1 engineering electromagnetics essentials chapter 1 vector calculus expressions for gradient,...

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Engineering Electromagnetics

EssentialsChapter 1

Vector calculus expressions forgradient, divergence, and curl

Introduction

Chapter 2and

2

The subject is based on the fundamental principles of electricity and magnetism

Finds applications in electrical engineering, electronics and communication engineering, and related disciplines.

Electromagnetics

3

• To make the subject easy to understand

• To give the students the overall essence of the subject

• To uncover in a reasonable period of time the elementary concepts of the subject as is required to appreciate the engineering application of the subject

• To make studying the subject enjoyable and entertaining.

Objective of the Book

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PrerequisitesElectromagnetics is simple because it is mathematical, and it is mathematics that makes the subject simple.

However, very little mathematics is required to develop the understanding of the subject

Required BackgroundHigh-school-level algebra, trigonometry, and calculus

Definition of vector and scalar quantities

Meaning of dot and cross products of two vector quantities

5

It is worth developing the vector calculus expressions at the outset:

Gradient of a scalar quantity Divergence of a vector quantity Curl of a vector quantity

Vector Calculus Expressions

Let us develop these expressions in generalized curvilinear system of coordinates.But why to choose generalized curvilinear system of coordinates?

6

Why to choose curvilinear system of coordinates for vector calculus expressions?Very easy to deduce vector calculus expressions and remember them

Much easier to deuce than in cylindrical or spherical system of coordinates

Can be written by permuting the first term of a vector calculus expression

Can be easily translated in rectangular, cylindrical or spherical system of coordinates

7

Point P (x,y,z) represented in rectangular system of coordinates

8

Point P (r,,z) represented in cylindrical system of coordinates

zzryrx

sincos

9

Point P (r,,) represented in spherical system of coordinates

cosrzsinsincossin

ryrx

10

zzryrx

sincos

Cylindrical vis-à-vis rectangular coordinate

cosrzsinsincossin

ryrx

Spherical vis-à-vis rectangular coordinate

11

xa

ya

za

a

in the directions of increasing x, y, and z respectively

,

,

,

,

UNIT VECTORS

Rectangular system

za in the directions of increasing r, , and z respectively

Cylindrical system

ra

a

in the directions of increasing r, , and respectively

Spherical system

ra

a

12

zyx aaa

zr aaa

aaar

(Rectangular system)

(Cylindrical system)

(Spherical system)

CROSS PRODUCTS OF UNIT VECTORS

13

Element of volume in different systems of coordinates

14

),,( zyx

),,( zdyyx

),,( dzzyx

),,( zydxx

A

B

C

D

),,( zdyydxx E

),,( dzzdyyx F

),,( dzzydxx G

),,( dzzdyydxx H

Rectangular system of coordinates

dxdydzd Element of volumeElement of volume

15

),,( zr A

B ),,( zdr

C ),,( dzzr

D ),,( zdrr

E ),,( zddrr

),,( dzzdr F

G ),,( dzzdrr

H ),,( dzzddrr

Cylindrical system of coordinates

Element of volume dzrddrd

16

A

B

H

G

X

Y

Z

(c)

DE

F

C

d

r

dr

rd

d

dsinr

dsinr

Spherical system of coordinatesA ),,( r

B ),,( dr

C ),,( dr

D ),,( drr

E ),,( ddrr

F ),,( ddr

G ),,( ddrr

H ),,( dddrr Element of volume drrddrd sin

17

A

F

G

H

B

D

C

E

1 2 3( , , )u u u

1 1

2 2

3 3

( ,,

)

u duu duu du

2 2h du

3 3h du

1 1h du

1a

2a

3a

Generalized curvilinear system of coordinates

Element of volume 332211 duhduhduhd

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Element of volume 332211 duhduhduhd Generalized curvilinear system

dxdydzd Rectangular system Element of volume

zuyuxuhhh 321321 ,,;1,1,1

dzrddrd Cylindrical system Element of volume

zuuruhrhh 321321 ,,;1,,1

321321 ,,;sin,,1 uururhrhh

Spherical system Element of volume drrddrd sin

Element of distance vector directed from one point (A) to another (H)

zyx adzadyadxRd

Rectangular system of coordinates

zr adzardadrRd

Cylindrical system of coordinates

arardadrRd r

sin

Spherical system of coordinates

333222111 aduhaduhaduhRd

Rd

Generalized curvilinear system of coordinates

A

F

G

H

B

D

C

E

1 2 3( , , )u u u

1 1

2 2

3 3

( ,,

)

u duu duu du

2 2h du

3 3h du

1 1h du

1a

2a

3a

20

Gradient of a scalar quantity

The gradient of a scalar quantity such as electric potential V at a point is a vector quantity.

In general, the scalar quantity, here V, varies in space with different rates in different directions.

The magnitude of the gradient of the scalar quantity, here V, is the maximum rate of variation of the quantity in space and the direction of the gradient is the direction in which this maximum rate of variation takes place.

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Gradient of a scalar quantity in curvilinear system

A

H

V V+dV

dR

dR

dn1 2 3( , , )u u u

1 1 2 2 3 3( , , )u du u du u du

na

ψN

Elemental distance vector Rd

directed from A to H situated on the two equipotentials V and V+dV


AH

dn

dR

AN

AN cosAH

cosdRdn

Rdadn n

.

22

A

H

V V+dV

dR

dR

dn1 2 3( , , )u u u

1 1 2 2 3 3( , , )u du u du u du

na

ψN

Rdadn n

.

RdanVdn

nVdV n

nanVVV

grad

RdVdV

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RdVdV

332211 )()()( aVaVaVV


][])()()[( 333222111332211 aduhaduhaduhaVaVaVdV

333222111 )()()( duhVduhVduhVdV

Also,

33

22

11

duuVdu

uVdu

uVdV

24

333

222

111

)(

)(

)(

hVuV

hVuV

hVuV

Equating the right hand sides

333222111 )()()( duhVduhVduhVdV

Also,

33

22

11

duuVdu

uVdu

uVdV

25

333

222

111

)(

)(

)(

hVuV

hVuV

hVuV

333

222

111

1)(

1)(

1)(

uV

hV

uV

hV

uV

hV

332211 )()()( aVaVaVV

333

222

111

111 auV

ha

uV

ha

uV

hV

(Gradient of scalar V in generalized curvilinear system of coordinates)

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333

222

111

111 auV

ha

uV

ha

uV

hV

(Curvilinear system of coordinates)

yyx ayVa

yVa

xVV

(Rectangular system of coordinates)

),,;1,1,1( 321321 zuyuxuhhh

zr azVaV

ra

rVV

1

(Cylindrical system of coordinates)

),,;1,,1( 321321 zuuruhrhh

aV

raV

ra

rVV r

sin11

(Spherical system of coordinates)

),,;sin,,1( 321321 uururhrhh

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Divergence of a vector quantity

The divergence of a vector quantity such as electric field at a point is a scalar quantity.

E

The divergence of a vector quantity such as electric field at a point is the outward flux of through the surface of an elemental or differential volume enclosing the point (closed surface integral of over the surface of the enclosure) divided by the volume element in the limit the volume element tending to zero, thereby the volume element shrinking to the point.

E

S

n dSaELt

EE

0div

E

E

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Divergence of a vector quantity in curvilinear system

A

F

G

H

B

D

C

E

1 2 3( , , )u u u

1 1

2 2

3 3

( ,,

)

u duu duu du

2 2h du

3 3h du

1 1h du

1a

2a

3a

S

n dSaELt

EE

0div

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Outward flux through the faces normal to the unit vector 1a

A

F

G

H

B

D

C

E

1 2 3( , , )u u u

1 1

2 2

3 3

( ,,

)

u duu duu du

2 2h du

3 3h du

1 1h du

1a

2a

3a

Flux through the face ABFC in the direction of 1a

EF

Inward flux through the face ABFCEF

Outward flux through the face ABFC EF

1111332211 )).(( dSEadSaEaEaEFE

))(()AC)(AB( 33221 duhduhdS

33221 duhduhEFE

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Flux through the face DEHG in the direction of 1a

11

duuFF E

E

Outward flux through the face DEHG

)()( 3211

32111

11

hhEu

dudududuuFdu

uFFF EE

EE

33221 duhduhEFE

Outward flux through the face ABFC EF

Outward flux through the faces ABFC and DEHG each normal to the unit vector 1a

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Outward flux through the faces each normal to the unit vector 1a

)()( 3211

321 hhEu

dududu

Outward flux through the faces each normal to the unit vector2a

)()( 1322

321 hhEu

dududu

Outward flux through the faces each normal to the unit vector 3a

)()( 2133

321 hhEu

dududu

Outward flux through all the faces of the differential volume

)()()()( 2133

1322

3211

321 hhEu

hhEu

hhEu

dududu

32

S

n dSaELt

EE

0div

Outward flux through all the faces of the differential volume

)()()()( 2133

1322

3211

321 hhEu

hhEu

hhEu

dududu

.)()()()(

0),,( 332211

2133

1322

3211

321

321 duhduhduh

hhEu

hhEu

hhEu

dududu

dududuLt

E

3

321

2

213

1

132

321

)()()(1uEhh

uEhh

uEhh

hhhE

33

3

321

2

213

1

132

321

)()()(1uEhh

uEhh

uEhh

hhhE


),,;1,1,1( 321321 zuyuxuhhh

zE

yE

xE

E zyx


),,;1,,1( 321321 zuuruhrhh

zEE

rrrE

rzrEE

rrE

rE zrzr

1)(1)()(1


(Spherical system

of coordinates)

)(sin1)(sin

sin1)(1

)()sin()sin(sin1

2

2

2

2

Er

Err

Err

rEErrEr

rE

r

r


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The curl of a vector quantity such as electric field at a point is a vector quantity.

E

Curl of a vector quantity

332211 )()()( aEaEaEE

E

321 )(,)(,)( EEE

333 0

)(S

ldE

SLt

E l

: Components in the directions of respectively 321 ,, aaa

22113 duhduhS

35

3

A)B(B)E(E)D(D)A(

33 0

)(dS

dIdIdIdIdSLt

E EEEE

333 0

)(S

ldE

SLt

E l

22113 duhduhS

EdI stands for line integral.

36

3

A)B(B)E(E)D(D)A(

33 0

)(dS

dIdIdIdIdSLt

E EEEE

EdI stands for line integral.

)( 111D)A( aduhEdIE

LduhEaduhaEaEaE 111111332211 )).((

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)( 111D)A( aduhEdIE

LduhEaduhaEaEaE 111111332211 )).((

22

E)B( duuLLdIE

)( 22

B)E( duuLLdIE

38

LduhEaduhaEaEaEdIE 222222332211B)A( )).((

LdIE A)B(

11

E)D( duuLLdIE

A)B(B)E(E)D(D)A( EEEE dIdIdIdI

22

11

22

11

)( duuLdu

uLLdu

uLLdu

uLLL

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A)B(B)E(E)D(D)A( EEEE dIdIdIdI

22

11

22

11

)( duuLdu

uLLdu

uLLdu

uLLL

3

A)B(B)E(E)D(D)A(

33 0

)(dS

dIdIdIdIdSLt

E EEEE

212

1121

1

22A)B(B)E(E)D(D)A(

)()(dudu

uEh

duduuEh

dIdIdIdI EEEE

111 duhEL

222 duhEL

40

3

A)B(B)E(E)D(D)A(

33 0

)(dS

dIdIdIdIdSLt

E EEEE

212

1121

1

22A)B(B)E(E)D(D)A(

)()(dudu

uEh

duduuEh

dIdIdIdI EEEE

2

11

1

22

213

)()(1)(uEh

uEh

hhE

Similarly,

2

22

1

33

211

)()(1)(uEh

uEh

hhE

1

33

3

11

132

)()(1)(uEh

uEh

hhE

41

332211 )()()( aEaEaEE

2

22

1

33

211

)()(1)(uEh

uEh

hhE

1

33

3

11

132

)()(1)(uEh

uEh

hhE

2

11

1

22

213

)()(1)(uEh

uEh

hhE

32

11

1

22

212

1

33

3

11

131

3

22

2

33

32

)()(1)()(1)()(1 auEh

uEh

hha

uEh

uEh

hha

uEh

uEh

hhE

332211

321

332211

321

1

EhEhEhuuu

ahahah

hhhE

42

32

11

1

22

212

1

33

3

11

131

3

22

2

33

32

)()(1)()(1)()(1 auEh

uEh

hha

uEh

uEh

hha

uEh

uEh

hhE


),,;1,1,1( 321321 zuyuxuhhh

zxy

yzx

xyz a

yE

xE

axE

zEa

zE

yEE


),,;1,,1( 321321 zuuruhrhh


zrzr

rz aE

rrE

ra

rE

zEa

zrEE

rE

)(1)(1

zrzr

rz aE

rrE

ra

rE

zEa

zEE

r

)(11

aE

rrE

ra

rrEE

raEE

rE rr

r

)(1)(sin

11)(sinsin1

),,;sin,,1( 321321 uururhrhh (Spherical system of coordinates)

43

332211

321

332211

321

1

EhEhEhuuu

ahahah

hhhE


zyx

zyx

EEEzyx

aaa

E

),,;1,1,1( 321321 zuyuxuhhh


44

332211

321

332211

321

1

EhEhEhuuu

ahahah

hhhE


zr

zr

ErEEzr

aara

rE

1

),,;1,,1( 321321 zuuruhrhh


45

332211

321

332211

321

1

EhEhEhuuu

ahahah

hhhE


ErrEE

r

arara

rE

r

r

sin

sin

sin1

2



46

Laplacian of scalar and vector quantities

The gradient of a scalar quantity is a vector quantity.

The divergence of a vector quantity is a scalar quantity.

Then if we take as the scalar quantity thenV

VV

grad

becomes a vector quantity and its divergence a scalar quantity.

In other words,VV 2

is a scalar quantity called the Laplacian of the scalar quantity here .V

Laplacian of a scalar quantity

47

3

321

2

213

1

132

321

)()()(1uEhh

uEhh

uEhh

hhhE

332211 aEaEaEE

332211 )()()( aVaVaVV

Interpreting as in the following expression E

V

3

321

2

213

1

132

321

2 )()()(1uVhh

uVhh

uVhh

hhhVV

we get

Let us take the following two vector quantities:

48

.1)(33

21

322

13

211

32

1321

2

uV

hhh

uuV

hhh

uuV

hhh

uhhhVV

333

222

111

1)(

1)(

1)(

uV

hV

uV

hV

uV

hV

3

321

2

213

1

132

321

2 )()()(1uVhh

uVhh

uVhh

hhhVV

49

.1)(33

21

322

13

211

32

1321

2

uV

hhh

uuV

hhh

uuV

hhh

uhhhVV


2

2

2

2

2

22

zV

yV

xV

zV

zyV

yxV

xV

),,;1,1,1( 321321 zuyuxuhhh (Rectangular system of coordinates)

2

2

2

2

22 1111

zVV

rrVr

rrzVr

zV

rrVr

rrV

VVrVr

rrV

sin1sinsin

sin1 2

22

.sin1sin

sin11

2

2

2222

2

V

rV

rrVr

rr

),,;1,,1( 321321 zuuruhrhh (Cylindrical system of coordinates)



50

Laplacian of a vector quantity

Laplacian of the vector is the vector sum of the Laplacian of the vector components. For the vector E

332

222

1122 )()()( aEaEaEE

32

22

12 ,, EEE

are obtained by replacing by in the expression for .

V2321 ,, EEEV

where

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Electromagnetics, a subject that helps in developing the understanding of many concepts in electrical, electronics and communication engineering, is based on the fundamental principles of electricity and magnetism.

Vector calculus expressions can concisely describe the concepts of the subject.

You can hit the right spot by very easily deriving, in the curvilinear system of coordinates, the expressions for the gradient of a scalar quantity, divergence of a vector quantity, curl of a vector quantity, and Laplacian of a scalar or a vector quantity.

Vector symmetry of the terms of the vector calculus expressions must have fascinated you in that from one of the terms of an expression you can permute to write the remaining terms of the expression.

Vector calculus expressions in the curvilinear system of coordinates can easily be translated to write the corresponding expressions in the rectangular, cylindrical and spherical systems of coordinates.

Vector calculus expressions immensely help us develop concisely the concepts of electromagnetics.

Summarising Notes

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Readers are encouraged to go through Chapter 2 of the book for greater details of vector calculus expressions used extensively in the rest of the book.

1 engineering electromagnetics essentials chapter 1 vector calculus expressions for gradient,...

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