EC23301

Electromagnetic Fields
UNIT – I INTRODUCTION
Electromagnetic model, Units and constants, of
vector algebra, Rectangular, cylindrical and spherical
coordinate systems, Line, surface and volume
integrals, Gradient of a scalar field, Divergence of a
vector field, Divergence theorem, Curl of a vector
field, Stoke's theorem
 Review of Vectors
What is EM?
• Branch of Electrical engineering or physics in which

electric and magnetic phenomena are studied.

• The study of the interactions between electric charges at
rest/stationary and in motion.
• It focus/entails on the analysis, synthesis, physical
interpretation, and application of electric and magnetic
fields. 3
 Applications
 Microwaves
 Antennas
 Electric machines
 Satellite communications
 Bio electromagnetics, (How EM interact with and influence
Biological processes)
 Plasmas (a gas in which a portion of the particles are ionized ),
 Nuclear research
 Fiber optics
 Electromechanical energy conversion
 Radar meteorology and remote sensing
 In physical medicine, for example, EM power, either in the form of
shortwaves or microwaves, is used to heat deep tissues and to
stimulate certain physiological responses to relieve certain
pathological conditions.
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 The Concept Fields
Field:
• A field is a function that specifies a particular
quantity everywhere in a region.
• It can be scalar or vector.
• Scalar fields: temperature distribution in a
building, sound intensity in a theatre, electric
potential in a region, etc.
• Gravitational force on a body in space and the
velocity of raindrops in the atmosphere are
examples of vector fields.
 Scalar and Vector Fields

• A scalar is a quantity that has only magnitude.

• Quantities such as time, mass, distance,
temperature, entropy, electric potential, and
population are scalars.
• A vector is a quantity that has both magnitude
and direction.
• Vector quantities include velocity, force,
displacement, and electric field intensity.
• To distinguish between a scalar and a vector it is
customary to represent a vector by a letter with an
arrow on top of it.
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• A unit vector 𝒂𝑨 along A is defined as a vector whose magnitude is unity and
its direction is along A is given by:

• A vector A in Cartesian (or rectangular) coordinates may be represented as

(Ax, Ay, Az) or Axax +Ayay + Azaz .
• The magnitude of vector A is given by:-

• A unit vector along vector A is given by:-

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Orthogonal coordinate system
 Coordinate System and Transformation
• In general, the physical quantities in EM are functions of space
and time.
• In order to describe the spatial variations of the quantities, we
must be able to define all points uniquely in space in a suitable
manner
• This requires using an appropriate coordinate system.
• Like the Cartesian (or rectangular), the circular cylindrical,
the spherical, the elliptic cylindrical, the parabolic
cylindrical, the conical, the prolate spheroidal, the oblate
spheroidal, and the ellipsoidal.

12
 Orthogonal coordinate system
(Syllabus)

• Cartesian (or rectangular) coordinate

system
• Cylindrical coordinate system
• Spherical coordinate system

13
 Cartesian Coordinates (X, Y, Z)
• A point P can be represented as (x, y, z).
• The ranges of the coordinate variables x, y, and z are
-∞< X <∞
-∞< Y<∞
-∞ < Z< ∞
• A vector A in Cartesian (otherwise known as rectangular)
coordinates can be written as
(Ax, Ay, Az) or Axax + Ayay + Azaz
• where ax, ay, and az are unit vectors along the x, y, and z-
directions.

14
• There are two types of vector multiplication:
 Scalar (or dot) product: A • B
 Vector (or cross) product: A X B
The dot product of two vectors A and B, written as A • B. is defined
geometrically as the product of the magnitudes of A and B and the cosine of
the angle between them.
Thus: A • B = |A||B| cos (𝚹AB)
Where, 𝚹AB is the smaller angle between A and B.
• The result of A • B is called either the scalar product because it is scalar, or the
dot product due to the dot sign..
• If A = (Ax, Ay, Az) and B = (Bx, By, Bz), then

A • B = AxBx + AyBy + AzBz

ax • ay = ay • az = az • ax = 0 and ax • ax = ay • ay = az • az = 1
• which is obtained by multiplying A and B component by
component.

16
• The cross product of two vectors A and B. written as A X B. is a vector
quantity whose magnitude is the area of the parallelogram formed by A and B
and is in the direction of advance of a right-handed screw as A is turned into
B.

• Where an is a unit vector normal to the plane containing A and B.

• The direction of an is taken as the direction of the right thumb when the

• A X B = AB sin 𝚹AB an it is called vector product because the result is a

fingers of the right hand rotate from A to B.

vector.
• If A = (Ax, Ay, Az) and B = (Bx, By, Bz) then

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• Example 1: Find the magnitude of the given vector D and
unit vector

18
• The cross product of A and B is a vector with magnitude equal to the area of the
parallelogram and direction as indicated or Direction of AXB and an using (a) right-hand
rule, (b) right-handed screw rule. 19
• Exampe 2. If A = 10ax - 4ay + 6az and B = 2 ax + ay, find: (a) the
component of A along ay, (b) the magnitude of 3A - B, (c) a unit
vector along A + 2B.

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Example 3

21
• Example 4: Points P and Q are located at (2, 2, 1) and (-2, 0, 4). Calculate
 The position vector P
 The distance vector from P to Q
 The distance between P and Q
 A vector parallel to PQ with magnitude of 10

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• Solve Example 2-5.page 24 of cheng

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Circular Cylindrical Coordinates (ρ, ∅, Z)
•Very convenient for problems having cylindrical symmetry.
•This coordinate system is useful in dealing with systems that take the shape
of a cylinder. For example, the flow of water through a pipe.
•A point P in cylindrical coordinates is represented as (ρ, ∅, z) .
•ρ is the radius of the cylinder passing through P or the radial distance from
the z-axis:
•∅, called the azimuthal angle, is measured from the x-axis in the xy-plane;
and z is the same as in the Cartesian system.

24
 • A vector A in cylindrical coordinates can be written as

• Where 𝑎𝑝, 𝑎∅, 𝑎𝑧 are unit vectors in the 𝑝,∅,z directions

• 𝑎𝑝 points in the direction of increasing ρ, 𝑎∅ a in the

and the magnitude of A is:

direction of increasing ∅, and 𝑎𝑧 in the positive z-direction.

Thus,

• The relationships between the variables (x, y, z) of the Cartesian coordinate system
and those of the cylindrical system (p, ∅, z) are easily obtained
25
Find the
direction of rho
vector
• The relationships between (𝑎𝑥, 𝑎𝑦, 𝑎𝑧) and (𝑎𝑝, 𝑎∅, 𝑎𝑧)
are obtained geometrically from

27
Spherical Coordinates (r, Ө, Φ)
 Appropriate for problems having a degree of spherical symmetry.

 A point P can be represented as (r, Ө, Φ )

 r is defined as the distance from the origin to point P or the radius of a sphere
centered at the origin and passing through P;
 Ө (called the colatitude) is the angle between the z-axis and the position
vector of P; and
 Φ is measured from the x-axis (the same azimuthal angle in cylindrical
coordinates). According to these definitions, the ranges of the variables are

28
 A vector A in spherical coordinates may
be written as

• Where ar, aӨ, and aΦ are unit vectors along the r, Ө, and Φ-directions. The
magnitude of A is

29
• The space variables (x, y, z) in Cartesian coordinates can be related to variables (r, Ө, Φ) of a
spherical coordinate system.

Ө, ∅)
For point transformation from Cartesian to spherical coordinate system (x, y, z) → (r,

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31
32
• Differential Length, Area and Volume
(for all coordinate system)
 Differential Length, Area and Volume

Where:
• dl is the differential displacement
• dS is the differential normal area
• dV is the differential Volume
• dl and dS are vectors, whereas dV is a scalar quantity

dS dS a n

an is an outward normal vector

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 Differential Length, Area and Volume
• Differential elements in Cartesian co-ordinate system
Differential displacement dl is given by:
dl  dx a x  dy a y  dz a z

dS dy dz a x
dx dz a y
dx dy a z

dV  dx dy dz
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 Differential Length, Area and Volume (Cont’d)
• Differential elements in cylindrical co-ordinate system

dl  d a   d a   dz a z

dS  d dz a 
d dz a
  d d a z

dV  d  d dz
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 Differential Length, Area and Volume (Cont’d)
• Differential elements in spherical co-ordinate system
dl  dr a r  r d a  r sin  d a 

dS r 2 sin  d d a r
r sin  dr d a
r dr d a 

dV  r sin  dr d d
2

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• Solve Prolems
 Line, Surface and Volume Integrals
Given a vector field A :
• and a Path L, the integral of A around L is given by the equation:
b
A dl
L
 A cos dl
a

• For path of integration of a closed loop:

A dl
L
  A cos dl
L

• For a smooth surface S, the Surface Integral is:

 A dS   A cos dS

S S

• For a scalar function,  v, the Volume Integral

over a differential volume dV is given by:

 .dV
v

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 Del Operator

• The del operator  is defined as follows in all the three coordinate

systems:
  
• Cartesian  ax  a y  az
x y z

 1  
• Cylindrical  a  a  a z
   z

• Spherical  1  1 
 ar  a  a
r r  r sin  

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 Del Operator (Cont’d)

• Properties of Del operator 

• The del operator of a vector filed is a scalar-
• The del operator of a scalar field is a vector-
• Transformation between Cartesian and Cylindrical

y
  x y2 2
; tan   ; z  z
x

• Hence,
  sin  
 cos  
x     

z z
  cos  
 sin  
y   
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 Del Operator (Cont’d)
• Transformation between Cartesian and Spherical

x2  y2 y
r  x 2  y 2  z 2 ; tan   ; tan  
z x

• Hence,

  cos cos   sin  

 sin  cos   
x r r  r 
  cos sin   cos  
 sin  sin   
y r r  r 

  sin  
 cos  
z r r 
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 Del Operator (Cont’d)

The operator is useful in defining:

– The gradient of a scalar V, written as  V
– The divergence of a vector A, • V
– The curl of a vector A,  x V

– And the Laplacian of a vector V, 2 V

Each will be discussed as follows!

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 Gradient of a Scalar field
• The Gradient of a Scalar field V is a vector that represents both the magnitude & the
direction of the maximum space rate of increase of V

grad V  V
V V V
• Cartesian V  ax  ay  az
x y z

V 1 V V
• Cylindrical V  a  a  az
   z

• Spherical V 1 V 1 V
V  ar  a  a
r r  r sin  

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 Gradient of a Scalar field (Cont’d)
• Consider the figure: the field dV between points Pl and P2 where V1, V2, and
V3 are contours on which V is constant. mathematical expression for the
gradient is:

Let

Then

Or

dl is the differential displacement from P1 to P2 and  is the angle between G and dl.
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 Gradient of a Scalar field (Cont’d)

• dV/dl is a maximum when  = 0, that is, when dl is in the direction of G.

Hence,

• Thus G has its magnitude and direction as those of the maximum rate of
change of V. By definition, G is the gradient of V. Therefore:

Fundamental properties of the gradient of a scalar field V:

1. The magnitude of V equals the maximum rate of change in V per unit distance.
2. V points in the direction of the maximum rate of change in V.
3. V at any point is perpendicular to the constant V surface that passes through that point
4. The projection (or component) of V in the direction of a unit vector a is V • a
and is called the directional derivative of V along a.
5. If A = V , V is said to be the scalar potential of A. 46
 Divergence of a Vector field
• The Divergence of a Vector field A at a point P is defined as the outward flux
per unit volume as the volume shrinks about P.

A.dS
S
div A   A  lim
v  0 v

(a) Positive divergence (b) negative divergence (c) zero divergence

To obtain an expression for div A in Cartesian

coordinates from the earlier eq at point P(xo,yo, zo)
the point is enclosed by a differential volume as
shown in the Figure . The surface integral of the
eq. is obtained from the following eqn. by working
through the 3-dimensional Taylor series expansion.

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 Divergence of a Vector field (Cont’d)
Physical meaning:

Consider a flow of fluid:

• Figure a) shows a compressed air capped on one side. A similar cap has just
been removed from the other end and air is rushing out.
 A 0
• If the mass of fluid (or charge) coming out of a domain is the same as that
entering the domain (see figure (b)), then, no divergence, and

 A 0

NB. A is the fluid velocity (or the current) field vector

 Divergence of a Vector field (Cont’d)

• Cartesian: A x A y A z
 A   
x y z

1 A A z
• Cylindrical:  A 
1 
 
 
Aρ 
 

z

• Spherical:

1  2 1  1 A
A  2
r r
r Ar   
r sin  
A sin  
r sin  

49
 Divergence of a Vector field (Cont’d)

Properties:
• It produces a scalar field(because scalar product is involved).
• The divergence of a scalar V, div V, makes no sense
 A  B    A   B
 V A   V  A  A V
From the definition of the divergence of A in the earlier equation:

A dS  A dv
S v

This is called the divergence theorem, otherwise known as the Gauss-

Ostrogradsky theorem.
The divergence theorem states that the total outward flux of a vector field
A through the closed surface S is the same as the volume integral of the
divergence of A.
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 Curl of A Vector and Stokes's Theorem
Recall: the circulation of a vector field A around a closed path L is: A dl
L

• The Curl of A (rot A) is an axial (or rotational) vector

– whose magnitude is the maximum circulation of A per unit area as the
area tends to zero and
– whose direction is the normal direction of the area when the area is
oriented so as to make the circulation maximum.

• Where: area S is bounded

by the curve L and an is the
unit vector normal to the
surface S and is
determined using the right
hand rule.
Curl of A Vector and Stokes's Theorem (Cont’d)

• Cartesian
ax ay az
  
 A 
x y z
Ax A y Az

 A z A y   A x A z   A y A x 
 A    a x  
   a y  
  a z

 y z   z x   x y 

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Curl of A Vector and Stokes's Theorem (Cont’d)

• Cylindrical
a a az
1   
 A 
   z
A A A z

 1 A z A   A  A z 
 A    a   
    a

   z   z  

 

1   A 
A  
a z
    

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Curl of A Vector and Stokes's Theorem (Cont’d)

• Spherical

ar ra r sin  a
1   
 A 
r 2 sin  r  
Ar rA A

 A 
1


  A sin  A 
ar  
 a
1  1 A r  rA

r sin      
r  sin   r  
  
1  rA  A r 
    a
r  r  

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 Curl of A Vector and Stokes's Theorem (Cont’d)
Properties of the Curl of a vector field:

• The curl of a vector field is another vector

• The curl of a scalar field V, ∇ X V, makes no sense
• The divergence of the curl of a vector field vanishes, ∇ • (∇ X A) = 0
• The curl of the gradient of a scalar field vanishes, that is, ∇ X ∇ V = 0.

For Vector fields A and B and a scalar field V

∇ X (A + B) = ∇ X A + ∇ X B

∇ X (A X B) = A(∇ • B) - B(∇ • A) + (B • ∇ A -
(A • ∇)B
∇ X (VA) = V ∇ X A + ∇ V
XA

Please see to other properties of the curl are in Appendix A of the text book 55
 Curl of A Vector and Stokes's Theorem (Cont’d)
Stokes’s Theorem

Stokes's theorem states that the circulation of a

vector field A around a (closed) path-L is equal lo
the surface integral of the curl of A over the open
surface S bounded by L (see the Figure) provided
that A and ∇ X A are continuous on S.

A dl  A dS
L S

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 Laplacian of A Scalar

The Laplacian of a scalar field V, written as ∇2V, is the

divergence of the gradient of V.

Laplacian V  V 2V

      V V V 
 a x  a y  a z  . ax  ay  az 
 x y z   x y z 

that is,

 2
V  2
V  2
V
V  2  2  2
2

x y z

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 Laplacian of A Scalar (Cont’d)
Similarly, for the cylindrical and the spherical coordinate
systems:
1   V  1  2
V  2
V
V
2
    2  2
       2
z

1   2 V  1   V 
V  2  2  sin 
2
r 
r r  r  r sin     
1  2V
 2
r sin 2   2
• A scalar field V is said to be harmonic in a given region if its
Laplacian vanishes in that region. i.e.
 2V  0
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 Laplacian of A Scalar (Cont’d)
The Laplacian operator of a vector field A is defined as the
gradient of the divergence of A minus the curl of the curl of
A. i.e.,
 A A  A    A
2

• Cartesian
2 A 2 A x a x  2 A y a y  2 A z a z
• Where  2 2 A x 2 A x 2 A x
 A x  2  2
 2
  x  y  z
  2
A y  2
A y  2
Ay
2
 A y  2  2
 2
 x y z
 2 2 2
 Az  Az  Az 2
 A z  2  2
 2
  x  y  z 59
 Classification of Vector fields
• Vector fields can be classified as:
(a) ∇ • A = 0, ∇ XA= 0
(b) ∇ • A ≠ 0, ∇ X A = 0
(c) ∇ • A = 0, ∇ X A ≠ 0
(d) ∇ • A ≠ O, ∇ X A ≠ O

60
 Classification of Vector fields (Cont’d)
All Vector fields can be classified in terms of their vanishing
or non-vanishing divergence or curl:
1. Solenoidal (or divergenceless) if div A = 0

Solenoidal fields have neither source nor sink with in the region

Examples of solenoidal fields are incompressible fluids,

magnetic fields, and conduction current density under steady
state conditions.
An irrotational field can always be expressed in terms of a scalar
field V :

2. Irrotational (or Potential) if x A = 0

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