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RB IITG NPTEL
Web Course
On
ELECTROMAGNETIC THEORY
Ratnajit Bhattacharjee
Department of ECE IIT Guwahati
Guwahati- 781039
Syllabus
1
2
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 Stoke’s Theorem
Static Electric Fields

Coulomb’s Law

Electric Field & Electric Flux Density

Gauss’s Law with Application

Electrostatic Potential, Equipotential Surfaces

Boundary Conditions for Static Electric Fields

Capacitance and Capacitors

Electrostatic Energy

Laplace’s and Poisson’s Equations

Uniqueness of Electrostatic Solutions

Method of Images

Solution of Boundary Value Problems in Different
Coordinate Systems.
3
Steady Electric Currents
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
Current Density and Ohm’s Law

Electromotive Force and Kirchhoff’s Voltage Law,
Continuity Equation and Kirchhoff’s Current Law
4

Power Dissipation and Joule’s Law

Boundary Conditions for Current Density.
Static Magnetic Fields

Biot-Savart Law and its Application

Ampere’s Law and its Application

Magnetic Dipole

Behavior of Magnetic Materials

Vector Magnetic Potential

Magnetic Boundary Conditions

Inductances and Inductors, Inductance Calculation
for Common Geometries

5
6
Energy Stored in a Magnetic Field.
Time Varying Fields & Maxwell’s Equations

Faraday’s law of Electromagnetic Induction

Maxwell’s Equations

Electromagnetic Boundary Conditions

Wave Equations and Their Solutions

Time Harmonic Fields.
Electromagnetic Waves

Plane Waves in Lossless Media

Plane Waves in Lossy Media
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
Poynting Vector and Power Flow in Electromagnetic
Field
7

Skin Effect

Wave Polarisation

Normal and Oblique Incidence of Plane wave at a
(i)
Plane Conducting Boundary
(ii)
Plane Dielectric Boundary
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
Laplace’s
and
Poisson’s 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.
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CHAPTER-1
1. Introduction
Electromagnetic theory is a discipline concerned with the study of
charges at rest and in motion. Electromagnetic principles are
fundamental to the study of electrical engineering and physics.
Electromagnetic theory is also indispensable to the understanding,
analysis and design of various electrical, electromechanical and
electronic systems. Some of the branches of study where
electromagnetic principles find application are:
RF communication
Microwave Engineering
Antennas
Electrical Machines
Satellite Communication
Atomic and nuclear research
Radar Technology
Remote sensing
EMI EMC
Quantum Electronics
VLSI
Electromagnetic theory is a prerequisite for a wide spectrum of studies
in the field of Electrical Sciences and Physics. Electromagnetic theory
can be thought of as generalization of circuit theory. There are certain
situations that can be handled exclusively in terms of field theory. In
electromagnetic theory, the quantities involved can be categorized as
source quantities and field quantities. Source of electromagnetic
field is electric charges: either at rest or in motion. However an
electromagnetic field may cause a redistribution of charges that in turn
change the field and hence the separation of cause and effect is not
always visible.
Electric charge is a fundamental property of matter. Charge exist only
in positive or negative integral multiple of electronic charge, -e,
e  1.60 1019 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 the existence of Quarks have been experimentally verified.]
Principle of conservation of charge states that the total charge
(algebraic sum of positive and negative charges) of an isolated system
remains unchanged, though the charges may redistribute under the
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influence of electric field. Kirchhoff’s Current Law (KCL) is an assertion
of the conservative property of charges under the implicit assumption
that there is no accumulation of charge at the junction.
Electromagnetic theory deals directly with the electric and magnetic
field vectors where as circuit theory deals with the voltages and
currents. Voltages and currents are integrated effects of electric and
magnetic fields respectively. Electromagnetic field problems involve
three space variables along with the time variable and hence the
solution tends to become correspondingly complex. Vector analysis is a
mathematical tool with which electromagnetic concepts are more
conveniently expressed and best comprehended. Since use of vector
analysis in the study of electromagnetic field theory results in real
economy of time and thought, we first introduce the concept of vector
analysis.
2. Vector Analysis
The quantities that we deal in electromagnetic theory may be either
scalar or vectors [There are other class of physical quantities called
Tensors: scalars and vectors are special cases]. Scalars are quantities
characterized by magnitude only and algebraic sign. A quantity that
has direction as well as magnitude is called a vector. Both scalar and
vector quantities are function of time and position. A field is a function
that specifies a particular quantity everywhere in a region. Depending
upon the nature of the quantity under consideration, the field may be a
vector or 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
of vector field.
Fundamentals Vector Algebra:
A vector is represented by a directed line segment: length of the line is
proportional to magnitude and the orientation of the directed line
segment with respect to some reference gives the vector.
ˆ , where A  A is the magnitude
A vector A can be written as A  aA
and aˆ 
A
is the unit vector which has unit magnitude and direction
A
same as that of A .
Two vectors A and B are added together to give another vector C .
We have:
C  A B
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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 of
vector B and  is a scalar.
Some important laws of vector algebra are:
A B  B  A
A  ( B  C )  ( A  B)  C
Commutative law
Associative law
 ( A  B)   A   B
Distributive law
The position vector rP of a point P is the directed distance from the
origin ( O ) to P i.e. rP  OP .
Figure 3: Distant vector
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If rP  OP and rQ  OQ are the position vectors of the points P and Q
then the distance vector PQ  OQ  OP  rQ  rP
Product of Vectors
When two vectors A and B are multiplied, the result is either a scalar or a
vector depending how the two vectors were multiplied. The two types of
vector multiplication are:
Scalar product (or dot product) A B gives a scalar
Vector product (or cross product) A  B gives a vector
The dot product between two vectors is defined as A B  AB cos AB .
Figure: Dot product
The dot product is commutative i.e. A B  B A and distributive i.e.
A ( B  C )  A B  A C . 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 A and B , the
magnitude is given by ABSin AB and direction is given by right hand rule as
explained in Figure.
Figure: Vector cross product
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A  B  aˆn ABSin AB , where
aˆn 
aˆ n
is the unit vector given by
A B
A B
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)
Vector triple product
A  ( B  C )  B( A C )  C ( A B)
3. COORDINATE SYSTEMS
In order to describe the spatial variations of the quantities, we require
using appropriate co-ordinate system. A point or vector can be
represented in a curvilinear coordinate system that may be orthogonal or
non-orthogonal. An orthogonal system is one in which the 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 a coordinate system, the surfaces may be curved surfaces in general. Further,
let aˆu , aˆv and aˆ w be the unit vectors in the three co-ordinate directions
(base vectors). In a general right-handed orthogonal curvilinear system,
the vectors satisfy the following relations.
aˆu  aˆv  aˆw
aˆv  aˆw  aˆu
aˆw  aˆu  aˆv
These equations are not independent and specification of one will
automatically imply the other two.
Furthermore, the following relations hold
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aˆu .aˆv  aˆv .aˆw  aˆw .aˆu  0
aˆu .aˆu  aˆv .aˆv  aˆw .aˆw  1
A vector can be represented as sum of its orthogonal components
A  Au aˆu  Av aˆv  Awaˆw
In general u , v and w may not represent length. We multiply u , v and w by
conversion factors h1 , h2 and h3 respectively to convert differential changes
du , dv and dw to corresponding changes in length dl1 , dl2 and dl3 .
Therefore
dl  aˆu dl1  aˆv dl2  aˆwdl3
 h1duaˆu  h2 dvaˆv  h3dwaˆw
In the same manner, differential volume dv can be written as
dv  h1h2 h3dudvdw and differential area ds1 normal to aˆu is given by
ds1  dl2 dl3 = h2 h3dvdw and ds1  h2 h3dvdwaˆu . In the same manner, differential
areas normal to unit vectors aˆv and aˆ w can be defined.
In the following sections we discuss three most commonly used orthogonal
co-ordinate systems, viz:
1. Cartesian (or rectangular) co-ordinate system
2. Cylindrical co-ordinate system
3. Spherical polar co-ordinate system
Cartesian Co-ordinate System
In Cartesian co-ordinate system, we have, (u, v, w)  ( x, y, z ) . A point
P ( x0 , y0 , z0 ) in Cartesian co-ordinate system is represented as intersection
of three planes x  x0 , y  y0 and z  z0 . The unit vectors satisfies the
following relation:
aˆ x  aˆ y  aˆ z
aˆ y  aˆ z  aˆ x
aˆ z  aˆ x  aˆ y
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aˆ x .aˆ y  aˆ y .aˆ z  aˆ z .aˆ x  0
aˆ x .aˆ x  aˆ y .aˆ y  aˆ z .aˆ z  1
Figure: Cartesian coordinate system
OP  aˆ x x0  aˆ y y0  aˆ z z0
In cartesian coordinate system, a vector A can be written as
A  aˆ x Ax  aˆ y Ay  aˆ z Az . The dot and cross product of two vectors A and B
can be written as follows:
A B  Ax Bx  Ay By  Az Bz
A  B  aˆ x ( Ay Bz  Az By )  aˆ y ( Az Bx  Ax Bz )  aˆ z ( Ax By  Ay Bx )
aˆ x
aˆ y
aˆ z
 Ax
Ay
Az
Bx
By
Bz
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Since x , y and z all represent lengths, h1  h2  h3  1 . The differential
length, are and volume are defined respectively as
dl  dxaˆ x  dyaˆ y  dzaˆ z
ds x  dydzaˆ x
ds y  dxdzaˆ y
ds z  dxdyaˆ x
dv  dxdydz
Cylindrical Co-ordinate System
For cylindrical coordinate systems we have (u, v, w)  (  ,  , z ) and a point
P( 0,0 , z0 ) 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
z  z0 as shown in figure.
Fig.
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In cylindrical coordinate system, the unit vectors satisfy the following
relations:
aˆ   aˆ  aˆ z
aˆ  aˆ z  aˆ 
aˆ z  aˆ   aˆ
A vector A can be written as
A  A aˆ   A aˆ  Az aˆ z .
The differential length is defined as
dl  aˆ  d    d aˆ  dzaˆ z
h1  1, h2   h3  1
Differential volume element in cylindrical coordinates
Differential areas are
ds   d dzaˆ 
ds  d  dzaˆ
dsz   d d  aˆ z
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Differential volume
dv   d  d dz
Transformation between Cartesian and Cylindrical coordinates
Let us consider a vector A  aˆ  A  aˆ A  aˆ z Az is to be expressed in
Cartesian coordinates as A  aˆ x Ax  aˆ y Ay  aˆ z Az . In doing so we note that
Ax  A aˆ x  (aˆ  A  aˆ A  aˆ z Az ) aˆ x and it applies for other components as
well.
Unit vectors in Cartesian and Cylindrical coordinates
From the figure we note that:
aˆ  aˆ x  cos 
aˆ  aˆ y  sin 

aˆ aˆ x  cos(  )   sin 
2
aˆ aˆ y  cos 
Therefore we can write
Ax  A aˆ x  A cos   A sin 
Ay  A aˆ x  A sin   A cos 
Az  A aˆ z  Az
These relations can be put conveniently in the matrix form as:
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 Ax  cos 
 A    sin 
 y 
 Az   0
 sin 
cos 
0
0  A 
0  A 
1   Az 
A , Az and Az themselves may be functions of  ,  and z . These
x , y and z
x   cos 
variables are transformed in terms of
as:
y   sin 
zz
The inverse relation ships are:
  x2  y 2
  tan 1
y
x
zz
Thus we see that a vector in one coordinate system is transformed to
another coordinate system through two-step process: Finding the
component vectors and then variable transformation.
Spherical polar coordinate
For spherical polar coordinate system, we have,
point P(r0 , 0 , 0 ) is represented as the intersection of
(i)
(ii)
(iii)
(u, v, w)  (r , ,  ) . A
Spherical surface r  r0
Conical surface    0 , and
Half plane containing z -axis making angle   0 with the
xz plane as shown in the figure.
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Figure:
The unit vectors satisfy the following relations
aˆr  aˆ  aˆ
aˆ  aˆ  aˆ
aˆ  aˆr  aˆ
The orientation of the unit vectors are shown in the figure
Figure:
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A
vector
in
spherical
A  Ar aˆr  A aˆ  A aˆ
and
polar
co-ordinate
is
written
as:
dl  aˆr dr  aˆ rd  aˆ r sin  d .
For
spherical polar coordinate system we have h1  1 , h2  r sin  and h3  r .
Figure:
With reference to the Figure, the elemental areas are:
dsr  r 2 sin  d d aˆr
ds  r sin  drd aˆ
ds  rdrd aˆ
and elemental volume is given by
Coordinate
transformation
spherical polar
dv  r 2 sin  drd d
between
rectangular
With reference to the Figure we can write the following:
aˆr aˆ x  sin  cos 
aˆr aˆ y  sin  sin 
aˆr aˆ z  cos 
and
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aˆ aˆ x  cos  cos 
aˆ aˆ y  cos  sin 

aˆ aˆ z  cos(  )   sin 
2

aˆ aˆ x  cos(  )   sin 
2
aˆ aˆ y  cos 
aˆ aˆ z  0
Figure
Given a vector A  Ar aˆr  A aˆ  A aˆ in the Spherical Polar coordinate
system, its component in the Cartesian coordinate system can be
found out as follows:
Ax  A.aˆ x  Ar sin  cos   A cos cos   A sin 
Similarly,
Ay  A.aˆ y  Ar sin  sin   A cos sin   A cos 
Az  A.aˆz  Ar cos  A sin 
The above expressions may be put in a compact form:
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 Ax  sin  cos  cos cos 
 Ay    sin  sin  cos sin 
  
 Az   cos
 sin 
 sin    Ar 
cos    A 
0   A 
The components Ar , A and A themselves will be function of r , 
and  . r ,  and  are related to x , y and z as
x  r sin  cos 
y  r sin  sin 
z  r cos
and conversely,
r  x2  y2  z 2
z
  cos 1
  tan 1
x2  y2  z 2
y
x
Using the variable transformation listed above, the vector
components, which are functions of variables of one coordinate
system, can be transformed to functions of variables of other
coordinate system and a total transformation can be done.
Line, surface and volume integrals
In electromagnetic theory we come across integrals, which contain
vector functions. Some representative integrals are listed below:
 Fdv
  dl
 F .dl
 F .ds etc.
V
C
C
S
In the above integrals, F and  respectively represent vector and
scalar function of space coordinates. C , S & V represent path,
surface and volume of integration. All these integrals are evaluated
using extension 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
b

a
f ( x)dx 
Lim
n
 fx
n   i 1
i
i
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where the interval (a, b) is subdivided into n continuous interval of
lengths  x1........... xn .
Line integral
Line integral
 E.dl
is the dot product of a vector with a specified
C
path C ; in other words it is the integral of the tangential component
of E along the curve C .
Figure: Line integral
As shown in the figure, given a vector field E around C , we define

the integral E dl 

b
a
E cos dl as the line integral of E along the
C
curve C . If the path of integration is a closed path as shown in
Figure, the line integral becomes a closed line integral and is called
the circulation of E around C and denoted as
 E dl
C
Figure: Closed line integral
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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
   A cos ds   A aˆn ds   A ds
S
S
S
Figure: Computation of surface integral
If the surface integral is carried out over a closed surface, then we
write:

 A ds which gives the net outward flux of
A from S .
S
Volume integrals
We define
 fdv
or
V
 fdv as
the volume integral of the scalar function
v
f (function of spatial coordinates) over the volume V . Evaluation of
integral of the form
 Fdv
can be carried out as a sum of three scalar
V
volume 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. Hamilton
and later on developed by P. G. Tait.
Mathematically the vector differential operator can be written in the
general form as:

1 
1 
1 
aˆu 
aˆv 
aˆw
h1 u
h2 v
h3 w
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In Cartesian coordinates:




aˆ x  aˆ y  aˆ z
x
y
z
In cylindrical coordinates:


1 

aˆ  
aˆ  aˆ z

 
z
and in spherical polar coordinates:


1 
1

aˆr 
aˆ 
aˆ
r
r 
r sin  
Gradient of a Scalar