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Dr. Rakhesh Singh Kshetrimayum
1. Introduction
Dr. Rakhesh Singh Kshetrimayum
2/20/20131 Electromagnetic FieldTheory by R. S. Kshetrimayum

1.1 Electromagnetic theory in a nutshell
Electromagnetic field theory is the study of fields produced
by electric charges at
rest or
in motion
Electromagnetic theory can be divided into three sub-
divisions
2/20/2013Electromagnetic FieldTheory by R. S. Kshetrimayum2
divisions
electrostatics,
magnetostatic and
time-varying fields as depicted in Fig. 1.1
depending on whether the charge which is the source of
electromagnetic field is at rest or motion

Electromagnetic
theory
Electrostatics Magentostatics Time-varying fields
Fig. 1.1 Electromagnetic theory in a nutshell

Electrostatic fields are produced by static electric charges
Magnetostatic fields are produced by electric charges moving
with uniform velocity also known as direct current
Time-varying fields are produced by accelerated or
decelerated charges or time-varying currents or alternating
decelerated charges or time-varying currents or alternating
currents
An accelerated or decelerated charge also produces radiation

1.2 Computational electromagnetics
Computational electromagnetics (CEM) is an
interdisciplinary field where
we apply numerical methods and
use computers
to solve
to solve
practical
and real-life electromagnetic problems
which usually do not have simple analytical solutions

1.2.1Why do we need Computational electromagnetics?
Maxwell’s equations along with
the electromagnetic boundary conditions
describe any kind of electromagnetic phenomenon in nature
excluding quantum mechanics
excluding quantum mechanics
Due to the linearity of the four Maxwell’s equations in the
differential forms,
it may appear rather easy to solve them analytically

But the boundary and interface conditions make them hard to
solve analytically for many practical electromagnetic
engineering problems
Hence one has to resort to use
experimental,
experimental,
approximate or
computational methods
to solve them

An advantage of this is that it is possible to simulate a
device/experiment/phenomenon
any number of times
as per our requirements
In that way,
In that way,
we can try to achieve the best or optimal result
before actually doing the experiments
Sometime experiments are dangerous to perform

1.2.2 Computational electromagnetics in a nutshell
For any computational solution in Computational
electromagnetics,
it is necessary to develop the required equations and
solve them using a computer also known as equation solvers
There are two types of equations:
integral or
differential equations and
correspondingly two solvers:
integral or
differential equation solvers

Integral equations are equations in which the unknown is
under an integral sign just like
in differential equation your unknown function is under a
differential sign
For example, for a given potential of V on a wire of
For example, for a given potential of V on a wire of
unknown line charge density λ
It is an integral equation since the unknown λ is under an
integral sign
0
( ') '
( )
4 ( , ')
x dx
V x
r x x
λ
πε
= ∫

Similarly,
is a differential equation because the unknown function f(x) is
under a differential sign
Sometimes, complex equations can constitute both integral
2
2
2
( )d f x
x
dx
− =
Sometimes, complex equations can constitute both integral
as well as differential equations also known as integro-
differential equation
In general, all the available Computational electromagnetics
methods may be classified broadly into two categories:
a) differential equation solvers and
b) integral equation solvers

Computational
electromagnetics
Integral
equation solver
Time domain integral
Differential
equation solver
Time domain integral
equation solver
Frequency domain
integral equation solver
Fig. 1.2 Computational electromagnetics in a nutshell
Time domain differential
equation solver
Frequency domain
differential equation solver

Time Domain Integral Equation (TDIE) solver: solves
complex electromagnetic engineering problems in the form
of integral equations in time domain
Frequency Domain Integral Equation (FDIE) solver: solves
of integral equations in frequency domain
A suitable example for this is Method of Moments (MoM)

Time Domain Differential Equation (TDDE) solver: solves
of differential equations in time domain
A possible example for this is Finite DifferenceTime Domain
(FDTD)
(FDTD)
Frequency Domain Differential Equation (FDDE) solver:
solves complex electromagnetic engineering problems in the
form of differential equations in frequency domain

1.3 General curvilinear coordinate system
1.3.1 Coordinate systems
Note that it is possible to develop one general expressions
also known as general curvilinear coordinate system for
divergence,
curl and
curl and
other vector operations
of the three orthogonal coordinate systems viz.
Rectangular,
Cylindrical and
Spherical coordinate systems

A point in space represented by a1, a2 and a3 in the
general curvilinear coordinate system

Differential elements can be expressed as dl1=s1da1,
dl2=s2da2, dl3=s3da3 where s1, s2 and s3 are the scale
factors

θ
θ
ˆr
ˆz
ˆφ
ρ
φ
θ
ˆθ
ˆx
ˆy
φ θ
ˆρ
(a) (b)

ˆφ
ˆyˆz
θ
ˆrˆz
ˆρφ
ˆx
ˆθ
ˆφ ˆρ
(c) (d)

Fig. 1.4
(a) Coordinate systems and their variables
(b) Geometry relationship between the Rectangular and
Spherical coordinate systems
(c) Geometry relationship between the Rectangular and
(c) Geometry relationship between the Rectangular and
Cylindrical coordinate systems and
(d) Geometry relationship between the Spherical and
Cylindrical coordinate systems

1.3.2 Direction cosines
Direction cosines of a vector are the cosines of the angles
between the vector and three coordinate axes
For instance, the direction cosines of a vector
ˆ ˆ ˆ( , , )A x y z A x A y A z= + +
r
with the x-, y- and z- axes are:
ˆ ˆ ˆ( , , ) x y zA x y z A x A y A z= + +
2 2 2
ˆ( , , )
cos
( , , )
x
x y z
AA x y z x
A x y z A A A
α
•
= =
+ +
r
r

2 2 2
ˆ( , , )
cos
( , , )
y
x y z
AA x y z y
A x y z A A A
β
•
= =
+ +
r
r
2 2 2
ˆ( , , )
cos
( , , )
z
x y z
AA x y z z
A x y z A A A
γ
•
= =
+ +
r
r
where α, β and γ are respectively the angles vector makes
with the x-, y- and z- axes
( , , ) x y z
A x y z A A A+ +
A
r

In a more general sense, direction cosine refers to the cosine
of the angle between any two vectors
They are quite useful for converting one coordinate system
to another (or coordinate transformation)
(a) Spherical and Rectangular
(a) Spherical and Rectangular
ˆ ˆˆ ˆ ˆ ˆsin cos cos cos sin
ˆ ˆˆ ˆ ˆ ˆsin sin cos sin cos
ˆ ˆˆˆ ˆ ˆcos sin 0
x r x x
y r y y
z r z z
θ φ θ θ φ φ φ
θ φ θ θ φ φ φ
θ θ θ φ
• = • = • = −
• = • = • =
• = • = − • =

(b) Cylindrical and Rectangular
(c) Spherical and Cylindrical
ˆˆˆ ˆ ˆ ˆcos sin 0
ˆˆˆ ˆ ˆ ˆsin cos 0
ˆˆˆ ˆ ˆ ˆ0 0 1
x x x z
y y y z
z z z z
ρ φ φ φ
ρ φ φ φ
ρ φ
• = • = − • =
• = • = • =
• = • = • =
(c) Spherical and Cylindrical
ˆ ˆˆ ˆ ˆˆ sin cos 0
ˆ ˆ ˆ ˆ ˆˆ 0 0 1
ˆ ˆˆˆ ˆ ˆcos sin 0
r
r
z r z z
ρ θ ρ θ θ ρ φ
φ φ θ φ φ
θ θ θ φ
• = • = • =
• = • = • =
• = • = − • =

1.3.3 Coordinate transformations
(a) Spherical to Rectangular and vice versa
( ) [ ] ( )
sin cos cos cos sin
sin sin cos sin cos , , , ,
x r
y sr
A A
A A A x y z T A rθ
θ φ θ φ φ
θ φ θ φ φ θ φ
   − 
    = ⇒ =    
   
r r
( ) [ ] ( )
cos sin 0
y sr
z
AA
θ
φθ θ
    
    −    
sin cos sin sin cos
cos cos cos sin sin
sin cos 0
xr
y
z
AA
A A
A A
θ
φ
θ φ θ φ θ
θ φ θ φ θ
φ φ
    
    = −    
    −    

(b) Cylindrical to Rectangular and vice versa
( ) [ ] ( )
cos sin 0
sin cos 0 , , , ,
0 0 1
x
y cr
zz
AA
A A A x y z T A z
AA
ρ
φ
φ φ
φ φ ρ φ
   − 
    = ⇒ =    
        
r r
cos sin 0
sin cos 0
0 0 1
x
y
z z
A A
A A
A A
ρ
φ
φ φ
φ φ
    
    = −    
        

(c) Spherical to Cylindrical and vice versa
( ) [ ] ( )
sin cos 0
0 0 1 , , , ,
cos sin 0
r
sc
z
A A
A A A z T A r
AA
ρ
φ θ
φ
θ θ
ρ φ θ φ
θ θ
    
    = ⇒ =    
    −    
r r
sin 0 cos
cos 0 sin
0 1 0
r
z
AA
A A
A A
ρ
θ φ
φ
θ θ
θ θ
    
    = −    
        

Vector calculus
Vector differential
calculus
Vector integral
calculus
Gradient
Divergence
Fig. 1.3Vector calculus
Curl Laplacian
Divergence
theoremStoke’s
theorem

1.4 Vector differential calculus
1.4.1 Gradient of a scalar function:
The gradient of any scalar function Ψ is a vector whose
components in any direction are given by the spatial rate
change of Ψ along that direction
1 2 3
1 1 2 2 3 3
a a a
s a s a s a
ψ ψ ψ
ψ
∂ ∂ ∂
∇ = + +
∂ ∂ ∂
) ) )

How to memorize this formula?
Note that in each of the three terms in the gradient of scalar
function above,
we have a unit vector,
partial differential of the scalar function with respect to the
partial differential of the scalar function with respect to the
corresponding variable and
divide by the corresponding scale factor

1.4.2 Divergence of a vector:
( ) ( ) ( )2 3 1 1 3 2 1 2 3
1 2 3 1 2 3
1
A s s A s s A s s A
s s s a a a
 ∂ ∂ ∂
∇• = + + 
∂ ∂ ∂ 
It is a measure of how much the vector spreads out
(diverge) from the point in question

Note that in the expression of divergence of a vector above,
outside the third bracket, we have division by product of all
scale factors, and
inside the third bracket there are three terms
Each term contains a
partial differential w.r.t. one of the variable
to the product of corresponding vector component and
scale factors of the remaining two axes

Divergence

1.4.3 Curl of a vector:
1 1 2 2 3 3
1 2 3 1 2 3
1 1 2 2 3 3
1
s a s a s a
A
s s s a a a
s A s A s A
∂ ∂ ∂
∇× =
∂ ∂ ∂
) ) )
How much that vector curls around the point in
question?
1 1 2 2 3 3s A s A s A

Note that in the expression of curl of a vector above,
outside the determinant, we have division by product of all scale
factors
Also note that inside the determinant, in row one and three, we
Also note that inside the determinant, in row one and three, we
have multiplied by corresponding scale factors to unit vectors
and vector components respectively

Curl

Fig. 1.5 (a) No divergence and curl

Fig. 1.5 (b) Positive divergence and curl around z-axis

Scalar triple product
( ) ( ) ( )
x y z
x y z
x y z
A A A
A B C B B B B C A C A B
C C C
• × = = • × = • ×
r r r r r rr r r
Note that the above three vector scalar triple products
are the same from the definition of scalar triple product
Vector triple product (“bac-cab” rule)
( ) ( ) ( )A B C B A C C A B× × = • − •
r r r r r rr r r

Some useful vector identities:
This means curl of a gradient of scalar function is always
zero
( ) 0ψ∇× ∇ =
( )∇ • ∇× =
r
This means divergence of a curl of vector is always zero
( ) 0A∇ • ∇× =
r
( ) ( ) ( )A B B A A B∇• × = • ∇× − • ∇×
r r rr r r

This means that divergence of cross product of two vectors is
equal to
the dot product of second vector and curl of first vector
minus dot product of first vector and curl of second vector

1.4.4 Laplacian of a scalar or vector function:
Laplacian is an operator which can operate on a scalar or
vector
Laplacian of a scalar function:
Laplacian of a vector function:
2
ψ ψ∇ = ∇•∇
2 3 1 3 1 2
1 2 3 1 1 1 2 2 2 3 3 3
1 s s s s s s
s s s a s a a s a a s a
ψ ψ ψ     ∂ ∂ ∂ ∂ ∂ ∂
= + +     
∂ ∂ ∂ ∂ ∂ ∂      
( ) ( )A A A∇×∇× =∇ ∇• − ∇•∇Q ( ) 2
A A=∇ ∇• −∇
( )2
A A A∇ = ∇ ∇• −∇×∇×

Note that in the expression of Laplacian of a scalar function above,
outside the third bracket, we have division by product of all scale
factors, and
Each term is a partial differential with respect to a variable of the
expression in a first bracket
Inside first bracket, you have multiplication of scale factors of the
remaining two axes divide by the scale factor of the same variable
into partial differential of the scalar function with the same
variable

Laplacian

1.5 Vector integral calculus
1.5.1 Scalar line integral of a scalar function
where is the scalar function
and is the vector line element
( ) ( )( )1 2 3 1 2 3 1 1 1 2 2 2 3 3 3, , , ,a a a dl a a a s da a s da a s da aψ ψ= + +∫ ∫
r ) ) )
ψ
dl
r
1.5.2 Scalar line integral of a vector field
dl
r
( )
( ) ( ) ( ){ } ( )
1 2 3
1 1 2 3 1 2 1 2 3 2 3 1 2 3 3 1 1 1 2 2 2 3 3 3
, ,
, , , , , ,
A a a a dl
A a a a a A a a a a A a a a a s da a s da a s da a
•
= + + • + +
∫
∫
rr
) ) ) ) ) )

where is the vector field and
1.5.3 Scalar surface integral of a vector field
A ds A nds• = •∫ ∫
r rr )
A
r
dl
r
where is the vector field and
is the normal to surface element ds
A ds A nds• = •∫ ∫
A
r
ˆn

1.5.4 DivergenceTheorem
It is also known as Green’s or Gauss’s theorem
Consider a closed surface S in presence of a vector field as
shown in Fig. 1.8 (a)
Let the volume enclosed by this closed surface be given byV
Let the volume enclosed by this closed surface be given byV
Then according to the Divergence theorem
( )dvAsdA
S V
∫ ∫∫∫ •∇=•
rrr

Fig. 1.8 (a) Divergence theorem (Converts closed surface
integrals to the volume integrals)
A
r
V
ˆnds
r
da

Fig. 1.8 (b) Stoke’s theorem
(Converts closed line integrals to surface integrals)
A
r
S
S
C
da ˆn
dl
r
ds
r

1.5.5 Stokes theorem
Consider a closed curve C enclosing an area S in presence of
a vector field
Then, Stokes theorem can be written as
( )∫ ∫∫ •×∇=•
C S
sdAldA
rrrr

1.6 Summary
Vector calculus
Vector differential calculus
Gradient
Vector integral calculus
Curl
Stoke’s theorem
( )∫ ∫∫ •×∇=•
C S
sdAldA
rrrr
a a a
ψ ψ ψ
ψ
∂ ∂ ∂
∇ = + +
) ) ) 1 1 2 2 3 3s a s a s a
) ) )
Divergence
Fig. 1.9Vector calculus in a nutshell
Laplacian
Divergence theorem
( )dvAsdA
S V
∫ ∫∫∫ •∇=•
rrr
1 2 3
1 1 2 2 3 3
a a a
s a s a s a
ψ ψ ψ
ψ
∂ ∂ ∂
∇ = + +
∂ ∂ ∂
) ) )
( ) ( ) ( )2 3 1 1 3 2 1 2 3
1 2 3 1 2 3
1
A s s A s s A s s A
s s s a a a
 ∂ ∂ ∂
∇• = + + 
∂ ∂ ∂ 
1 1 2 2 3 3
1 2 3 1 2 3
1 1 2 2 3 3
1
s a s a s a
A
s s s a a a
s A s A s A
∂ ∂ ∂
∇× =
∂ ∂ ∂
2 3 1 3 1 2
1 2 3 1 1 1 2 2 2 3 3 3
1 s s s s s s
s s s a s a a s a a s a
ψ ψ ψ     ∂ ∂ ∂ ∂ ∂ ∂
= + +     
∂ ∂ ∂ ∂ ∂ ∂      
2
ψ ψ∇ = ∇•∇

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