FIELD THEORY

Sub Code : EC44 IA Marks : 25Hrs/Week : 04 Exam Hrs. : 03Total Hrs. : 52 Exam Marks : 100

1. Electric Fields 18 hoursa. Coulomb’s law and Electric field intensityb. Electric flux density, Gauss law and divergencec. Energy and potentiald. Conductors, dielectrics and capacitancee. Poisson’s and Laplace’s equations

2. Magnetic fields 14 hoursa. The steady magnetic fieldb. Magnetic forces, materials and inductance

3. Time varying fields and Maxwell’s equations 5 hours

4. Electromagnetic waves 15 hours

Text Books :

William H Hayt Jr and John A Buck, “Engineering Electromagnetics”, Tata McGraw-Hill,6th Edition, 2001

Reference books :

John Krauss and Daniel A Fleisch, “Electromagnetics with Application”, McGraw-Hill, 5th

Edition, 1999Guru and Hiziroglu, Electromagnetics Field theory fundamentals, Thomson Asia Pvt. Ltd IEdition, 2001Joseph Edminster, “Electromagnetics”, Schaum Outline Series, McGraw-HillEdward C Jordan and Keith G Balmain, “Electromagnetic Waves and Radiating Systems”, Prentice-Hall of India, II Edition, 1968, Reprint 2002.David K Cheng, “Field and Wave Electromagnetics”, Pearson Education Ais II Edition, 1989, IndianRepr-01

Introduction to Field Theory

The behavior of a physical device subjected to electric field can be studied either by Field approachor by Circuit approach. The Circuit approach uses discrete circuit parameters like RLCM, voltageand current sources. At higher frequencies (MHz or GHz) parameters would no longer be discrete.They may become non linear also depending on material property and strength of v and iassociated. This makes circuit approach to be difficult and may not give very accurate results.

Thus at high frequencies, Field approach is necessary to get a better understanding of performance ofthe device.

FIELD THEORY

The ‘Vector approach’ provides better insight into the various aspects of Electromagneticphenomenon. Vector analysis is therefore an essential tool for the study of Field Theory.

The ‘Vector Analysis’ comprises of ‘Vector Algebra’ and ‘Vector Calculus’.

Any physical quantity may be ‘Scalar quantity’ or ‘Vector quantity’. A ‘Scalar quantity’ is specifiedby magnitude only while for a ‘Vector quantity’ requires both magnitude and direction to bespecified.

Examples :

Scalar quantity : Mass, Time, Charge, Density, Potential, Energy etc.,Represented by alphabets – A, B, q, t etc

Vector quantity : Electric field, force, velocity, acceleration, weight etc., represented by alphabetswith arrow on top.

etc.,B,E,B,A

Vector algebra : If C,B,A

are vectors and m, n are scalars then

(1) Addition

laweAssociativC)BA()CB(A

laweCommutativABBA

(2) Subtraction)B(-AB-A

(3) Multiplication by a scalar

lawveDistributiBmAm)BA(m

lawveDistributiAnAmAn)(m

laweAssociativ)A(mn)A(nm

laweCommutativmAAm

A ‘vector’ is represented graphically by a directed line segment.

A ‘Unit vector’ is a vector of unit magnitude and directed along ‘that vector’.

Aa is a Unit vector along the direction of A

.

Thus, the graphical representation of A

and Aa are

AaAorA /AaAlso

actor Unit veAVector

A

AA

A

Product of two or more vectors :

(1) Dot Product ( . )

πθ0,B}θCOSA{ORθCOSB(AB.A

B

B

θCosA

A

θCosB

A

A . B = B . A (A Scalar quantity)

(2) CROSS PRODUCT (X)

C = A x B = nθSINBA

C xAB xA)CB( xA

A xB-B xA

vectorsofsystemhandedrightaformCBAsuch thatdirected

BandAofplanelar toperpendicurunit vectoisnand

)θ0(BandAbetweenangleis'θ'where

Ex.,

π

CO-ORDINATE SYSTEMS :

For an explicit representation of a vector quantity, a ‘co-ordinate system’ is essential.

Different systems used :

Sl.No. System Co-ordinate variables Unit vectors1. Rectangular x, y, z ax , ay , az2. Cylindrical ρ, , z aρ , a , az3. Spherical r, , ar , a , a

These are ‘ORTHOGONAL‘ i.e., unit vectors in such system of co-ordinates are mutuallyperpendicular in the right circular way.

r,z,zyxi.e.,

RECTANGULAR CO-ORDINATE SYSTEM :

Zx=0 plane

azp

y=0 Yplane ay

ax z=0 planeX

yxz

xzy

zyx

xzzyyx

aa xa

aa xa

aa xa

0a.aa.aa.a

az is in direction of ‘advance’ of a right circular screw as it is turned from ax to ay

Co-ordinate variable ‘x’ is intersection of planes OYX and OXZ i.e, z = 0 & y = 0

Location of point P :

If the point P is at a distance of r from O, then

If the components of r along X, Y, Z are x, y, z then

arazaya xr rzyx

Equation of Vector AB :

ZandYX,alongBofcomponentsareB&B,Band

ZandYX,alongAofcomponentsareA&A,Awhere

A-BABorBABA

thenaBaBaBBOBand

aAaAaAAOAIf

zys

zys

zzyyxx

zzyyxx

Dot and Cross Products :

get wegrouping,andby term termproducts'Cross'Taking

)aBaBa(B x)aAaAa(AB xA

CABABA)aBaBa(B.)aAaAa(AB.A

zzyyxxzzyyxx

zzyyxxzzyyxxzzyyxx

zyx

zyx

zyx

BBBAAAaaa

B xA

)AB.(ABABABlengthVector

where

AB

ABa

ABalongVectorUnit

CandB,Asidesofoidparallelopaof volume therepresents)C xB(.A(ii)

parallelareBandA0θ0θSin then0B xA

larperpendicuareBandA90θi.e.,0θCos then0B.A(i)

vectors,zerononareCandB,AIf

CCCBBBAAA

)C xB(.A

AB

0

zyx

zyx

zyx

B

B

AB

0 A

A

Differential length, surface and volume elements in rectangular co-ordinate systems

zyx

zyx

adzadyadxrd

dzzrdyrdx

xrrd

azaya xr

y

Differential length 1-----]dzdydx[rd 1/2222

Differential surface element, sd

1. zadxdy:z tor2. zadxdy:z tor ------ 23. zadxdy:z tor

Differential Volume element

dv = dx dy dz ------ 3

zdx p’

p dzdy

r

rdr

0 y

xOther Co-ordinate systems :-

Depending on the geometry of problem it is easier if we use the appropriate co-ordinate system thanto use the Cartesian co-ordinate system always. For problems having cylindrical symmetrycylindrical co-ordinate system is to be used while for applications having spherical symmetryspherical co-ordinate system is preferred.

Cylindrical Co-ordiante systems :-z

P(ρ, , z) x = ρ Cos y = ρ Sin

az r ρ z = z

ap r y

zzy / x tanφ

yxρ1-

22

ρ

x

0

1zrhar

ρrh;aρaraCosρaSinρ-r

1ρrh;aha

ρraSinaCos

ρr

1------dzzrdrρd

ρrrd

azaSinρaCosρr

azaya xr

zz

yx

ρρρyx

zyx

zyx

z

Thus unit vectors in (ρ, , z) systems can be expressed in (x,y,z) system as

2222

z

zzz

y

xyxρ

(dz))dρ(ρdrdand

2------adzadρaρdrd,Furtherorthogonalareaanda,a;aa

aCosaSinaaCosaSin-a

aSinaCosaaSinaCosa

yx

Differential areas :

zz

adz)ρ(dads3-------a.)dρ((dz)ads

a.)dρ()ρ(dads

Differential volume :

4-----dzdρdρdor(dz))dρ()ρ(dd

Spherical Co-ordinate Systems :-Z X = r Sin Cos

Y = r Sin Sin z p Z = r Cos

R r

0 y Y

x r Sin

X

dddrSinr vd

ddrrSdddrSinrSd

ddSinrSd

adSinradradrRd

dRdRdrrRRd

aCosaSin-R/Ra

aSinaSinCosaCosCosR/Ra

aCosaSinSinaCosSinrR/

rRa

aCosraSinSinraCosSinrR

2

2

2r

r

yx

zyx

zyxr

zyx

General Orthogonal Curvilinear Co-ordinates :-z u1 a3 u3

a1u2

a2y

x

Co-ordinate Variables : (u1 , u2, u3) ;Hereu1 is Intersection of surfaces u2 = C & u3 = Cu2 is Intersection of surfaces u1 = C & u3 = Cu3 is Intersection of surfaces u1 = C & u2 = C

33

22

11

321

333222111

33

22

11

321zyx

133221

321321

uRh,

uRh,

uRh

;factorsscaleareh,h,hwhereaduhaduhaduh

duuRdu

uRdu

uRRdthen

u&u,uoffunctionsarezy, x,&azaya xRIf

0a.a&0a.a,0a.aifOrthogonalisSystemu&u,u tol tangentiaorsubnit vectarea,a,a

Co-ordinate Variables, unit Vectors and Scale factors in different systems

Systems Co-ordinate Variables Unit Vector Scale factors

General u1 u2 u3 a1 a2 a3 h1 h2 h3

Rectangular x y z ax ay az 1 1 1

Cylindrical ρ z a ρ a az 1 ρ 1

Spherical r ar a a 1 r r sin

Transformation equations (x,y,z interms of cylindrical and spherical co-ordinate systemvariables)

Cylindrical : x = ρ Cos , y = ρ Sin , z = z ; ρ 0, 0 2 - < z <

Sphericalx = r Sin Cos , y = r Sin Cos , z = r Sinr 0 , 0 , 0 2

)u,u,(uAAand)u,u,(uAA)u,u,(uAA wherefieldVectoraisaAaAaAA&

fieldScalara)u,u,u(VVwhere

AhAhAhuuu

ahahah

hhh1A x

)Ah(hu

)Ah(hu

)Ah(huhhh

1A.

au v

h1a

u v

h1a

u v

h1V

3213332122

32111332211

321

332211

321

332211

321

3213

2312

1321321

333

222

111

Vector Transformation from Rectangular to Spherical :

z

y

x

zyx

zyx

rzryrxr

zyxr

rr

RrrRS

zzyyxxR

AAA

a.aa.aa.aa.aa.aa.aa.aa.aa.a

AAA

asA,A,A torelatedareA,A,AwhereaAaAaA

a)a.A(a)aA(a)aA(A:Spherical

aAaAaAA:rRectangula

R

Field Theory

A ‘field’ is a region where any object experiences a force. The study of performance in the presenceof Electric field )E(

, Magnetic field () is the essence of EM Theory.

P1 : Obtain the equation for the line between the points P(1,2,3) and Q (2,-2,1)

zyx a2-a4-aPQ P2 : Obtain unit vector from the origin to G (2, -2, 1)

Problems on Vector Analysis

Examples :-1. Obtain the vector equation for the line PQ between the points P (1,2,3)m and Q (2, -2, 1) m

ZPQ P (1,2,3)

Q(2,-2,-1)0

Y

X

)a2-a4-a(

a3)-(-1a2)-(-2a1)-(2

a)z-(za)y-(ya) x-(xPQ vectorThe

zyx

zyx

zpqypqxpq

2. Obtain unit vector from origin to G (2,-2,-1)

G

G

0

)a0.333-a0.667-a(0.667a

3(-1)(-2)2G

GGa,orunit vectThe

)a-a2-a(2

a0)-(za)0-(ya)0-(xG vectorThe

zyxg

222

g

zyx

zgygxg

3. Given

zyx

zyx

a5a2-a4-B

aa3-a2A

B xA(2)andB.A(1)find

Solution :)a5a2-a(-4.)aa3-a(2B.A(1) zyxzyx

= - 8 + 6 + 5 = 3Since ax . ax = ay . ay = az . az = 0 and ax ay = ay az = az ax = 0

(2)524132aaa

B xAzyx

= (-13 ax -14 ay - 16 az)

4. Find the distance between A( 2, /6, 0) and B = ( 1, /2, 2)Soln : The points are given in Cylindrical Co-ordinate (ρ,, z). To find the distance between twopoints, the co-ordinates are to be in Cartesian (rectangular). The corresponding rectangular co-ordinates are (ρ Cos, ρ Sin, z)

2.6421.73AB)(

a2a1.73-

a0)-(2a1)-(1a1.73-

a)A-(Ba)A-(Ba)A-(BAB

a2aa2a2

Sina6

CosB&

aa1.73a6

Sin2a6

Cos2A

22

zx

zyx

zzzyyyxxx

zyzyx

yxyx

5. Find the distance between A( 1, /4, 0) and B = ( 1, 3/4, )Soln : The specified co-ordinates (r, , ) are spherical. Writing in rectangular, they are (r Sin

Cos , r Sin Sin , r Cos ).Therefore, A & B in rectangular co-ordinates,

1.7320.5)0.5(2

)AB.AB(AB

a(-0.707)a0.707)(-a1.414-

a)A-(Ba)A-(Ba)A-(BAB

)a0.707a0.707(

)a4

3CosaSin4

3SinaCos4

3Sin(B

)a0.707a0.707(

)a4

Cos1a0Sin4

Sin1a0Cos4

Sin(1A

1/2

1/2

zyx

zzzyyyxxx

yx

zyx

yx

zyx

6. Find a unit vector along AB in Problem 5 above.

AB

ABaAB = [ - 1.414 ax + (-0.707) ay + (-0.707) az]1.732

1

= )a408.0a0.408-a0.816-( zyx

7. Transform ordinates.-ColCylindricainFinto)a6a8-a(10F zyx

Soln :

a)a.F(a)a.F(a)a.F(F zzppCyl

Field Theory Notes eBook

Publisher : VTU eLearning Author :

Type the URL : http://www.kopykitab.com/product/1851

Get this eBook