto statics, Maxwell’s equations, Lorentz transformation, electromagnetic scattering, polarization and dipoles, electromagnetic duality, integral equations and moment methods. In addition, this conclusion will cover applications of the principles and theories of electromagnetic to our everyday world.

The advent of electromagnetic fields and waves theory resulted to quantum leap in various fields of human endeavours. Although the application of this theory of electromagnetic cuts across diverse areas, but this assignment restricts itself to the following areas of applications: (a) Wireless communications, (b) Oil and gas exploration/survey (c) Electrical Machines and drives, (d) Power generation (e) Remote sensing.

WIRELESS COMMUNICATION APPLICATION:

Prior to the advent of electromagnetic theory, communications was purely based on wired networks. This wired form of communication networks made communication both costly and cumbersome. As examined in the introduction part of this assignment, the paper presented by James Clerk Maxwell to Royal Academy of Science in 1863 showed that electric and magnetic fields can be propagated in form of waves through a wireless medium (free space) with aid of the displacement current. This stroke of theoretical genius marked a watershed in communication engineering. This theory was practically demonstrated in 1897 by Guglielmo Marconi in his first transatlantic mobile communication. Since 1897, wireless communication technique and services has continued to develop and grow by leaps and bounds due to enabling technologies. Between 1960s and 1970s, Bell Laboratories developed the concept of cellular wireless communication. This discovery with the development of miniaturized and highly reliable radio

frequency hardware gave rise to the explosion in wireless communication as is experienced today.

OIL AND GAS EXPLORATION/SURVEY APPLICATION:

Developments in electromagnetics have brought new techniques in oil and gas survey technology. The controlled source Electromagnetic Method (CSEM) is used to determine the existence or otherwise of an oil reservoir. This process is called electromagnetic survey technology. The principle used here is the concept of remote sensing. Here, electromagnetic wave signal is sent through the soil subsurface down across the various rock layers . These signals once they encounter an obstacle (Oil deposits) they will be reflected back in form of a scattered field to a receiver. The received signals are interpreted accordingly. Research shows that electromagnetic waves are sensitive to resistivity of various materials, phenomenon is used to identify the availability of hydrocarbon materials or oil reservoir within an area under consideration.

ELECTRICAL MACHINE APPLICATION:

The era of industrial revolution in the early 19th century was driven by the inventions of various forms of electrical and mechanical machines and drives. Applications of electromagnetic engineering gave rise to the invention of certain electrical machines such as transformers, induction and synchronous machines, electric motors, electric generators etc. The discovery made by Hans Oersted a lecturer at a university in Copenhghan in 1820 about the magnetic effect of an electric current and also the invention of Michael Faraday about electromagnetic induction in 1831, provided the foundation for the construction of various forms of electrical rotating machines. Similarly, electrical machines such as transformers are based on the principles of electromagnetic induction. Used to transform voltages, transformers are now as important as air to mankind today as it concerns power transmission and distribution. Also, other electrical machines such as DC motors and generators are today useful in vehicle manufacture and other related fields.

POWER GENERATION APPLICATION:

Among other things, electromagnetic principles have wider applications in power generation especially in hydropower generation. Based on the principles of electromagnetic induction as propagated by Michael Faraday, electricity can be generated in a turbine system arrangement of hydro electricity generator.

REMOTE SENSING APPLICATION:

Environmental remote sensing is totally dependent on electromagnetic wave theory. The application of electromagnetic theory in environmental remote sensing entails detection of objects at a distance without physically making contacts with such objects. This is possible because of the property of the electromagnetic fields to be reflected and scattered upon hitting a target.

Once electromagnetic field is radiated in the direction of a target, upon hitting that target, the electromagnetic field is then reflected and scattered . these scattered fields are then received by a receiver for interpretation. The interpretation of the reflected signal reveals the location of the target and other characteristics of the target too. This particular application of electromagnetic theory is common in both military (RADAR) operations and aviation technology. It is also found in SONAR systems application for measuring sea depth and detection of oil deposits.

It is important to state here that although this assignment covered a wide range of concepts in electromagnetic as contained in the curriculum, this field of study was not exhausted. Indeed, research in this area remains for the student a continous exercise.

2.11 BIBILIOGRAPHY

1.Constantine .A. Balanis, Advanced Engineering Electromagnetics.

2.Bio Thide (2001), Electromagnetic fields theory

3.George Frese, Hans Engeles, (April 2003), Magnetic Resonance and imaging (MRI) and electromagnetic fields .

4.Digital energy Journal (2011), Latest with Electromagnetics.

5.Thomes Penick (1998), The Birth of electromagnetic Engineering.

6. Theodore S. Rapport (2002), Wireless communications principles and practice.