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1. Introduction

.

2/20/20131 Electromagnetic Field Theory by R. S. Kshetrimayum

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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-

2/20/2013Electromagnetic Field Theory 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 ofelectromagnetic field is at rest or motion

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1.1 Electromagnetic theory in a nutshell Electrostatic fields are produced by static electric charges

Magnetostatic fields are produced by electric charges movingwith uniform velocity also known as direct current

Time-varying fields are produced by accelerated or


-

currents

An accelerated or decelerated charge also produces radiation

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1.2 Computational electromagnetics

Computational electromagnetics (CEM) is an

interdisciplinary field where

we apply numerical methods and

use computers

to solve


practical

and real-life electromagnetic problems

which usually do not have simple analytical solutions

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1.2.1 Why do we need Computational electromagnetics?

Maxwell’s equations along with the electromagnetic boundary conditions

describe any kind of electromagnetic phenomenon in nature


exc u ng quan um mec an cs Due to the linearity of the four Maxwell’s equations in the

differential forms,

it may appear rather easy to solve them analytically

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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


,

approximate or

computational methods

to solve them

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An advantage of this is that it is possible to simulate a

device/experiment/phenomenon

any number of times

as per our requirements


,

we can try to achieve the best or optimal result

before actually doing the experiments

Sometime experiments are dangerous to perform

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1.2.2 Computational electromagnetics in a nutshell

For any computational solution in Computationalelectromagnetics,

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

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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


,

unknown line charge density λ

It is an integral equation since the unknown λ is under an

integral sign

0

( ') '( )

4 ( , ')

x dxV x

r x x

λ

πε = ∫

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Similarly,

is a differential equation because the unknown function f(x) isunder a differential sign

Sometimes com lex e uations can constitute both inte ral

2

2

2

( )d f x x

dx− =


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

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Computationalelectromagnetics

Integral

equation solver

Differential

equation solver


equation solver

Frequency domainintegral equation solver

Fig. 1.2 Computational electromagnetics in a nutshell

Time domain differentialequation solver

Frequency domain

differential equation solver

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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)

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Time Domain Differential Equation (TDDE) solver: solves

complex electromagnetic engineering problems in the form

of differential equations in time domain

A possible example for this is Finite Difference Time Domain

(FDTD)


Frequency Domain Differential Equation (FDDE) solver:solves complex electromagnetic engineering problems in the

form of differential equations in frequency domain

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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,


other vector operations

of the three orthogonal coordinate systems viz.

Rectangular,

Cylindrical and

Spherical coordinate systems

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A point in space represented by a1, a2 and a3 in the

general curvilinear coordinate system


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Differential elements can be expressed as dl1=s1da1,

dl2

=s2

da2

, dl3

=s3

da3

where s1

, s2

and s3

are the scale

factors


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θ

r̂ ˆ z

φ̂


φ

θ

ˆθ

ˆ x

ˆ y

φ θ

ˆ

(a)(b)

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φ̂

ˆ yˆ zθ

r̂ ˆ z


ˆ ρ φ

ˆ xθ̂

φ̂ ˆ

(c) (d)

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Fig. 1.4

(a) Coordinate systems and their variables

(b) Geometry relationship between the Rectangular and

Spherical coordinate systems


c eometry re at ons p etween t e ectangu ar an

Cylindrical coordinate systems and

(d) Geometry relationship between the Spherical and

Cylindrical coordinate systems

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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 vectorr


with the x-, y- and z- axes are:

( , , ) x y z

x y z x y z= + +

2 2 2

ˆ( , , )cos

( , , )

x

x y z

A A x y z x

A x y z A A Aα

•= =

+ +

r

r

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2 2 2

ˆ( , , )cos

( , , )

y

x y z

A A x y z y

A x y z A A A

β •

= =

+ +

r

r

2 2 2

ˆ( , , )cos z

A A x y z z

A xγ

•= =

r

r


where α, β and γ are respectively the angles vector makes

with the x-, y- and z- axes

x y z

r

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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)


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

• = • = • = −

• = • = • =

• = • = − • =

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(b) Cylindrical and Rectangular

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 Cylindricalˆ 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

• = • = • =

• = • = • =

• = • = − • =

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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

A A

A A A x y z T A r θ

θ φ θ φ φ

θ φ θ φ φ θ φ

−

= ⇒ =r r


cos sin 0 z A A φ θ θ

−

sin cos sin sin cos

cos cos cos sin sin

sin cos 0

xr

y

z

A A

A A

A A

θ

φ

θ φ θ φ θ

θ φ θ φ θ

φ φ

= −

−

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(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

A A

ρ

φ θ

φ

θ θ φ θ φ

θ θ

= ⇒ =

−

r r


sin 0 cos

cos 0 sin

0 1 0

r

z

A

A

A

ρ

θ φ

φ

θ θ

θ θ

= −

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Vector calculus

Vector differential

calculusVector integral

calculus


Gradient

Divergence

Fig. 1.3 Vector calculus

Curl Laplacian

Divergence

theoremStoke’s

theorem

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1.4 Vector differential calculus1.4.1 Gradient of a scalar function:

The gradient of any scalar functionΨ

is a vector whosecomponents in any direction are given by the spatial rate

change ofΨ along that direction


1 2 3

1 1 2 2 3 3

a a as a s a s a

ψ ψ ψ ψ

∂ ∂ ∂∇ = + +

∂ ∂ ∂

) ) )

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1.4 Vector differential calculusHow to memorize this formula?

Note that in each of the three terms in the gradient of scalar

function above,

we have a unit vector,


corresponding variable and divide by the corresponding scale factor

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1.4 Vector differential calculus


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1.4 Vector differential calculus1.4.2 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

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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


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

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Divergence


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1.4 Vector differential calculus1.4.3 Curl of a vector:

1 1 2 2 3 3

1 2 3 1 2 3

1

s a s a s a

As s s a a a

∂ ∂ ∂∇ × =

∂ ∂ ∂

) ) )


How much that vector curls around the point in

question?

1 1 2 2 3 3

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Note that in the expression of curl of a vector above,

outside the determinant, we have division by product of all scale

factors


, ,

have multiplied by corresponding scale factors to unit vectorsand vector components respectively

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Curl


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

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1.4 Vector differential calculus 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 productsare 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

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1.4 Vector differential calculusSome useful vector identities:

This means curl of a gradient of scalar function is always

zero

( ) 0ψ ∇ × ∇ =


This means divergence of a curl of vector is always zero

( ) 0 A∇ • ∇ × =r

( ) ( ) ( ) A B B A A B∇ • × = • ∇ × − • ∇ ×r r rr r r

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1.4 Vector differential calculus 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


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1.4 Vector differential calculus1.4.4 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:

2ψ 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 ss 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∇ = ∇ ∇ • − ∇ × ∇ ×

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1 5 Vector integral calculus

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1.5 Vector integral calculus

1.5.1 Scalar line integral of a scalar function

where is the scalar function

( ) ( )( )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


and is the vector line element

1.5.2 Scalar line integral of a vector field

d l

( )

( ) ( ) ( ){ } ( )

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

) ) ) ) ) )


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where is the vector field and

and is the vector line element

1.5.3 Scalar surface integral of a vector field

A ds A nds• = •r rr )

Ar

d lr


where is the vector field and

is the normal to surface element ds

r

n̂


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1.5.4 Divergence Theorem

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)


et t e vo ume enc ose y t s c ose sur ace e g ven y

Then according to the Divergence theorem

dv Asd A

S V

∫ ∫∫∫ •∇=•rrr


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Fig. 1.8 (a) Divergence theorem (Converts closed surface

integrals to the volume integrals)

Ar


V

n̂dsr

da

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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

sd Ald A rrrr

1 6 SVector calculus

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1.6 Summary

Vector differential calculus

Gradient

Vector integral calculus

CurlStoke’s theorem

)∫ ∫∫ •×∇=•

C S

sd Ald A rrrr

∂ ∂ ∂ s a s a s a) ) )


Divergence

Fig. 1.9 Vector calculus in a nutshell

Laplacian

Divergence theorem( )dv Asd A

S V

∫ ∫∫∫ •∇=•rrr

1 2 3

1 1 2 2 3 3

a a as a s a s a

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 2 3 1 2 3

1 1 2 2 3 3

1 A

s s s a a as 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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