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Dr. Rakhesh Singh Kshetrimayum

1. Introduction

Dr. Rakhesh Singh Kshetrimayum

2/20/20131 Electromagnetic Field Theory 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 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 of electromagnetic field is at rest or motion

1.1 Electromagnetic theory in a nutshell

Electromagnetic theory


Electrostatics Magentostatics Time-varying fields

Fig. 1.1 Electromagnetic theory in a nutshell

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


1.2.1 Why 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 dxV 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 xx

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 electromagneticsmethods may be classified broadly into two categories: � a) differential equation solvers and � b) integral equation solvers

1.2 Computational electromagneticsComputational 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 complex electromagnetic engineering problems in the form


complex electromagnetic engineering problems in the form of integral equations in frequency domain� A suitable example for this is Method of Moments (MoM)


� 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)


(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

rz

φ


ρ

φ

θ

θ

x

y

φ θρ

(a) (b)


φ

yzθ

rz


ρφ

xθ

φ ρ

(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 Aα

•= =

+ +

r

r


2 2 2

ˆ( , , )cos

( , , )

y

x y z

AA x y z y

A x y z A A Aβ

•= =

+ +

r

r

2 2 2

ˆ( , , )cos

( , , )

z

x y z

AA x y z z

A x y z A A Aγ

•= =

+ +

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+ +

Ar


� 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

zAA

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. 1.3 Vector calculus

Curl Laplacian

Divergence theoremStoke’s

theorem

1.4 Vector differential calculus1.4.1 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 as a s a s a

ψ ψ ψψ

∂ ∂ ∂∇ = + +

∂ ∂ ∂

) ) )

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,

� 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 Vector differential calculus


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

1A 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 Vector differential calculus1.4.3 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

As 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

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 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

1.4 Vector differential calculusSome 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

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


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 s

s s s a s a a s a a s a

ψ ψ ψ ∂ ∂ ∂ ∂ ∂ ∂= + +

∂ ∂ ∂ ∂ ∂ ∂

( ) ( )A A A∇×∇× =∇ ∇• − ∇•∇Q ( ) 2A A=∇ ∇• −∇

( )2A 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

1.4 Vector differential calculusLaplacian


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

d lr


� and is the vector line element

1.5.2 Scalar line integral of a vector field

d lr

( )

( ) ( ) ( ){ } ( )

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

� 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

A ds A nds• = •∫ ∫

Ar

n


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)

� Let the volume enclosed by this closed surface be given by V


� Let the volume enclosed by this closed surface be given by V

� Then according to the Divergence theorem

( )dvAsdAS V

∫ ∫∫∫ •∇=•rrr


Fig. 1.8 (a) Divergence theorem (Converts closed surface integrals to the volume integrals)

Ar


V

ndsr

da


Fig. 1.8 (b) Stoke’s theorem

(Converts closed line integrals to surface integrals)

Ar

S


S

C

da n

dlr

dsr


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

sdAldArrrr

1.6 SummaryVector calculus

Vector differential calculus

Gradient

Vector integral calculus

CurlStoke’s theorem

( )∫ ∫∫ •×∇=•C S

sdAldArrrr

a a aψ ψ ψ

ψ∂ ∂ ∂

∇ = + +) ) ) 1 1 2 2 3 3

s a s a s a) ) )


Divergence

Fig. 1.9 Vector calculus in a nutshell

Laplacian

Divergence theorem

( )dvAsdAS V

∫ ∫∫∫ •∇=•rrr

1 2 3

1 1 2 2 3 3

a a as a s a s a

ψ ψ ψψ

∂ ∂ ∂∇ = + +

∂ ∂ ∂

) ) )

( ) ( ) ( )2 3 1 1 3 2 1 2 3

1 2 3 1 2 3

1A 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

As 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 ∂ ∂ ∂ ∂ ∂ ∂= + +

∂ ∂ ∂ ∂ ∂ ∂

2ψ ψ∇ = ∇ • ∇

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