7/31/2019 ch_1_formod

1/21

RB IITG NPTEL

Web CourseOn

ELECTROMAGNETIC THEORY

Ratnajit BhattacharjeeDepartment of ECE IIT Guwahati

Guwahati- 781039

Syllabus

1 Mathematical Fundamentals

Coordinate Systems and Transformations

Vector Analysis

Differential Length, Area and Volume

Line, Surface and Volume Integrals

Gradient, Divergence and Curl

Divergence Theorem and Stokes Theorem

2

3

Static Electric Fields

Coulombs Law

Electric Field & Electric Flux Density

Gausss Law with Application

Electrostatic Potential, Equipotential Surfaces

Boundary Conditions for Static Electric Fields

Capacitance and Capacitors

Electrostatic Energy

Laplaces and Poissons Equations

Uniqueness of Electrostatic Solutions

Method of Images

Solution of Boundary Value Problems in Different

Coordinate Systems.

Steady Electric Currents

7/31/2019 ch_1_formod

2/21

RB IITG NPTEL

4

Current Density and Ohms Law

Electromotive Force and Kirchhoffs Voltage Law,

Continuity Equation and Kirchhoffs Current Law

Power Dissipation and Joules Law Boundary Conditions for Current Density.

Static Magnetic Fields

Biot-Savart Law and its Application

Amperes Law and its Application

Magnetic Dipole

Behavior of Magnetic Materials

Vector Magnetic Potential

Magnetic Boundary Conditions

Inductances and Inductors, Inductance Calculation for

Common Geometries

Energy Stored in a Magnetic Field.

5 Time Varying Fields & Maxwells Equations

Faradays law of Electromagnetic Induction

Maxwells Equations

Electromagnetic Boundary Conditions

Wave Equations and Their Solutions

Time Harmonic Fields.

6 Electromagnetic Waves

Plane Waves in Lossless Media

Plane Waves in Lossy Media

Poynting Vector and Power Flow in Electromagnetic

7/31/2019 ch_1_formod

3/21

RB IITG NPTEL

Field

Skin Effect

Wave Polarisation

Normal and Oblique Incidence of Plane wave at a(i) Plane Conducting Boundary

(ii) Plane Dielectric Boundary

7 Fundamentals of Antennas and Radiating systems

Fundamentals of Radiation

Radiation Field of an Hertzian Dipole Basic Antenna Parameters like Directivity, Gain,

Beamwidth, Radiation Resistance, Effective Aperture

and Effective Height etc.

Half-wave Dipole Antenna

Quarter-wave Monopole Antenna

Small-Loop Antennas

Basics of Antenna Arrays

.8 Introduction to Numerical Techniques in

Electromagnetics

Finite Difference Equivalent of Laplaces and Poissons

Equation and Solution of Boundary-Value Problems.

Basic Concepts of the Method of Moments

Formulation of Integral Equations

Application of Method of Moments to Wire Antennas

and Scatterers.

7/31/2019 ch_1_formod

4/21

RB IITG NPTEL

CHAPTER-1

1. Introduction

Electromagnetic theory is a discipline concerned with the study ofcharges at rest and in motion. Electromagnetic principles arefundamental to the study of electrical engineering and physics.Electromagnetic theory is also indispensable to the understanding,analysis and design of various electrical, electromechanical andelectronic systems. Some of the branches of study whereelectromagnetic principles find application are:

RF communicationMicrowave EngineeringAntennasElectrical MachinesSatellite CommunicationAtomic and nuclear researchRadar TechnologyRemote sensingEMI EMCQuantum ElectronicsVLSI

Electromagnetic theory is a prerequisite for a wide spectrum of studiesin the field of Electrical Sciences and Physics. Electromagnetic theorycan be thought of as generalization of circuit theory. There are certainsituations that can be handled exclusively in terms of field theory. Inelectromagnetic theory, the quantities involved can be categorized assource quantities and field quantities. Source of electromagneticfield is electric charges: either at rest or in motion. However anelectromagnetic field may cause a redistribution of charges that in turnchange the field and hence the separation of cause and effect is notalways visible.

Electric charge is a fundamental property of matter. Charge exist only inpositive or negative integral multiple of electronic charge, -e,

191.60 10e = coulombs. [It may be noted here that in 1962, Murray Gell-Mann hypothesized Quarks as the basic building blocks of matters.Quarks were predicted to carry a fraction of electronic charge and theexistence of Quarks have been experimentally verified.] Principle ofconservation of charge states that the total charge (algebraic sum ofpositive and negative charges) of an isolated system remainsunchanged, though the charges may redistribute under the influence of

electric field. Kirchhoffs Current Law (KCL) is an assertion of the

7/31/2019 ch_1_formod

5/21

RB IITG NPTEL

conservative property of charges under the implicit assumption thatthere is no accumulation of charge at the junction.

Electromagnetic theory deals directly with the electric and magnetic fieldvectors where as circuit theory deals with the voltages and currents.

Voltages and currents are integrated effects of electric and magneticfields respectively. Electromagnetic field problems involve three spacevariables along with the time variable and hence the solution tends tobecome correspondingly complex. Vector analysis is a mathematical toolwith which electromagnetic concepts are more conveniently expressedand best comprehended. Since use of vector analysis in the study ofelectromagnetic field theory results in real economy of time andthought, we first introduce the concept of vector analysis.

2. Vector Analysis

The quantities that we deal in electromagnetic theory may be eitherscalar or vectors [There are other class of physical quantities calledTensors: scalars and vectors are special cases]. Scalars are quantitiescharacterized by magnitude only and algebraic sign. A quantity that hasdirection as well as magnitude is called a vector. Both scalar and vectorquantities are function of time and position. A field is a function thatspecifies a particular quantity everywhere in a region. Depending uponthe nature of the quantity under consideration, the field may be a vectoror a scalar field. Example of scalar field is the electric potential in a

region while electric or magnetic fields at any point is the example ofvector field.

Fundamentals Vector Algebra:

A vector is represented by a directed line segment: length of the line isproportional to magnitude and the orientation of the directed linesegment with respect to some reference gives the vector.

A vector A can be written as A aA= , where A A= is the magnitude and

A

aA

= is the unit vector which has unit magnitude and direction same as

that ofA .

Two vectors A and B are added together to give another vector C. We

have:

C A B= +

7/31/2019 ch_1_formod

6/21

RB IITG NPTEL

Figure 1: Vector addition

Vector subtraction is similarly carried out as:

( )D A B A B= = +

Figure 2: Vector subtraction

Scaling of a vector is defined as C B= , where C is a scaled version ofvector B and is a scalar.

Some important laws of vector algebra are:

A B B A+ = + Commutative law

( ) ( )A B C A B C+ + = + + Associative law

( )A B A B + = + Distributive law

The position vector Pr of a point P is the directed distance from theorigin (O ) to P i.e. Pr OP=

r.

Figure 3: Distant vector

7/31/2019 ch_1_formod

7/21

RB IITG NPTEL

If Pr OP=r

and Qr OQ=r

are the position vectors of the points P and Q

then the distance vector Q PPQ OQ OP r r= = r r

Product of Vectors

When two vectors A and B are multiplied, the result is either a scalar ora vector depending how the two vectors were multiplied. The two typesof vector multiplication are:

Scalar product (or dot product) A Bg gives a scalarVector product (or cross product) A B gives a vector

The dot product between two vectors is defined as cos ABA B AB =g .

Figure: Dot productThe dot product is commutative i.e. A B B A=g g and distributive i.e.

( )A B C A B A C+ = +g g g . Associative law does not apply to scalar product.

The vector or cross product of two vectors A and B is denoted by A B .

A B is a vector perpendicular to the plane containing

Aand

B, the

magnitude is given by ABABSin and direction is given by right hand ruleas explained in Figure.

Figure: Vector cross product

n AB

A B a ABSin = , where na is the unit vector given by

7/31/2019 ch_1_formod

8/21

RB IITG NPTEL

A Bn A B

a

= r r

The following relations hold for vector product.

A B B A = i.e. cross product is non commutative

( )A B C A B A C + = + i.e. cross product is distributive

( ) ( )A B C A B C i.e. cross product is non associative

Scalar and vector triple product

Scalar triple product ( ) ( ) ( )A B C B C A C A B = = g g g

Vector triple product ( ) ( ) ( )A B C B A C C A B = g g

3. COORDINATE SYSTEMS

In order to describe the spatial variations of the quantities, we requireusing appropriate co-ordinate system. A point or vector can berepresented in a curvilinear coordinate system that may beorthogonal or non-orthogonal. An orthogonal system is one in whichthe co-ordinates are mutually perpendicular. Non-orthogonal co-ordinate

systems are also possible, but their usage is very limited in practice.

Let u =constant, v =constant and w =constant represent surfaces in aco-ordinate system, the surfaces may be curved surfaces in general.Further, let ua , va and wa be the unit vectors in the three co-ordinatedirections (base vectors). In a general right-handed orthogonalcurvilinear system, the vectors satisfy the following relations.

u v w

v w u

w u v

a a a

a a a

a a a

=

=

=

These equations are not independent and specification of one willautomatically imply the other two.Furthermore, the following relations hold

. . . 0

. . . 1

u v v w w u

u u v v w w

a a a a a a

a a a a a a

= = =

= = =

A vector can be represented as sum of its orthogonal components

u u v v w wA A a A a A a= + +

7/31/2019 ch_1_formod

9/21

RB IITG NPTEL

In general u ,v and w may not represent length. We multiply u ,v and w by conversion factors 1h , 2h and 3h respectively to convert differential

changes du , dv and dw to corresponding changes in length 1dl , 2dl and

3dl . Therefore

1 2 3

1 2 3

u v w

u v w

dl a dl a dl a dl

h dua h dva h dwa

= + +

= + +

In the same manner, differential volume dv can be written as

1 2 3dv h h h dudvdw= and differential area 1ds normal to ua is given by

1 2 3ds dl dl = = 2 3h h dvdw and 1 2 3 uds h h dvdwa= . In the same manner, differential

areas normal to unit vectors va and wa can be defined.

In the following sections we discuss three most commonly used

orthogonal co-ordinate systems, viz:

1. Cartesian (or rectangular) co-ordinate system2. Cylindrical co-ordinate system3. Spherical polar co-ordinate system

Cartesian Co-ordinate System

In Cartesian co-ordinate system, we have, ( , , ) ( , , )u v w x y z = . A point

0 0 0( , , )P x y z in Cartesian co-ordinate system is represented as intersection

of three planes 0x x= , 0y y= and 0z z= . The unit vectors satisfies thefollowing relation:

x y z

y z x

z x y

a a a

a a a

a a a

=

=

= . . . 0

. . . 1

x y y z z x

x x y y z z

a a a a a a

a a a a a a

= = =

= = =

7/31/2019 ch_1_formod

10/21

RB IITG NPTEL

Figure: Cartesian coordinate system

0 0 0

x y zOP a x a y a z = + +

In cartesian coordinate system, a vector A can be written as

x x y y z zA a A a A a A= + + . The dot and cross product of two vectors A and B can be written as follows:

( ) ( ) ( )

x x y y z z

x y z z y y z x x z z x y y x

x y z

x y z

x y z

A B A B A B A B

A B a A B A B a A B A B a A B A B

a a a

A A A

B B B

= + +

= + +

=

gr r

Since x , y and z all represent lengths, 1 2 3 1h h h= = = . The differentiallength, are and volume are defined respectively as

7/31/2019 ch_1_formod

11/21

RB IITG NPTEL

x y z

x x

y y

z x

dl dxa dya dza

ds dydzads dxdza

ds dxdya

dv dxdydz

= + +

==

=

=

uur

uur

uur

Cylindrical Co-ordinate System

For cylindrical coordinate systems we have ( , , ) ( , , )u v w z = and a point

0, 0 0( , )P z is determined as the point of intersection of a cylindrical

surface 0 = , half plane containing the z-axis and making an angle

0 = with the xz plane and a plane parallel to xy plane located at 0z z=

as shown in figure.

Fig.

In cylindrical coordinate system, the unit vectors satisfy the followingrelations:

7/31/2019 ch_1_formod

12/21

RB IITG NPTEL

z

z

z

a a a

a a a

a a a

=

=

=A vector A can be written as z zA A a A a A a = + + .

The differential length is defined as

1 2 3 1, 1zdl a d d a dza h h h = + + = = =

Differential volume element in cylindrical coordinates

Differential areas are

z z

ds d dza

ds d dza

ds d d a

=

=

=

uuur

uuur

Differential volume dv d d dz =

7/31/2019 ch_1_formod

13/21

RB IITG NPTEL

Transformation between Cartesian and Cylindrical coordinates

Let us consider a vector z zA a A a A a A = + + is to be expressed in

Cartesian coordinates as x x y y z zA a A a A a A= + + . In doing so we note that

( )x x z z x

A A a a A a A a A a = = + +g g and it applies for other components as

well.

Unit vectors in Cartesian and Cylindrical coordinates

From the figure we note that:

cos

sin

cos( ) sin2

cos

x

y

x

y

a a

a a

a a

a a

=

=

= + =

=

g

g

g

g

Therefore we can write

cos sin

sin cos

x x

y x

z z z

A A a A A

A A a A A

A A a A

= =

= = +

= =

gr

gr

g

These relations can be put conveniently in the matrix form as:

7/31/2019 ch_1_formod

14/21

RB IITG NPTEL

cos sin 0

sin cos 0

0 0 1

x

y

z z

A A

A A

A A

=

A , zA and zA themselves may be functions of , and z. These

variables are transformed in terms ofx , y and z as:

cos

sin

x

y

z z

=

=

=The inverse relation ships are:

2 2

1tan

x y

y

x

z z

= +

=

=

Thus we see that a vector in one coordinate system is transformed toanother coordinate system through two-step process: Finding thecomponent vectors and then variable transformation.

Spherical polar coordinate

For spherical polar coordinate system, we have, ( , , ) ( , , )u v w r = . A point

0 0 0( , , )P r is represented as the intersection of

(i) Spherical surface 0r r=

(ii) Conical surface 0 = , and

(iii) Half plane containing z-axis making angle 0 = with the xz

plane as shown in the figure.

7/31/2019 ch_1_formod

15/21

RB IITG NPTEL

Figure:

The unit vectors satisfy the following relations

r

r

a a a

a a a

a a a

=

=

=

The orientation of the unit vectors are shown in the figure

Figure:

7/31/2019 ch_1_formod

16/21

RB IITG NPTEL

A vector in spherical polar co-ordinate is written as:

r rA A a A a A a = + + and sinrdl a dr a rd a r d = + + . For

spherical polar coordinate system we have 1 1h = , 2 sinh r = and 3h r= .

Figure:

With reference to the Figure, the elemental areas are:

2 sin

sin

r rds r d d a

ds r drd a

ds rdrd a

=

=

=

uuur

uuur

and elemental volume is given by2 sindv r drd d =

Coordinate transformation between rectangular and sphericalpolar

With reference to the Figure we can write the following:

sin cos

sin sin

cos

r x

r y

r z

a a

a a

a a

=

=

=

g

g

g

7/31/2019 ch_1_formod

17/21

RB IITG NPTEL

cos cos

cos sin

cos( ) sin2

x

y

z

a a

a a

a a

=

=

= + =

g

g

g

cos( ) sin2

cos

0

x

y

z

a a

a a

a a

= + =

=

=

g

g

g

Figure

Given a vector r rA A a A a A a = + + in the Spherical Polar coordinate

system, its component in the Cartesian coordinate system can befound out as follows:

. sin cos cos cos sinx x rA A a A A A = = +

Similarly,

. sin sin cos sin cosy y rA A a A A A = = + +

. cos sinz z rA A a A A = = The above expressions may be put in a compact form:

7/31/2019 ch_1_formod

18/21

RB IITG NPTEL

sin cos cos cos sin

sin sin cos sin cos

cos sin 0

Ax Ar

Ay A

Az A

=

The components rA ,A and A themselves will be function of r, and

. r, and are related to x ,y and z as

sin cos

sin sin

cos

x r

y r

z r

=

=

=and conversely,

2 2 2

1

2 2 2

1

cos

tan

r x y z

z

x y z

y

x

= + +

=+ +

=

Using the variable transformation listed above, the vectorcomponents, which are functions of variables of one coordinatesystem, can be transformed to functions of variables of othercoordinate system and a total transformation can be done.

Line, surface and volume integrals

In electromagnetic theory we come across integrals, which containvector functions. Some representative integrals are listed below:

VFdv C dl .CF dl .SF ds etc.

In the above integrals, Fand respectively represent vector and

scalar function of space coordinates. , &C S V represent path, surface

and volume of integration. All these integrals are evaluated usingextension of the usual one-dimensional integral as the limit of a sum,

i.e., if a function ( )f x is defined over arrange a to b of values of x,

then the integral is given by

1

( )b n

i i

ia

Limf x dx f x

n

=

=

7/31/2019 ch_1_formod

19/21

RB IITG NPTEL

where the interval ( , )a b is subdivided into n continuous interval of

lengths 1........... nx x .

Line integral

Line integral.

C

E dl is the dot product of a vector with a specifiedpath C; in other words it is the integral of the tangential component

ofE along the curve C.

Figure: Line integral

As shown in the figure, given a vector field Earound C, we define

the integral cosb

aC

E dl E dl= r

g as the line integral of E along the

curve C. If the path of integration is a closed path as shown inFigure, the line integral becomes a closed line integral and is called

the circulation ofE around C and denoted asC

E dl g

Figure: Closed line integral

7/31/2019 ch_1_formod

20/21

RB IITG NPTEL

Surface integral

Given a vector field A , continuous in a region containing the smooth

surface S, we define the surface integral or the flux of A through S

as

cos nS S S

A ds A a ds A ds = = = g g

Figure: Computation of surface integral

If the surface integral is carried out over a closed surface, then we

write:S

A ds = g which gives the net outward flux of A from S.Volume integrals

We defineV

fdv orv

fdv as the volume integral of the scalar function f(function of spatial coordinates) over the volume V . Evaluation of

integral of the formV

Fdv can be carried out as a sum of three scalarvolume integrals, where each scalar volume integral is a component of

the vector F.

The Del Operator

The vector differential operator was introduced by Sir W. R. Hamiltonand later on developed by P. G. Tait.

Mathematically the vector differential operator can be written in thegeneral form as:

1 2 3

1 1 1

u v wa a ah u h v h w

= + +

In Cartesian coordinates:

7/31/2019 ch_1_formod

21/21

RB IITG NPTEL

x y za a a

x y z

= + +

In cylindrical coordinates:

1

za a az

= + +

and in spherical polar coordinates:

1 1 sin

ra a ar r r

= + +

Gradient of a Scalar