electromagnetics and radiationjnujprdistance.com/assets/lms/lms jnu/mba/mba - telecom mana… ·...

Electromagnetics and Radiation

This book is a part of the course by Jaipur National University, Jaipur.This book contains the course content for Electromagnetics and Radiation.

JNU, JaipurFirst Edition 2013

The content in the book is copyright of JNU. All rights reserved.No part of the content may in any form or by any electronic, mechanical, photocopying, recording, or any other means be reproduced, stored in a retrieval system or be broadcast or transmitted without the prior permission of the publisher.

JNU makes reasonable endeavours to ensure content is current and accurate. JNU reserves the right to alter the content whenever the need arises, and to vary it at any time without prior notice.

I/JNU OLE

Index

ContentI. ...................................................................... II

List of FiguresII. ........................................................... V

List of TablesIII. ...........................................................VI

AbbreviationsIV. ....................................................... VII

ApplicationV. ............................................................... 92

BibliographyVI. ........................................................... 98

Self Assessment AnswersVII. ................................... 100

Book at a Glance

II/JNU OLE

Content

Chapter I ....................................................................................................................................................... 1Coulomb’s Law and Electric Field Intensity ............................................................................................. 1Aim ................................................................................................................................................................ 1Objectives ...................................................................................................................................................... 1Learning outcome .......................................................................................................................................... 11.1 Introduction .............................................................................................................................................. 21.2 Coulomb’s Law ........................................................................................................................................ 21.3 Principle of Superposition ........................................................................................................................ 31.4 Electric Field Intensity ............................................................................................................................. 3 1.4.1 Field due to Continuous Volume Distribution ......................................................................... 3 1.4.2 Field due to Line Charge.......................................................................................................... 5 1.4.3 Field of Sheet .......................................................................................................................... 71.5 Electric Flux Density ............................................................................................................................... 81.6 Gauss’s Law ........................................................................................................................................... 101.7 Gauss’s Law and the Divergence Theorem .............................................................................................11 1.7.1 Divergence ..............................................................................................................................11 1.7.2 Divergence and Maxwell’s First Equation in Point Form ..................................................... 13 1.7.3 The Divergence Theorem ....................................................................................................... 13Summary ..................................................................................................................................................... 14References ................................................................................................................................................... 14Recommended Reading ............................................................................................................................. 14Self Assessment ........................................................................................................................................... 15

Chapter II ................................................................................................................................................... 17Energy and Potential ................................................................................................................................. 17Aim .............................................................................................................................................................. 17Objectives .................................................................................................................................................... 17Learning outcome ........................................................................................................................................ 172.1 Introduction to Energy and Potential ..................................................................................................... 182.2 Energy Expended in Moving a Point Charge in an Electric Field ......................................................... 182.3 The Line Integral .................................................................................................................................... 192.4 Definition of Potential Difference and Potential ................................................................................... 202.5 The Potential Field of a Point Charge .................................................................................................... 222.6 The Potential Field of a System of Charges: Conservative Property ..................................................... 242.7 Potential Gradient .................................................................................................................................. 25Summary ..................................................................................................................................................... 28References ................................................................................................................................................... 28Recommended Reading ............................................................................................................................. 28Self Assessment ........................................................................................................................................... 29

Chapter III .................................................................................................................................................. 31Poisson’s and Laplace’s Equations ........................................................................................................... 31Aim .............................................................................................................................................................. 31Objectives .................................................................................................................................................... 31Learning outcome ........................................................................................................................................ 313.1 Introduction ............................................................................................................................................ 323.2 Poisson’s and Laplace’s Equations ........................................................................................................ 32 3.2.1 2∇ Operation in Different Co-ordinate Systems .................................................................... 333.3 Uniqueness Theorem.............................................................................................................................. 333.4 Procedure for Solving Laplace’s Equation ............................................................................................ 35Summary ..................................................................................................................................................... 38References ................................................................................................................................................... 38Recommended Reading ............................................................................................................................. 38

III/JNU OLE

Self Assessment ........................................................................................................................................... 39

Chapter IV .................................................................................................................................................. 41Magnetic Forces, Materials and Inductance ........................................................................................... 41Aim .............................................................................................................................................................. 41Objectives .................................................................................................................................................... 41Learning outcome ........................................................................................................................................ 414.1 Introduction to Magnetic Flux ............................................................................................................... 424.2 Magnetic Flux and Magnetic Flux Density ........................................................................................... 424.3 Magnetic Force on a Moving Charge .................................................................................................... 424.4 Magnetic Force on Current .................................................................................................................... 434.5 Torque on a Current Loop ...................................................................................................................... 454.6 Magnetisation ......................................................................................................................................... 484.7 Magnetic Materials ................................................................................................................................ 514.8 Magnetic Boundary Conditions ............................................................................................................. 524.9 Inductors and Inductance ....................................................................................................................... 554.10 Mutual Inductance Calculations .......................................................................................................... 564.11 Internal and External Inductance ......................................................................................................... 574.12 Magnetic Forces on Magnetic Materials .............................................................................................. 574.13 Magnetic Circuits ................................................................................................................................. 59Summary ..................................................................................................................................................... 60References ................................................................................................................................................... 60Recommended Reading ............................................................................................................................. 60Self Assessment ........................................................................................................................................... 61

Chapter V .................................................................................................................................................... 63Time –Varying Fields And Maxwell’s Equations .................................................................................... 63Aim .............................................................................................................................................................. 63Objectives .................................................................................................................................................... 63Learning outcome ........................................................................................................................................ 635.1 Faraday’s Law ........................................................................................................................................ 645.2 Displacement Current ............................................................................................................................ 655.3 Maxwell’s Equations in Point Form ...................................................................................................... 685.4 Maxwell’s Equations in Integral Form .................................................................................................. 69Summary ..................................................................................................................................................... 72References ................................................................................................................................................... 72Recommended Reading ............................................................................................................................. 72Self Assessment ........................................................................................................................................... 73

Chapter VI .................................................................................................................................................. 75Antenna Fundamentals ............................................................................................................................. 75Aim .............................................................................................................................................................. 75Objectives .................................................................................................................................................... 75Learning outcome ........................................................................................................................................ 756.1 Introduction ........................................................................................................................................... 766.2 How an Antenna Radiates ...................................................................................................................... 766.3 Near and Far Field Regions ................................................................................................................... 776.4 Antenna Performance Parameters .......................................................................................................... 77 6.4.1 Radiation Pattern .................................................................................................................... 78 6.4.2 Directivity .............................................................................................................................. 79 6.4.3 Input Impedance .................................................................................................................... 79 6.4.4 Voltage Standing Wave Ratio (VSWR) ................................................................................. 80 6.4.5 Return Loss (RL) ................................................................................................................... 81 6.4.6 Antenna Efficiency................................................................................................................. 81 6.4.7 Antenna Gain ......................................................................................................................... 81

IV/JNU OLE

6.4.8 Polarisation ............................................................................................................................ 82 6.4.9 Bandwidth .............................................................................................................................. 836.5 Types of Antennas .................................................................................................................................. 83 6.5.1 Half Wave Dipole................................................................................................................... 84 6.5.2 Monopole Antenna ................................................................................................................. 85 6.5.3 Loop Antenna ......................................................................................................................... 85 6.5.4 Helical Antennas .................................................................................................................... 87 6.5.5 Horn Antennas ....................................................................................................................... 88Summary ..................................................................................................................................................... 89References ................................................................................................................................................... 89Recommended Reading ............................................................................................................................. 89Self Assessment ........................................................................................................................................... 90

V/JNU OLE

List of Figures

Fig. 1.1 Coulomb interaction between two charges ....................................................................................... 2Fig. 1.2 Evaluation of the E field due to a volume charge distribution ......................................................... 4Fig. 1.3 Evaluation of the E field due to a line charge ................................................................................... 6Fig. 1.4 Evaluation of the E field due to an infinite sheet of charge.............................................................. 7Fig. 1.5 Flux lines .......................................................................................................................................... 9Fig. 1.6 The vector differential area, ....................................................................................................... 10Fig. 1.7 Example of interpretation of the divergence .................................................................................. 12Fig. 2.1 A graphical interpretation of line integral in a uniform field .......................................................... 20Fig. 2.2 (a) Circular path (b) A radial path .................................................................................................. 21Fig. 2.3 A general path between general points B and A in the field of a point charge Q at the origin ....... 23Fig. 2.4 A potential field is shown by its equipotential surfaces .................................................................. 26Fig. 3.1 Concentric right cylinders .............................................................................................................. 36Fig. 4.1 Direction of the charge in an electric and magnetic field ............................................................... 43Fig. 4.2 Magnetic force on current ............................................................................................................... 43Fig. 4.3 Example of force between line currents ......................................................................................... 44Fig. 4.4 x-y plane ......................................................................................................................................... 46Fig. 4.5 Forces due to Bz .............................................................................................................................. 46Fig. 4.6 Forces due to By .............................................................................................................................. 47Fig. 4.7 Loop area ........................................................................................................................................ 47Fig. 4.8 Loop area ........................................................................................................................................ 48Fig. 4.9 Magnetic moments ......................................................................................................................... 48Fig. 4.10 Magnetic flux density ................................................................................................................... 49Fig. 4.11 Magnetic moments ........................................................................................................................ 50Fig. 4.12 B-H curve ..................................................................................................................................... 52Fig. 4.13 Tangential magnetic field ............................................................................................................. 52Fig. 4.14 Tangential magnetic field ............................................................................................................. 54Fig. 4.15 Normal magnetic flux density ...................................................................................................... 54Fig. 4.16 Example mutual inductance between coaxial loops ..................................................................... 56Fig. 4.17 Magnetic forces on magnetic materials ........................................................................................ 57Fig. 4.18 Magnetic field ............................................................................................................................... 58Fig. 4.19 Electric and magnetic circuits ....................................................................................................... 59Fig. 5.1 A filamentary conductor forms a loop connecting the two plates of a parallel-plate capacitor ..... 68Fig. 6.1 Radiation from an antenna .............................................................................................................. 76Fig. 6.2 Field regions around an antenna ..................................................................................................... 77Fig. 6.3 Radiation pattern of a generic directional antenna ......................................................................... 78Fig. 6.4 Equivalent circuit of transmitting antenna ...................................................................................... 80Fig. 6.5 A linearly (vertically) polarised wave ............................................................................................. 82Fig. 6.6 Commonly used polarisation schemes ........................................................................................... 82Fig. 6.7 Measuring bandwidth from the plot of the reflection coefficient ................................................... 83Fig. 6.8 Half wave dipole ............................................................................................................................. 84Fig. 6.9 Radiation pattern for half wave dipole ........................................................................................... 84Fig. 6.10 Monopole antenna ........................................................................................................................ 85Fig. 6.11 Radiation pattern for the monopole antenna ................................................................................. 85Fig. 6.12 Loop antenna ................................................................................................................................ 86Fig. 6.13 Radiation pattern of small and large loop antenna ....................................................................... 86Fig. 6.14 Helix antenna ................................................................................................................................ 87Fig. 6.15 Radiation pattern of helix antenna ................................................................................................ 87Fig. 6.16 Types of horn antenna ................................................................................................................... 88

VI/JNU OLE

List of Tables

Table 4.1 Difference between magnetisation and polarisation .................................................................... 50Table 4.2 Difference between inductors and capacitors .............................................................................. 55Table 4.3 Difference between electric and magnetic circuits ...................................................................... 59

VII/JNU OLE

Abbreviations

Emf – Electromotive forceRL – Return Loss VSWR – Voltage Standing Wave Ratio

1/JNU OLE

Chapter I

Coulomb’s Law and Electric Field Intensity

Aim

The aim of this chapter is to:

describe Coulomb’s law•

explainelectricfluxintensityandelectricfluxdensity•

illustrate Gauss’s law and divergence theorem•

Objectives

The objective of this chapter is to:

examinevariousfieldsofintensity•

learn Gauss’s law•

calculatedivergenceandMaxwell’sfirstequation•

Learning outcome

At the end of this chapter, the students will be able to:

recall Coulomb’s law•

understand • electricfluxintensityandelectricfluxdensity

categorise • Gauss’s law, divergence theorem and Maxwell’s equation


2/JNU OLE

1.1 IntroductionIt is a common experience that if we comb our hair, and subsequently bring the comb close to tiny bits of paper, the comb attracts pieces of paper. The phenomenon involved is the production of static electric charges, which has been known to mankind as long back as 600 BC.

If you rub a glass rod with a piece of silk, the rod develops an electric charge. When another glass rod, similarly rubbed, is brought near it, the two repel each other. However, if the glass rod is brought towards an amber rod rubbed with fur, the two rods attract each other. Rods treated in this fashion are said to have developed electric charge, and, the force between such charged bodies is called electric force.

There are two types of observed electric charge, which we designate as positive and negative. The convention was derived from Benjamin Franklin’s experiments. He rubbed a glass rod with silk and called the charges on the glass rod positive. He rubbed sealing wax with fur and called the charge on the sealing wax negative. Like charges repel and opposite charges attract each other. The unit of charge is called the Coulomb (C).

The smallest unit of “free” charge known in nature is the charge of an electron or proton, which has a magnitude of, (1)

Charge of any ordinary matter is quantized in integral multiples of e. An electron carries one unit of negative charge, -e, while a proton carries one unit of positive charge, +e. In a closed system, the total amount of charge is conserved since charge can neither be created nor destroyed. A charge can, however, be transferred from one body to another.

1.2 Coulomb’s LawConsider a system of two point charges, q1 and q2, separated by a distance r in vacuum. Coulomb’s law states that the force F between two point charges q1 and q2 is:

Along the line joining them•Directly proportional to the product q• 1 to q2 of the chargesInversely proportional to the square of the distance r between them.•

The force exerted by q1 on q2 is given by Coulomb’s law,

12= (2)

Where ke is the Coulomb constant, and is a unit vector directed from q1 to q2,asillustratedinfig.below

Fig. 1.1 Coulomb interaction between two charges(Source: http://web.mit.edu/8.02t/www/802TEAL3D/visualizations/coursenotes/modules/guide02.pdf)

Note that electric force is a vector which has both magnitude and direction. In SI units, the Coulomb constant ke is given by

(3)

3/JNU OLE

Where,

(4)

This is known as the “permittivity of free space.” Similarly, the force on q1due to q2 is given by 21= 12 as illustratedinfig1.1(b).ThisisconsistentwithNewton’sthirdlaw.

1.3 Principle of SuperpositionCoulomb’s law applies to any pair of point charges. When more than two charges are present, the net force on any one charge is simply the vector sum of the forces exerted on it by the other charges. For example, if three charges are present, the resultant force experienced by q3 due to q1 and q2 will be

13+ 23 (5)

1.4 Electric Field IntensityIfwenowconsideronechargefixedinposition,sayq1. and move a second charge slowly around, we note that there exists everywhere a force on this second charge; in other words, this second charge is displaying the existence of a forcefield.Callthissecondchargeatestchargeqt. The force on it is given by Coulomb’s law,

(6)

Writing this force as force per unit charge

(7)

The quantity is a function of only q1 and the directed line segment from q1 to the position of the test charge. This describesavectorfieldandiscalledtheelectricfieldintensity.

Theelectricfieldintensityisdefinedasthevectorforceonaunitpositivetestcharge.Electricfieldintensitymustbemeasuredbytheunitnewtonspercoulombtheforceperunitcharge.UsingcapitalletterE.forelectricfieldintensity, we have

(8)

(9)

Theaboveequationdefineselectricfieldintensityduetoasinglepointchargeq1 in a vacuum.

1.4.1 Field due to Continuous Volume DistributionLet the volume charge distribution with uniform charge density pvbeasshowninfig4.8.ThechargedQassociatedwith the elemental volume dv is

(10)


4/JNU OLE

Fig. 1.2 Evaluation of the E field due to a volume charge distribution(Source:http://www.nu.edu.sa/userfiles/iaalyafrosi/chapter%2004.pdf)

And hence the total charge in a sphere of radius a is

(11)

(12)

TheelectricfielddEatP(0,0,z)duetotheelementaryvolumechargeis

(13)

Where Due to the symmetry of the charge distribution, the contributions to Ex or Ey. We are left with only Ez, given by,

(14)

Again, we need to derive expressions for dv, R2, and α

(15) Applyingthecosineruletofig.1.2,wehave

(16)

5/JNU OLE

(17)

It is convenient to evaluate the integral in eq (14) in terms of R and r’. Hence we express cosθ, cosα, and sinθ dθin terms of R and r', that is

(18)

(19)

Differentiating equation (19) with respect to keeping z and r′fixed,weobtain

(20)

Substituting equation (15) to equation (20) into equation (14) yields

(21)

(22)

(23)

(24)

or

(25)

ThisresultisobtainedforEatP(0,0,z).Duetothesymmetryofthechargedistribution,theelectricfieldatP(r,θ, φ) is readily obtained from equation (25) as

(26)

ThisisidenticaltotheelectricfieldatthesamepointduetoapointchargeQlocatedattheoriginorthecenterofthe spherical charge distribution

1.4.2 Field due to Line ChargeConsider a line charge with uniform charge density ρLextendingfromAtoBalongthez-axisasshowninfig.1.3.The charge element dQ associated with element dl = dz of the line is

(27)


6/JNU OLE

Fig. 1.3 Evaluation of the E field due to a line charge(Source:http://www.nu.edu.sa/userfiles/iaalyafrosi/chapter%2004.pdf)

And hence, the total charge Q is

(28)

TheelectricfieldintensityduetoeachofthechargedistributionspL, ps, and pv may be regarded as the summation ofthefieldcontributedbythenumerouspointchargesmakingupthechargedistribution.

(29)

TheelectricfieldintensityEatanarbitrarypointP(x,y,z)canbefoundusingequation(29).Thusfromfig.1.3

(30)

(31)

or

(32)

(33)

(34)

Substituting this in equation(29), we get

(35)

Toevaluatethis,itisconvenientthatwedefineα, α1 and α2 asinfig1.3.

(36)

(37)

7/JNU OLE

(38)

Hence, equation (35) becomes

(39)

(40)

Thus,forafinitelinecharge,

(41)

Asaspecialcase,foraninfinitelinecharge,pointBisat(0,0,∞) and A at (0, 0,‒∞) so that α1=π/2, α2= ‒π/2; the z-component vanishes and equation (41) becomes

(42)

Bearinmindthatequation(42)isobtainedforaninfinitelinechargealongthez-axissothatpandap have their usual meaning. If the line is not along the z-axis, p is the perpendicular distance from the line to the point of interest and apisaunitvectoralongthatdistancedirectedfromthelinechargetothefieldpoint.

1.4.3 Field of Sheet Consideraninfinitesheetofchargeinthexy-planewithuniformchargedensityps. The charge associated with elemental area dS is

And hence the total charge is,

(43)

ThecontributionofEfieldatpointP(0,0,h)bytheelementalsurfaceshowninFig.1.4is

(44)

Fig. 1.4 Evaluation of the E field due to an infinite sheet of charge(Source:http://www.nu.edu.sa/userfiles/iaalyafrosi/chapter%2004.pdf)


8/JNU OLE

Nowfromfig.1.4,

Substituting these terms in equation (44) gives

(45)

Due to the symmetry of the charge distribution, for every element 1, there is a corresponding element 2 whose contribution along apcancelsthatofelement1,asillustratedinfig.1.4.ThusthecontributionstoEp add up to zero so that E has only z-component. This can also be shown mathematically by replacing ap with cos φax + sin φay Integration of cosφ or sinφ over 0 < φ < 2π gives zero. Therefore

(46)

Thatis,Ehasonlyz-componentifthechargeisinthexy-plane.Ingeneral,foraninfinitesheetofcharge

(47)

Where anisaunitvectornormaltothesheet.Fromequation(46)or(47),wenoticethattheelectricfieldisnormalto the sheet and it is surprisingly independent of the distance between the sheet and the point of observation P. In aparallelplatecapacitor,theelectricfieldexistingbetweenthetwoplateshavingequalandoppositechargesisgiven by

(48)

1.5 Electric Flux DensityIn(approximately)1837,MichaelFaraday,beinginterestedinstaticelectricfieldsandtheeffectswhichvariousinsulatingmaterials(ordielectrics)hadonthesefields,devisedthefollowingexperiment:

9/JNU OLE

Faraday had two concentric spheres constructed in such a way that the outer one could be dismantled into two hemispheres. With the equipment taken apart, the inner sphere was given a known positive charge. Then, using about 2cm of “perfect” (ideal) dielectric material in the intervening space, the outer shell was clamped around the inner. Next, the outer shell was discharged by connecting it momentarily to ground. The outer shell was then carefully separated and the negative charge induced on each hemisphere was measured.

Faraday found that the magnitude of the charge induced on the outer sphere was equal to that of the charge on the inner sphere, irrespective of the dielectric used. He concluded that there was some kind of “displacement” from the inner to the outer sphere which was independent of the medium. This is now referred to severally as displacement, displacementflux,or,asweshalluse,electricflux.(ofcourse,theideaof“electricfluxlines”asentitiesstreamingaway from electric charge (i.e., streamlines) is simply an invention to aid our conceptualisation of the presence of anelectricfield).Thisflux,denotedbyΨ,isinSIunitsrelatedtothecharge,𝑄, producing it via a dimensionless proportionalityconstantofunity;i.e.,theelectricfluxincoulombsisgivenby

Ψ=QThus,thefluxisindependentofthepropertiesofthemediumbetweenthespheresanditsmagnitudedependsonlyon the size of the charge producing it.

Thepathofthefluxlinesisradiallyawayfromtheinnersphereasshowninfig.1.5

Fig. 1.5 Flux lines(Source:http://www.ie.itcr.ac.cr/acotoc/Maestria_en_Computacion/Sistemas_de_Comunicacion_II/Material/

Biblio2/chapt03.pdf)

Animportantentityinelectromagneticsistheideaofelectricfluxdensity, . For example, in the above illustration, at the surface of the inner sphere,

While at the outer surface,

Clearly, is measured in coulombs/metre2 (abbreviated C/m2). If we think of the inner sphere as shrinking to a point charge, 𝑄, then at a distance 𝑟 from the charge,

(49)

when considering 𝑄 to be at the origin so that it may be seen that, for free space,

(50)


10/JNU OLE

Equation (50) is one of the important so-called constitutive relations which are essential in solving electromagnetics problems – and it is not restricted to point charges

If the charge is distributed within a volume, such that the charge density leads to a volume integral

(51)

Here, as usual, 𝑅 = ∣𝑟−𝑟′∣ is the distance from the differential volume, 𝑑𝑣′,underconsiderationtothepointofobservation, and is the unit vector (𝑟− 𝑟′)/∣𝑟− 𝑟′∣ in that direction. If the region of interest is NOT effectively free space, then the permittivity, 𝜖0, must be replaced with the permittivity, 𝜖, of the region.

1.6 Gauss’s LawThinking back to Faraday’s experiment, it could be observed that the shape of the source charge inside the outer spherewouldnotbethecriticalfactorininducinga−𝑄 charge on the outer sphere. In fact, the inner charged body could take any shape and if it had a charge of +𝑄intotal,thiswouldinducea−𝑄 charge on the outer sphere. The totalamountoffluxinthedielectricatanydistancethatcompletelyenclosedtheinnercharged‘object’wouldthusbe the same irrespective of the object’s shape – it could be a cubical charge or even a charge on an irregularly shaped object.Ofcourse,thedistributionofthefluxlines(i.e.,the‘shape’ofthefieldorequivalentlythedistributionofthefluxdensity)inthedielectricwouldbeaffected,butnotthetotalflux.

The generalisation of Faraday’s experiments led to the following formalisation known as Gauss’s Law - “The electric flux passing through any closed surface is equal to the total free charge enclosed by that surface.”

In general, the closed surface may take any form we wish to visualize – which surface shape will be more convenient to consider for a particular application of Gauss’s law will usually depend on the shape of the charge distribution

Fig. 1.6 The vector differential area, (Source: http://www.ie.itcr.ac.cr/acotoc/Maestria_en_Computacion/Sistemas_de_Comunicacion_II/Material/


Sincethedifferentialflux,𝑑Ψ,crossingthedifferentialareamustbetheproductofthenormalcomponentof and the differential surface :

Therefore,

Gauss’s law (52)

SinceΨ=𝑄, measured in coulombs, where Q is the charge enclosed by 𝑆. Thus, too,

11/JNU OLE

(53)

Thismeans,inthestatementofGauss’slaw,thesurfaceoverwhichthefluxisintegratedsurroundsthecharge–thecharge is enclosed by the surface. This closed surface, used in the context of Gauss’s law, is often referred to as a Gaussian surface.

Of course, if the charge is due to a linear charge density, 𝜌𝐿, or a surface charge density, 𝜌𝑆 (surface on which the charge exists is not necessarily closed), the volume integral in equation (53) must be replaced by a line integral or a surface integral as follows:

For line charge,• where ρL is the linear charge density in C/m.For surface charge, • where ρs is the surface charge density in C/m2.

As intimated above, facilitating application of Gauss’s law is dependent on a suitable choice of the closed surface for integration. In fact, if the charge distribution is known, equation (53) can be used to obtain in an easy manner if it possible to choose a closed surface, 𝑆,whichsatisfiesthefollowingtwoproperties:

• is everywhere either normal or tangential to 𝑆 so that , respectively.On that portion of S where• , the magnitude, is constant.

Beforeconsideringafewimportantexamples,wenotethattheflux,Ψ,passingthroughanon-closedsurfaceisgiven simply as,

1.7 Gauss’s Law and the Divergence TheoremEach concept is discussed below in detail.

1.7.1 DivergenceIntheanalysisofthevectorandscalarfieldscommonlyfoundinthestudyofelectromagnetics,thereareseveraloperations which involve the so-called Del operator.In Cartesian coordinates, this operator takes the form

Wefirstconsiderthedotproduct,

Definition: The divergence of a vector function (i.e., inCartesiancoordinates)isdefinedas

Therefore,

(54)

Notice that the divergence of a vector is a scalar! The divergence will not have such a simple form in cylindrical and spherical coordinates.

Consider the example below


12/JNU OLE

The divergence of (sometimes written, div ), is

Interpretation of the divergenceFor the sake of simplicity, consider a vector function, , that has only an component. From (54)

With reference to the figure below and noting that the rectangular block is of differential volume ,thedefinitionofpartialdifferentiationgives

or

(55)

We next observe that,

Fig. 1.7 Example of interpretation of the divergence(Source:http://www.engr.mun.ca/~egill/index_files/5812_w10/5812_unit3_2011.pdf)

In(55), thefirst terminthenumeratoris theflowoutwhilethesecondis theflowintotheblockthrough𝑑𝑆𝑥. Therefore, (55) may be written as,

(56)

While recalling that points away from the block at both the right and left faces.

Making identical arguments for and , (56) may be generalised to include all surfaces by writing

(57)Where 𝑆 is now the whole surface of the block. Thus,

(58)

13/JNU OLE

andthedivergenceofavectorfield represents“thenetoutflowofthevector per unit volume as the volume shrinks to zero”. For example, couldbetheelectricfluxdensity, , so that wouldbethe“netfluxperunitvolume leaving” a particular region as the volume shrinks to zero. Hence the term divergence;

ifthereisnetoutflow(i.e.,asourceregion) andifthereisanetinflow(i.e.,asinkregion)

1.7.2 Divergence and Maxwell’s First Equation in Point FormInequation(58),wenowreplacethegeneralvectorfield withthespecificvectorfield,theelectricfluxdensity,

.This gives

(59)Since we immediately recognize from Gauss’s law that

Thus, equation (59) may be written as

Maxwell’sfirstequation (60)

ThisisthefirstoffourequationsreferredtoasMaxwell’sequationsastheyapplytoelectrostaticsandmagnetostatics.It is simply Gauss’s law rewritten on a per-unit-volume basis. Here it is said to be in “point form” as it indicates thatthefluxperunitvolumeleaving(i.e.,divergingfrom)avanishinglysmallvolumeisequaltothechargeperunit volume contained therein. If you simply consider the units on each side of the equation you will see this is a reasonable interpretation.

1.7.3 The Divergence TheoremNote that equation (53) may be written for any volume in which charge is enclosed and this volume may indeed be taken to have 𝑆 as the enclosing surface. Thus, we may write

where S surrounds v. Then, it can be noticed that equation (60) says,

From these two equations, on replacing 𝜌𝑣inthefirstwiththedivergenceof found in the second, we have

Divergence theorem (61)

Thistheoremistrueforanyvectorfield(notjust ). Its usefulness can be immediately seen when we consider that it is often easier to carry out a surface integral than a volume integral.

What does this integral say physically? “Itsaysthatthefluxexitingthroughaclosedsurfaceequalsthesumtotalofthedivergenceofthefluxdensitythroughout the volume which the surface encloses.” If we think of the total volume as consisting of small constituent volumes,thenasthefluxdivergesfromoneoftheconstituentsitconvergesonanotherandsoonuntilitreachesaconstituentwhichcontainsaportionoftheoutersurfacewhereatleastpartofthefluxmayescape–i.e.,diverge–fromthevolumealtogether.Itisthisescapingfluxthatcorrespondstothenetdivergencethroughoutthevolume.


14/JNU OLE

SummaryThere are two types of observed electric charge, which are designated as positive and negative, this convention •was derived from Benjamin Franklin’s experiments.An electron carries one unit of negative charge, -e, while a proton carries one unit of positive charge, +e.•Coulomb’s law states that the force F between two point charges q• 1 and q2 is:

along the line joining them directly proportional to the product q 1 to q2 of the chargesinversely proportional to the square of the distance r between them

Gauss’s Law states that “• The electric flux passing through any closed surface is equal to the total free charge enclosed by that surface”.The divergence of a vector function • (i.e., inCartesiancoordinates)isdefinedas,

ReferencesCoulomb’s Law.• Available at: <http://web.mit.edu/8.02t/www/802TEAL3D/visualizations/coursenotes/modules/guide02.pdf)> [Accessed 2 February 2011].Electrostatics• .Availableat:<http://www.nu.edu.sa/userfiles/iaalyafrosi/chapter%2004.pdf)>Accessedon2nd February, 2011.Unit 3 - Electric Flux Density, Gauss’s Law and Divergence• . Available at: <http://www.engr.mun.ca/~egill/index_files/5812_w10/5812_unit3_2011.pdf>[Accessed2February2011].

Recommended ReadingBalanis, C.A, 1989. • Advanced Engineering Electromagnetics. Wiley, Solution Manual edition.Cheng, D.K., 1992. • Fundamentals of Engineering Electromagnetics. Prentice Hall, 1st ed.Ida, N., 2004. • Engineering Electromagnetics. Springer, 2nd ed.

15/JNU OLE

Self Assessment

The unit of charge is called the ____________.1. Coulomba. Faradayb. Joulec. Henryd.

Coulomb’s law applies to any pair of ___________ charges.2. linea. pointb. vectorc. staticd.

The____________fieldintensityisdefinedasthevectorforceonaunitpositivetestcharge.3. magnetica. electronicb. staticc. electricd.

The electric _____________ passing through any closed surface is equal to the total free charge enclosed by 4. that surface.

curla. densityb. volumec. fluxd.

_____________ found that the magnitude of the charge induced on the outer sphere was equal to that of the 5. charge on the inner sphere, irrespective of the dielectric used.

Faradaya. Coulombb. Henryc. Jouled.

Which of the following is true?6. Divergencetheoremistrueforanyvectorfield.a. Gauss’stheoremistrueforanyvectorfield.b. Divergencetheoremisnottrueforanyvectorfield.c. Gauss’stheoremisnottrueforanyfield.d.

Which of the following is true?7. Faraday’s law applies to any pair of point charges.a. Coulomb’s law applies to any pair of line charges.b. Coulomb’s law applies to any pair of point charges.c. Coulomb’s law do not applies to any pair of point charges.d.


16/JNU OLE

Which of the following is true?8. A proton carries one unit of negative charge.a. A neutron carries one unit of negative charge.b. An atom carries one unit of negative charge.c. An electron carries one unit of negative charge.d.

The divergence of a vector function 9. (i.e., in Cartesian coordinates) is given by which of the following?

The force exerted by q10. 1 on q2 is given in which of the following Coulomb’s law?

a. 12=

b. 12=

c. 12=

d. 12=

17/JNU OLE

Chapter II

Energy and Potential

Aim


describeenergyexpandedinmovingapointchargeinanelectricfield•

explainthedefinitionofpotentialdifferenceandpotential•

describe• potential gradient

Objectives

The objectives of this chapter are to:

examineenergyexpandedinmovingapointchargeinanelectricfield•

elucidate line integral•

explain potential, potential difference and potential gradient•

Learning outcome


comprehend • line integral

recalldefinitionofpotentialdifferenceandpotential•

understan• daboutpotentialfieldofapointcharge


18/JNU OLE

2.1 Introduction to Energy and PotentialInthepreviouschapterwebecameacquaintedwithCoulomb’slawanditsuseinfindingtheelectricfieldaboutseveralsimpledistributionsofcharge,andalsowithGauss’slawanditsapplicationindeterminingthefieldaboutsome symmetrical charge arrangements. The use of Gauss’s law was invariably easier for these highly symmetrical distributions, because the problem of integration always disappeared when the proper closed surface was chosen.

However,ifwehadattemptedtofindaslightlymorecomplicatedfield,suchasthatoftwounlikepointchargesseparated by a small distance, we would have found it impossible to choose a suitable Gaussian surface and obtain an answer. Coulomb’s law, however, is more powerful and enables us to solve problems for which Gauss’s law is not applicable. The application of Coulomb’s law is lengthy, detailed, and often quite complex, the reason for thisbeingpreciselythefactthattheelectricfieldintensity,avectorfield,mustbefounddirectlyfromthechargedistribution. Three different integrations are needed in general, one for each component and the resolution of the vector into components usually adds to the complexity of the integrals.

Indeeditwouldbedesirableifwecouldfindsomeasyetundefinedscalarfunctionwithasingleintegrationandthendeterminetheelectricfieldfromthisscalarbysomesimplestraightforwardprocedure,suchasdifferentiation.Thisscalarfunctiondoesexistandisknownasthepotentialorpotentialfield.

2.2 Energy Expended in Moving a Point Charge in an Electric FieldTheelectricfieldintensityisdefinedastheforceonaunittestchargeatthatpointatwhichwewishtofindthevalueofthisvectorfield.Ifweattempttomovethetestchargeagainsttheelectricfield,wehavetoexertaforceequalandoppositetothatexertedbythefield,andthisrequiresustoexpendenergy,ordowork.

Ifwewishtomovethechargeinthedirectionofthefield,ourenergyexpenditureturnsouttobenegative;thefielddoes this work.

SupposewewanttomoveachargeQadistancedLinanelectricfieldE.TheforceonQduetotheelectricfieldis,

(1)

Wherethesubscriptremindsusthatthisforceisduetothefield.ThecomponentofthisforceinthedirectiondLmust overcome as ,

Where aL= a unit vector in the direction of dL.Theforcewhichmustbeappliedisequalandoppositetotheforceduetothefield,

and the expenditure of energy is the product of the force and distance. That is, differential work done by external source moving Q

or

(2)

where we have replaced aL dL by the simpler expression dL.

19/JNU OLE

This differential amount of work required may be zero under several conditions determined easily from equation(2). There are the trivial conditions for which E, Q, or dL is zero, and a much more important case in which E and dL areperpendicular.Herethechargeismovedalwaysinadirectionatrightanglestotheelectricfield.

Here,agoodanalogycanbedrawnbetweentheelectricfieldandthegravitationalfield,where,again,energymustbeexpendedtomoveagainstthefield.Slidingamassaroundwithconstantvelocityonafrictionlesssurfaceisaneffortless process if the mass is moved along a constant elevation contour; positive or negative work must be done in moving it to a higher or lower elevation, respectively.

Returningtothechargeintheelectricfield,theworkrequiredtomovethechargeafinitedistancemustbedeterminedby integrating,

(3)

Wherethepathmustbespecifiedbeforetheintegralcanbeevaluated.Thechargeisassumedtobeatrestatbothitsinitialandfinalpositions.Thisdefiniteintegralisbasictofieldtheory,andweshalldevotethefollowingsectionto its interpretation and evaluation.

2.3 The Line IntegralThe integral expression for the work done in moving a point charge Q from one position to another, equation (3), is an example of a line integral, which in vector analysis notation always takes the form of the integral along some prescribedpathofthedotproductofavectorfieldandadifferentialvectorpathlengthdL.Withoutusingvectoranalysis we should have to write

Where EL= component of E along dL.

A line integral is like many other integrals which appear in advanced analysis, including the surface integral appearing in Gauss’s law, in that it is essentially descriptive. It tells us to choose a path, break it up into a large number of very smallsegments,multiplythecomponentofthefieldalongeachsegmentbythelengthofthesegment,andthenaddthe results for all the segments. This is a summation, of course, and the integral is obtained exactly only when the numberofsegmentsbecomesinfinite.Thisprocedureisindicatedinfig.2.1,whereapathhasbeenchosenfromaninitialpositionBtoafinalpositionAandauniformelectricfieldselectedforsimplicity.Thepathisdividedintosixsegments,∆L1, ∆L2,........, ∆L6 and the components of E along each segment denoted by EL1, EL2,......, EL6 . The work involved in moving a charge Q from B to A is then approximately

or, using vector notation,

andsincewehaveassumedauniformfield,


20/JNU OLE

What is this sum of vector segments in the parentheses above? Vectors add by the parallelogram law, and the sum isjustthevectordirectedfromtheinitialpointBtothefinalpointA;LBA . Therefore,

(uniform E) (4)

Fig. 2.1 A graphical interpretation of line integral in a uniform field(Source: http://www.ie.itcr.ac.cr/acotoc/Maestria_en_Computacion/Sistemas_de_Comunicacion_II/Material/


Rememberingthesummationinterpretationofthelineintegral,thisresultfortheuniformfieldcanbeobtainedrapidly now from the integral expression

(5)

asappliedtoauniformfield.

(6)where the last integral becomes LBA and

(uniform E) (7)

Forthisspecialcaseofauniformelectricfieldintensity,notethattheworkinvolvedinmovingthechargedependsonly on Q, E, and LBA,avectordrawnfromtheinitialtothefinalpointofthepathchosen.Itdoesnotdependonthe particular path we have selected along which to carry the charge.

2.4 Definition of Potential Difference and Potential WearenowreadytodefineanewconceptfromtheexpressionfortheworkdonebyanexternalsourceinmovingachargeQfromonepointtoanotherinanelectricfieldE,

(8)

21/JNU OLE

Inmuchthesamewayaswedefinedtheelectricfieldintensityastheforceonaunittestcharge,wenowdefinepotential difference V as the work done (by an external source) in moving a unit positive charge from one point to anotherinanelectricfield,

(9)

We shall have to agree on the direction of movement, as implied by our language, and we do this by stating that VABsignifiesthepotentialdifferencebetweenpointsAandBandistheworkdoneinmovingtheunitchargefromB(lastnamed)toA(firstnamed).Thus,indeterminingVAB,BistheinitialpointandAisthefinalpoint.Thereasonforthissomewhatpeculiardefinitionwillbecomeclearershortly,whenitisseenthattheinitialpointBisoftentakenatinfinity,whereasthefinalpointArepresentsthefixedpositionofthecharge;pointAisthusinherentlymore important.

Potentialdifference ismeasured in joulesper coulomb, forwhich thevolt isdefinedas amorecommonunit,abbreviated as V. Hence the potential difference between points A and B is

(10)

and VAB is positive if work is done in carrying the positive charge from B to A.

Fig. 2.2 (a) Circular path (b) A radial path along which a charge of Q is carried in the field of an infinite line charge

(Source: http://www.ie.itcr.ac.cr/acotoc/Maestria_en_Computacion/Sistemas_de_Comunicacion_II/Material/Biblio2/chapt04.pdf)

Infig.2.2,workdoneintakingchargeQfromρ=b to ρ=a is given by

Thus, the potential difference between points at ρ=a and ρ=b is

(11)

WecantryoutthisdefinitionbyfindingthepotentialdifferencebetweenpointsAandBatradialdistancesrA and rB from a point charge Q. Taking origin as Q

and

We have (12)


22/JNU OLE

If rB > rA , the potential difference VAB is positive, indicating that energy is expended by the external source in bringing the positive charge from rB to rA .This agrees with the physical picture showing the two like charges repelling each other.

It is often convenient to speak of the potential, or absolute potential, of a point, rather than the potential difference between two points, but this means only that we agree to measure every potential difference with respect to a specifiedreferencepointwhichweconsidertohavezeropotential.Commonagreementmustbereachedonthezeroreferencebeforeastatementofthepotentialhasanysignificance.Apersonhavingonehandonthedeflectionplatesof a cathode-ray tube which are “at a potential of 50 V” and the other hand on the cathode terminal would probably be too shaken up to understand that the cathode is not the zero reference, but that all potentials in that circuit are customarily measured with respect to the metallic shield about the tube. The cathode may be several thousands of volts negative with respect to the shield.

Perhaps the most universal zero reference point in experimental or physical potential measurements is "ground,” by which we mean the potential of the surface region of the earth itself. Theoretically, we usually represent this surface byaninfiniteplaneatzeropotential,althoughsomelarge-scaleproblems,suchasthoseinvolvingpropagationacrossthe Atlantic Ocean, require a spherical surface at zero potential.

Anotherwidelyusedreference"point”isinfinity.Thisusuallyappearsintheoreticalproblemsapproximatingaphysical situation in which the earth is relatively far removed from the region in which we are interested, such as thestaticfieldnearthewingtipofanairplanethathasacquiredachargeinflyingthroughathunderhead,orthefieldinsideanatom.Workingwiththegravitationalpotentialfieldonearth,thezeroreferenceisnormallytakenatsealevel;foraninterplanetarymission,however,thezeroreferenceismoreconvenientlyselectedatinfinity.

Acylindricalsurfaceofsomedefiniteradiusmayoccasionallybeusedasazeroreferencewhencylindricalsymmetryispresentandinfinityprovesinconvenient.Inacoaxialcabletheouterconductorisselectedasthezeroreferenceforpotential. And, of course, there are numerous special problems, such as those for which a two-sheeted hyperboloid or an oblate spheroid must be selected as the zero-potential reference, but these need not concern us immediately.

If the potential at point A is VA and that at B is VB, then

(13) where we necessarily agree that VA and VB shall have the point.

2.5 The Potential Field of a Point ChargeIn the previous section we found an expression (12) for the potential difference between two points located at r=rA and r = rBinthefieldofapointchargeQplacedattheorigin,

(14)

It was assumed that the two points lay on the same radial line or had the same θ and φ coordinate values, allowing us to set up a simple path on this radial line along which to carry our positive charge. We now should ask whether different θ and φcoordinatevaluesfortheinitialandfinalpositionwillaffectouranswerandwhetherwecouldchoose more complicated paths between the two points without changing the results. Let us answer both questions atoncebychoosingtwogeneralpointsAandB(fig.1.3)atradialdistancesofrA and rB, and any values for the other coordinates.

The differential path length dL has r, θ, and φcomponents,andtheelectricfieldhasonlyaradialcomponent.Takingthe dot product then leaves us only

23/JNU OLE

Weobtainthesameanswerand,thereforethepotentialdifferencebetweentwopointsinthefieldofapointchargedepends only on the distance of each point from the charge and does not depend on the particular path used to carry our unit charge from one point to the other.

Fig. 2.3 A general path between general points B and A in the field of a point charge Q at the origin(Source: http://www.ie.itcr.ac.cr/acotoc/Maestria_en_Computacion/Sistemas_de_Comunicacion_II/Material/


ThesimplestpossibilityistoletV=0atinfinity.Ifweletthepointatr=rBrecedetoinfinitythepotentialatrA becomes

or, since there is no reason to identify this point with the A subscript,

(15)

ThisexpressiondefinesthepotentialatanypointdistantrfromapointchargeQattheorigin,thepotentialatinfiniteradius being taken as the zero reference. Returning to a physical interpretation, we may say that Q/4πε0r joules of workmustbedoneincarryinga1-CchargefrominfinitytoanypointrmetersfromthechargeQ.

AconvenientmethodtoexpressthepotentialwithoutselectingaspecificzeroreferenceentailsidentifyingrA as r once again and letting Q/4πε0rB be a constant. Then,

(16)

C1 may be selected so that V = 0 at any desired value of r. We could also select the zero reference indirectly by electing to let V be V0 at r = r0. It should be noted that the potential difference between two points is not a function of C1

Equation(15)or(16)representsthepotentialfieldofapointcharge.Thepotentialisascalarfieldanddoesnotinvolve any unit vectors.


24/JNU OLE

2.6 The Potential Field of a System of Charges: Conservative PropertyThepotentialatapointhasbeendefinedastheworkdoneinbringingaunitpositivechargefromthezeroreferenceto the point, and we have suspected that this work, and hence the potential, is independent of the path taken.

Toprovetheassertion,beginwith thepotentialfieldof thesinglepointchargeshowed, in the lastsection, theindependencewithregardtothepath,notingthatthefieldislinearwithrespecttochargesothatsuperpositionisapplicable. It will then follow that the potential of a system of charges has a value at any point which is independent of the path taken in carrying the test charge to that point.

Thus,thepotentialfieldofasinglepointcharge,whichweshallidentifyasQ1 and locate at r1, involves only the distance |r ‒ r1| j from Q1 to the point at r where we are establishing the value of the potential. For a zero reference atinfinity,wehave

The potential due to two charges, Q1 at r1 and Q2 at r2, is a function only of |r ‒ r1| and |r‒r2|, the distances from Q1 and Q2tothefieldpoint,respectively.

Continuingtoaddcharges,wefindthatthepotentialduetonpointchargesis

or (17)

If each point charge is now represented as a small element of a continuous volume charge distribution ρv ∆v, then

Asweallowthenumberofelementstobecomeinfinite,weobtaintheintegralexpression

(18)

Wehavecomequiteadistancefromthepotentialfieldofthesinglepointcharge,anditmightbehelpfultoexamine(18).ThepotentialV(r)isdeterminedwithrespecttoazeroreferencepotentialatinfinityandisanexactmeasureoftheworkdoneinbringingaunitchargefrominfinitytothefieldpointatrwherewearefindingthepotential.The volume charge density ρy(r′) and differential volume element dv′combine to represent a differential amount of charge ρv(r′)dv′ located at r′.

The distance |r‒r′| is thatdistance from thesourcepoint to thefieldpoint.The integral isamultiple (volume)integral.

25/JNU OLE

If the charge distribution takes the form of a line charge or a surface charge, the integration is along the line or over the surface:

(19)

(20)

The most general expression for potential is obtained by combining (17), (18), (19), and (20).

These integral expressions for potential in terms of the charge distribution should be compared with similar expressions fortheelectricfieldintensity,suchas(18)

Thepotentialagainisinversedistance,andtheelectricfieldintensity,inverse-squarelaw.

2.7 Potential GradientWenowhavetwomethodsofdeterminingpotential,onedirectlyfromtheelectricfieldintensitybymeansofalineintegral, and another from the basic charge distribution itself by a volume integral. Neither method is very helpful indeterminingthefieldsinmostpracticalproblems,however,forasweshallseelater,neithertheelectricfieldintensity nor the charge distribution is very often known. Preliminary information is much more apt to consist of a description of two equipotential surfaces, such as the statement that we have two parallel conductors of circular crosssectionatpotentialsof100andÀ100V.Perhapswewishtofindthecapacitancebetweentheconductors,orthe charge and current distribution on the conductors from which losses may be calculated. These quantities may beeasilyobtainedfromthepotentialfield,andourimmediategoalwillbeasimplemethodoffindingtheelectricfieldintensityfromthepotential.

We already have the general line-integral relationship between these quantities,

(21)

Butthisismucheasiertouseinthereversedirection:givenE,findV.

However, equation (21) may be applied to a very short element of length ∆L along which E is essentially constant, leading to an incremental potential difference ∆V,

(22)

LetusseefirstifwecandetermineanynewinformationabouttherelationofVtoEfromthisequation.Considerageneralregionofspace,asshowninfig.4.6,inwhichEandVbothchangeaswemovefrompointtopoint.Equation (22) tells us to choose an incremental vector element of length ∆L = ∆L aL and multiply its magnitude by the component of E in the direction of aL (one interpretation of the dot product) to obtain the small potential differencebetweenthefinalandinitialpointsofVL.

If we designate the angle between ∆L and E as θ, then


26/JNU OLE

We now wish to pass to the limit and consider the derivative dV/dL. To do this, we need to show that V may be interpreted as a function V(x, y, z). So far, V is merely the result of the line integral equation (21). If we assume aspecifiedstartingpointorzeroreferenceandthenletourendpointbe(x,y,z)weknowthattheresultoftheintegrationisauniquefunctionoftheendpoint(x,y,z)becauseEisaconservativefield.ThereforeVisasingle-valued function V(x, y, z). We may then pass to the limit and obtain

In which direction should ÁL be placed to obtain a maximum value of ∆V?RememberthatEisadefinitevalueatthe point at which we are working and is independent of the direction of ∆L. The magnitude ∆L is also constant, and our variable is aL, the unit vector showing the direction of ∆L. It is obvious that the maximum positive increment of potential, ∆Vmax, will occur when cosθ is ‒1, or ∆L points in the direction opposite to E. For this condition,

This little exercise shows us two characteristics of the relationship between E and V at any point:Themagnitudeoftheelectricfieldintensityisgivenbythemaximumvalueoftherateofchangeofpotential•with distance.This maximum value is obtained when the direction of the distance increment is opposite to E or, in other words, •the direction of E is opposite to the direction in which the potential is increasing the most rapidly.

Let us now illustrate these relationships in terms of potential. Fig. 1.4 is intended to show the information about somepotentialfield.

Fig. 2.4 A potential field is shown by its equipotential surfaces(Source: http://www.ie.itcr.ac.cr/acotoc/Maestria_en_Computacion/Sistemas_de_Comunicacion_II/Material/


It is done by showing the equipotential surfaces (shown as lines in the two-dimensional sketch). We desire informationabouttheelectricfieldintensityatpointP.StartingatP;welayoffasmallincrementaldistance∆L in various directions, hunting for that direction in which the potential is changing (increasing) the most rapidly. From the sketch, this direction appears to be left and slightly upward.

Fromthesecondcharacteristicabove,theelectricfieldintensityisthereforeoppositelydirected,ortotherightandslightly downward at P. Its magnitude is given by dividing the small increase in potential by the small element of length. It seems likely that the direction in which the potential is increasing most rapidly, is perpendicular to the equipotentials (in the direction of increasing potential), and this is correct, for if ∆L is directed along an equipotential, ∆V=0byourdefinitionofanequipotentialsurface.Butthen

27/JNU OLE

and since neither E nor ∆L is zero, E must be perpendicular to this ÁL or perpendicular to the equipotential.

Sincethepotentialfieldinformationismorelikelytobedeterminedfirst,letusdescribethedirectionof∆L which leadstoamaximumincreaseinpotentialmathematicallyintermsofthepotentialfieldratherthantheelectricfieldintensity. We do this by letting aN be a unit vector normal to the equipotential surface and directed toward the higher potentials.Theelectricfieldintensityisthenexpressedintermsofthepotential,

(23)

this shows that the magnitude of E is given by the maximum space rate of change of V and the direction of E is normal to the equipotential surface (in the direction of decreasing potential)

Since dV/dLmax occurs when ∆L is in the direction of aN , we may remind ourselves of this fact by letting

and

(24)

Equation(23)or(24)servetoprovideaphysicalinterpretationoftheprocessoffindingtheelectricfieldintensityfrom the potential. Both are descriptive of a general procedure, and we do not intend to use them directly to obtain quantitative information. This procedure leading from V to E is not unique to this pair of quantities, however, but hasappearedastherelationshipbetweenascalarandavectorfieldinhydraulics,thermodynamics,andmagnetics,andindeedinalmosteveryfieldtowhichvectoranalysishasbeenapplied.

The operation on V by which ‒Eisobtainedisknownasthegradient,andthegradientofascalarfieldTisdefinedas

(25)

where aN is a unit vector normal to the equipotential surfaces, and that normal is chosen which points in the direction of increasing values of T.

Using this new term, we now may write the relationship between V and E as

(26)


28/JNU OLE

SummaryCoulomb’s law is more powerful and enables us to solve problems for which Gauss’s law is not applicable.•The application of Coulomb’s law is lengthy, detailed, and often quite complex, the reason for this being precisely •thefactthattheelectricfieldintensity,avectorfield,mustbefounddirectlyfromthechargedistribution.Theelectricfieldintensityisdefinedastheforceonaunittestchargeatthatpointatwhichwewishtofindthe•valueofthisvectorfield.A line integral is like many other integrals which appear in advanced analysis, including the surface integral •appearing in Gauss’s law, in that it is essentially descriptive.Thepotentialatapointhasbeendefinedastheworkdoneinbringingaunitpositivechargefromthezero•reference to the point, and we have suspected that this work, and hence the potential, is independent of the path taken.Wehavetwomethodsofdeterminingpotential,onedirectlyfromtheelectricfieldintensitybymeansofaline•integral, and another from the basic charge distribution itself by a volume integral.

ReferencesEnergy and Potential• . Available at: <http://www.ie.itcr.ac.cr/acotoc/Maestria_en_Computacion/Sistemas_de_Comunicacion_II/Material/Biblio2/chapt04.pdf> [Accessed 13 February, 2011].Potential Energy• . Available at: <http://www.scar.utoronto.ca/~pat/fun/NEWT3D/PDF/ENERGY3D.PDF>. [Accessed 9 February 2011].

Recommended ReadingRyaben’kii, V.S., Kulman, N.K., (Translator), 2001. • Method of Difference Potentials and its Applications. Springer, 1st ed.Rao, N.N., 2004. Elements of Engineering Electromagnetics. Prentice Hall, 6th ed.•Ulaby, F.T., 2006. Fundamentals of Applied Electromagnetics. Prentice Hall, 5th ed.•

29/JNU OLE

Self Assessment

Coulomb’s law, is more powerful and enables us to solve problems for which _________ is not applicable.1. Divergence lawa. Faraday’s lawb. Stoke’s theoremc. Gauss’s lawd.

Themagnitudeoftheelectricfieldintensityisgivenbythe__________valueoftherateofchangeofpotential2. with distance.

minimuma. maximumb. fixedc. variable d.

Theelectricfield________wasdefinedastheforceonaunittestchargeatthatpointatwhichwewishtofind3. thevalueofthisvectorfield.

densitya. magnitudeb. intensityc. aread.

Potential difference is given by which of the following equations?4.

a.

b.

c.

d.

ThegradientofascalarfieldTisgivenbywhichequation?5.

a.

b.

c.

d.


30/JNU OLE

Which of the following are the two methods to determine potential?6. Directlyfromtheelectricfieldintensitybymeansofapointintegralandanotherfromthebasicchargea. distribution itself by a point integral.Directly from theelectricfield intensitybymeansofa line integralandanother from thebasicchargeb. distribution itself by a volume integral.Onedirectlyfromtheelectricfieldintensitybymeansofavolumeintegralandanotherfromthebasicchargec. distribution itself by a line integral.Onedirectlyfromtheelectricfieldintensitybymeansofalineintegralandanotherfromthebasiccharged. distribution itself by a line integral.

Which of the following is true?7. Thepotentialatapointhasbeendefinedasthedisplacementinbringingaunitpositivechargefromthea. zero reference to the point.Thepotentialatapointhasbeendefinedastheworkdoneinbringingaunitnegativechargefromthezerob. reference to the point.Thepotentialatapointhasbeendefinedastheworkdoneinbringingaunitpositivechargefromthezeroc. reference to the point.Thepotentialatapointhasbeendefinedastheworkdoneinbringingaunitpositivechargefromthefinald. point to the initial point.

Which of the following is true?8. Potential difference is measured in ohms per Coulomb.a. Potential difference is measured in joules per Henry.b. Potential difference is measured in volts per Coulomb.c. Potential difference is measured in joules per Coulomb.d.

Which of the following is true?9. The application of Coulomb’s law is lengthy, detailed and often quite complex.a. The application of Faraday’s law is lengthy, detailed and often quite complex.b. The application of Gauss’s law is lengthy, detailed and often quite complex.c. The application of Coulomb’s law is not lengthy, detailed and often quite easy.d.

Which of the following is true?10. Thepotentialfieldintensityisdefinedastheforceonaunittestchargeatthatpointatwhichwewishtoa. findthevalueofthisvectorfield.Theelectricfieldintensityisdefinedastheforceonaunittestchargeatthatpointatwhichwewishtofindb. thevalueofthisvectorfield.Theelectricfieldintensityisdefinedastheenergyonaunittestchargeatthatpointatwhichwewishtoc. findthevalueofthisvectorfield.Theelectricfieldintensityisdefinedastheforceonaunittestchargeatthatpointatwhichwewishtofindd. thevalueofthisvariablefield.

31/JNU OLE

Chapter III

Poisson’s and Laplace’s Equations

AimThe aim of this chapter is to:

explain Laplace’s equation•

illustrate Poisson’s equation•

state• uniqueness theorem

Objectives


derive Poisson’s and Laplace’s equations•

examine uniqueness theorem•

give• examples on Poisson’s and Laplace’s equations

Learning outcome


understand Poisson’s and Laplace’s equations•

determine • uniqueness theorem

identify • applications of these equations


32/JNU OLE

3.1 IntroductionIn earlier chapters, the and in the given region are obtained using Coulomb’s law and Gauss’s law. Using these laws is easy, if the charge distribution or potential throughout the region is known. Practically it is not possible in many situations, to know the charge distribution or potential variation throughout the region. Practically, charge and potential may be known at same boundaries of the region, only. From those values it is necessary to obtain potential and throughout the region. Such electrostatic problems are called boundary value problems. To solve such problems, Poisson’s and Laplace’s equations must be known. This chapter derives the Poisson’s and Laplace’s equations and explains its use in few practical situations.

3.2 Poisson’s and Laplace’s EquationsFrom the Gauss’s law in the point form, Poisson’s equation can be derived. Consider the Gauss’s law in the point form is given as,

(1)

Where,= Flux density

= Volume charge densityItisknownthatforahomogeneous,isotropicandlinearmedium,fluxdensityandelectricfieldintensityaredirectlyproportional. Thus

(2)

Therefore,

(3)

From the gradient relationship

(4)

Substituting (4) in (3),

(5)

Taking e− outside as constant,

(6)

Now operationiscalled‘delsquared’operationanddenotedas 2∇ .

Therefore, (7)

This equation (7) is called Poisson’s equation

If in a certain region, volume charge density is zero ( 0vr = ), which is true for dielectric medium then the Poisson’s equation takes the form,

33/JNU OLE

(For a charge free region) (8)

This is special case of Poisson’s equation and is called Laplace’s equation. The 2∇ operation is called the Laplacian

of V

The equation (7) is for homogeneous medium for which e is constant. But if e is not constant and the medium is inhomogeneous, the equation (5) must be used as Poisson’s equation for inhomogeneous medium.

3.2.1 2∇ Operation in Different Co-ordinate Systems

The potential V can be expressed in any of the three co-ordinate systems as V(x, y, z), V(r,θ,φ) or V(r,φ, z). Depending upon it, the 2∇ operation required for Laplace’s equation must be used.

In Cartesian co-ordinate systems,

(9)

The equation (9) is Laplace’s equation in Cartesian form

In cylindrical co-ordinate system

(10)

The equation (10) is Laplace’s equation in cylindrical form.

In spherical co-ordinate system,

(11)

The equation (11) is Laplace’s equation in spherical form.

3.3 Uniqueness TheoremThe boundary value problems can be solved by number of methods such as analytical, graphical, experimental etc. Thus there is a question that is the solution of Laplace’s equation solved by any method, unique? The answer to this question is the uniqueness theorem, which is proved by contradiction method.

Assume that the Laplace’s equation has two solutions say V1 and V2, both are function of the co-ordinates of the system used. These solutions must satisfy Laplace’s equation. So we can write,

∇2 V1=0 and ∇2 V2=0 (1)


34/JNU OLE

Both the solutions must satisfy the boundary conditions as well. At the boundary, the potentials at the different points are same due to equipotential surface then,

V1=V2 (2)

Let the difference between the two solutions is Vd.

Vd=V2‒V1 (3)

Using Laplace’s equation for the difference Vd,

(4)

Therefore,

(5)

On the boundary Vd=0 from the equations (2) and (3)

Now the divergence theorem states that,

vol s

Adv Ad S∇• =∫ ∫ (6)

Let A =Vd ∇ Vd and from vector algebra,

Now use this for ∇⋅(Vd ∇ Vd) with dVa = and ∇ Vd= .

But∇⋅∇=∇2

Hence,

(7)

Using (4)

(8)

To use this in (6), let hence

d d d dvol S

V V V V d S∇⋅ ∇ = ∇ ⋅∫ ∫ (9)

But Vd=0 on the boundary, hence right hand side of equation (9) is zero

35/JNU OLE

0d dvol

V V dv∇ ⋅∇ =∫ (10)

This is volume integral to be evaluated on the volume enclosed by the boundary.

It is known that, ,

20d

vol

V dv∇ =∫ as ∇ Vd is vector (11)

Now integration can be zero under two conditions,The quantity under integral sign is zero•The quantity is positive in some regions and negative in other regions by equal amount and hence zero•

But square term cannot be negative in any region hence; quantity under integral must be zero.

i.e.,

(12)

As the gradient of Vd=V2-V1 is zero means V2-V1 is constant and not changing with any co-ordinates. But considering boundary it can be proved that V2-V1 =constant=zero.

V2=V1 (13)

This proves that both the solutions are equal and cannot be different. Thus, Uniqueness Theorem can be stated as –“IfthesolutionofLaplace’sequationsatisfiestheboundaryconditionthenthissolutionisunique,bywhatevermethod it isobtained.”ThesolutionofLaplace’sequationgives thefieldwhich isunique,satisfying thesameboundary conditions, in a given region.

3.4 Procedure for Solving Laplace’s EquationThe procedure to solve a problem involving Laplace’s equation can be described as follows,Step 1: Solve the Laplace’s equation using the method of integration. Assume constants of integration as per the requirement.Step 2: Determine the constants applying the boundary conditions given or known for the region. The solution obtained in step 1 with constants obtained using boundary conditions is a unique solution.Step 3: Then canbeobtainedforthepotentialfieldVobtained,usinggradientoperation‒∇ V.Step 4: For homogeneous medium, can be obtained as .Step 5: At the surface, ρs=DN hence once is known, the normal component DN to the surface is known. Hence the charge induced on the conductor surface can be obtained as .Step 6: One the charge induced Q is known and potential V is known then the capacitance C of the system can be obtained.Let us have a look on few examples

Example 3.1: VerifythatthepotentialfieldgivenbelowsatisfiestheLaplace’sequation.

Solution:GivenfieldisinCartesiansystem,


36/JNU OLE

=

= 4‒6+2=0

As =0,thefieldsatisfiestheLaplace’sequation.

Example 3.2: The region between two concentric right cylinders contains a uniform charge densityρ. Solve the Poisson’s equation for the potential in the region.

Solution:Thecylindersareshowninthefig.3.1Selectthecylindricalco-ordinatesystem.Inco-axialcablelikestructure,theelectricfieldintensity is in radial direction from inner to outer cylinder.

Fig. 3.1 Concentric right cylinders

Hence and V both are function of only r and not of φ and z.

Therefore, is existing while and are zero.

According to Poisson’s equation,

, here ρv=ρ given

Integrating both sides

Where C1= Constant of integration

37/JNU OLE

Integrating both sides,

Where C2=Constant of integration

Knowing the boundary conditions, C1 and C2 can be obtained.


38/JNU OLE

Summary

Gauss’s law in the point form is given as • .For a homogeneous, isotropic and linearmedium, flux density and electric field intensity are directly•proportional.

Poisson’s equation is given by • .

Laplace’s equation in Cartesian form is given by • .

Laplace’s equation in cylindrical form is given by • .

Laplace’s equation in spherical form is given by•

.UniquenessTheoremcanbestatedas,”IfthesolutionofLaplace’sequationsatisfiestheboundarycondition•then this solution is unique, by whatever method it is obtained”.

ReferencesLaplace’s and Poisson’s Equations• . Available at: <http://www.ifm.liu.se/courses/TFYY67/Lect4.pdf> [Accessed 14 February 2010].Bakshi, U.A., Bakshi, A.V., 2009. • Electromagnetic Field Theory, Technical Publications.

Recommended ReadingKraus, J.D., Fleisch, D., 1999. • Electromagnetics, McGraw Hill Higher Education, 5th ed.Edminister, J., 2010. • Schaum’s Outline of Electromagnetics, McGraw-Hill. 3rd ed.Ida, N., 2004. • Engineering Electromagnetics. Springer, 2nd ed.

39/JNU OLE

Self Assessment

IfthesolutionofLaplace’sequationsatisfiestheboundaryconditionthenthissolutionis________,bywhatever1. method it is obtained.

uniquea. oneb. similarc. differentd.

Forahomogeneous,isotropicandlinearmedium,fluxdensityandelectricfieldintensityare____________2. proportional.

directlya. indirectlyb. inverselyc. laterallyd.

Gauss’s law in the point form is given by which of the following?3.

a.

b.

c.

d.

Poisson’s equation is given by which of the following?4.

Laplace’s equation in Cartesian form is given by which of the following?5.

a.

b.

c.

d.


40/JNU OLE

Laplace’s equation in cylindrical form is given by which of the following?6.

a.

b.

c.

d.

Coulomb’s law and Gauss’s law can be used easily, if the charge distribution or potential throughout the region 7. are _________.

similara. differentb. knownc. unknownd.

State which of the following statements is true.8. The a. 2∇ operation is called the Del operator.The b. ∇ operation is called the Laplacian of V.The c. 2∇ operation is called the operator of V.The d. 2∇ operation is called the Laplacian of V.

Which of the following is true?9. IfthesolutionofLaplace’sequationsatisfiestheboundaryconditionthenthissolutionisunique,bywhatevera. method it is obtained.If the solution of Laplace’s equation does not satisfy the boundary condition then this solution is unique, b. by whatever method it is obtained.IfthesolutionofLaplace’sequationsatisfiestheboundaryconditionthenthissolutionisfixed,bywhateverc. method it is obtained.IfthesolutionofPoisson’sequationsatisfiestheboundaryconditionthenthissolutionisunique,bywhateverd. method it is obtained.

Which of the following is true?10. Forahomogeneous, isotropicand linearmedium,fluxdensityandelectricfield intensityare indirectlya. proportional.For a homogeneous, isotropic and linearmedium,fluxdensity and electricfield intensity are directlyb. proportional.Forahomogeneous, isotropicand linearmedium,fluxdensityandelectricfield intensityare inverselyc. proportional.For a homogeneous, isotropic and linear medium, flux density and electric field intensity are not d. proportional.

41/JNU OLE

Chapter IV

Magnetic Forces, Materials and Inductance

Aim


analyse magnetic force on a moving charge•

explore the concept of magnetisation and permeability•

explain the role of• potential energy and forces on magnetic materials

Objectives


examine force between differential current element•

evaluate force and torque on a closed circuit•

give an overview of• inductance and mutual inductance

Learning outcome

At the end of this chapter students will be able to:

recall magnetisation and permeability•

comprehend the concept of • force and torque on a closed circuit

understand• inductance and mutual inductance


42/JNU OLE

4.1 Introduction to Magnetic FluxIn this chapter, we shall study the magnetic forces. We will discuss the concepts of magnetic torque, concepts of magnetisationalongwithpermeability.Similartotheboundaryconditionsinelectrostaticfields,weshallstudytheboundaryconditionsforthemagnetostaticfields.Weshallalsodiscussdifferentmagneticmaterials.

4.2 Magnetic Flux and Magnetic Flux DensityMagneticfluxis theamountofmagneticfield(ornumberoflinesofforce)producedbymagneticsource.ThesymbolformagneticfluxisΦ(Greekletter‘phi’).TheunitofmagneticfluxisWeber,Wb.Magneticfluxdensityistheamountoffluxpassingthroughadefinedareathatisperpendiculartothedirectionoftheflux.

ThesymbolformagneticfluxdensityisB.Theunitofmagneticfluxdensityisthetesla,T.Where 1T=1Wb/m2

Hence,

Where A(m2) is the area.

ExampleAmagneticpolefacehasarectangularsectionhavingdimension200mmby100mm.Ifthetotalfluxemergingfromthepoleis150µwb.Calculatethefluxdensity.Solution:Flux Φ=150µwb= 150×10-6WbCross-sectional area A=200×100= 20,000mm2=20000×10-6m2

Flux density=

=0.0075T or 7.5mT

4.3 Magnetic Force on a Moving ChargeElectricchargesmovinginamagneticfieldexperienceaforceduetothemagneticfield.GivenachargeQmovingwithvelocityuinamagneticfluxdensityB,thevectormagneticforceFm on the charge is given by

Notethattheforceisnormaltotheplanecontainingthevelocityvectorandthemagneticfluxdensityvector.Alsonote that the force is zero if the charge is stationary (u=0).

ExampleDetermine the vector magnetic force on a point charge +Q moving at a uniform velocity u=uoay in a uniform magnetic fluxdensityB=Boaz.Solution:

Givenachargemovinginanelectricfieldandamagneticfield,thetotalforceonthechargeisthesuperpositionoftheforcesduetotheelectricfieldandthemagneticfield.ThistotalforceequationisknownastheLorentzforceequation. The vector force component due to the electric

43/JNU OLE

Fig. 4.1 Direction of the charge in an electric and magnetic field

field(Fe) is given by

The total vector force (Lorentz force- F) is

The Lorentz force can also be written in terms of Newton’s law such that

where, m is the mass of the charged particle.

4.4 Magnetic Force on CurrentGiventhatchargemovinginamagneticfieldexperiencesaforce,acurrentcarryingconductorinamagneticfieldalso experiences a force. The current carrying conductor (line, surface or volume current) can be subdivided into current elements (differential lengths, differential surfaces or differential volumes). The charge-velocity product for a moving point charge can be related to an equivalent differential length of line current.

Fig. 4.2 Magnetic force on current(Source: http://www.ece.msstate.edu/~donohoe/ece3313notes8.pdf)

The equivalence of the moving point charge and the differential length of line current yield the equivalent magnetic force equation

z

Fm

B

u

x

y

+


44/JNU OLE

The equivalence of the moving point charge and the differential length of line current yield the equivalent magnetic force equation

(A-m)

(A-m)

Such that

)

The overall force on a line, surface and volume current is found by integrating over the current distribution

(Line current)

(Surface current)

(Volume current)

Example (Force between line currents)Determine the force/unit length on a line current I1duetothemagneticfluxproducedbyaparallellinecurrentI2 (separationdistance=d)flowingintheoppositedirection.

Fig. 4.3 Example of force between line currents

Themagneticfluxproducedonthez-axis(I1) due to the current I2 is.

The force on a length l of current I1duetothefluxproducedbyI2 is

z

d

x

y

µI1 I2

I

(x 0,y 0) (x 0,y d)

45/JNU OLE

The force per unit length on the current I1 is

The force on a length l of current I2duetothefluxproducedbyI1 is

The force per unit length on the current I2 is

Notethatthecurrentsrepeleachothergiventhecurrentsflowinginoppositedirections.Ifthecurrentsflowinthesame direction, they attract each another

4.5 Torque on a Current LoopGiven the change in current directions around a closed current loop, the magnetic forces on different portions of the loop vary in direction. Using the Lorentz force equation, we can show that the net force on a simple circular or rectangularloopisatorquewhichforcesthelooptoalignitsmagneticmomentwiththeappliedmagneticfield.

Consider the rectangular current loop shown below. The loop lies in the x-y plane and carries a DC current I. The loopliesinauniformmagneticfluxdensityBgivenby

The loop consists of four current segments carrying distinct vector current componentsI1=Iay I2=-Iax I3=-Iay I4=Iax.


46/JNU OLE

GivenauniformfluxdensityandaDCcurrentalongstraightcurrentsegments,themagneticforceoneachconductorsegmentcanbesimplifiedtothefollowingequation

Fig. 4.4 x-y plane(Source: http://www.ece.msstate.edu/~donohoe/ece3313notes8.pdf)

Theforcesonthecurrentsegmentscanbedeterminedforeachcomponentofthemagneticfluxdensity.

Forces due to Bz

Fig. 4.5 Forces due to Bz(Source: http://www.ece.msstate.edu/~donohoe/ece3313notes8.pdf)

Forces due to By

47/JNU OLE

Fig. 4.6 Forces due to By(Source: http://www.ece.msstate.edu/~donohoe/ece3313notes8.pdf)

Thevectortorqueontheloopisdefinedintermsoftheforcemagnitude(IByl2), the torque moment arm distance (l1/2),andthetorquedirection(definedbytherighthandrule):

Where A=l1l2isthelooparea.Thevectortorquecanbewrittencompactlybydefiningthemagneticmoment(m)of the loopm=IA (magnetic moment magnitude)m=IAan (vector magnetic moment)

whereanistheunitnormaltotheloop(definedbytherighthandruleasappliedtothecurrentdirection)

Fig. 4.7 Loop area(Source: http://www.ece.msstate.edu/~donohoe/ece3313notes8.pdf)

The magnitude of the torque in terms of the magnetic moment is


48/JNU OLE

The vector torque is then

Note that the torque on the loop tends to align the loop magnetic moment with the direction of the applied magnetic field.

4.6 MagnetisationJustasdielectricmaterialsarepolarisedundertheinfluenceofanappliedelectricfield,certainmaterialscanbemagnetisedundertheinfluenceofanappliedmagneticfield.Magnetisationformagneticfieldsisthedualprocesstopolarisationforelectricfields.Themagnetisationprocessmaybedefinedusingthemagneticmomentsoftheelectron orbits within the atoms of the material. Each orbiting electron can be viewed as a small current loop with an associated magnetic moment.

Fig. 4.8 Loop area(Source: http://www.ece.msstate.edu/~donohoe/ece3313notes8.pdf0

An unmagnetised material can be characterized by a random distribution of the magnetic moments associated with theelectronorbits.Theserandomlyorientedmagneticmomentsproducemagneticfieldcomponentsthattendtocanceloneanother(netH=0).Undertheinfluenceofanappliedmagneticfield,manyofthecurrentloopsaligntheirmagneticmomentsinthedirectionoftheappliedmagneticfield.

Fig. 4.9 Magnetic moments(Source: http://www.ece.msstate.edu/~donohoe/ece3313notes8.pdf)

Ifmostofthemagneticmomentsstayalignedaftertheappliedmagneticfieldisremoved,apermanentmagnetisformed.

49/JNU OLE

The bar magnet can be viewed as the magnetic analogy to the electric dipole. The poles of the bar magnet can be represented as equivalent magnetic charges separated by a distance l (the length of the magnet).

Fig. 4.10 Magnetic flux density(Source: http://www.ece.msstate.edu/~donohoe/ece3313notes8.pdf)

Themagneticfluxdensityproducedbythemagneticdipoleisequivalenttotheelectricfieldproducedbytheelectricdipole

The preceding equations assume the dipole is centred at the coordinate origin and oriented with its dipole moment along the z-axis. A current loop and a solenoid produce the same B as the bar magnet at large distances (in the far field)ifthemagneticmomentsofthesethreedevicesareequivalent.


50/JNU OLE

Fig. 4.11 Magnetic moments(Source: http://www.ece.msstate.edu/~donohoe/ece3313notes8.pdf)

If the magnetic moments of these three devices are equal

theyproduceessentiallythesamemagneticfieldatlargedistances(equivalentsources).Assumingthesamevectormagnetic moment, all three of these devices would experience the same torque when placed in a given magnetic field

The parameters associated with the magnetisation process are duals to those of the polarisation process. The magnetisation vector M is the dual of the polarisation vector P.

Let us see the difference between magnetisation and polarisation.

Magnetisation Polarisation

Table 4.1 Difference between magnetisation and polarisation

51/JNU OLE

NotethatthemagneticsusceptibilityPmisdefinedsomewhatdifferentlythantheelectricsusceptibilityPe.However,just as the electric susceptibility and relative permittivity are a measure of how much polarisation occurs in the material, the magnetic susceptibility and relative permeability are a measure of how much magnetisation occurs in the material.

4.7 Magnetic MaterialsMagneticmaterialscanbeclassifiedbasedonthemagnitudeoftherelativepermeability.Materialswitharelativepermeabilityofjustunderone(asmallnegativemagneticsusceptibility)aredefinedasdiamagnetic.Indiamagneticmaterials, the magnetic moments due to electron orbits and electron spin are very nearly equal and opposite such thattheycanceleachother.Thus,indiamagneticmaterials,theresponsetoanappliedmagneticfieldisaslightmagneticfieldintheoppositedirection.Superconductorsexhibitperfectdiamagnetism(Xm=‒1) at temperatures nearabsolutezerosuchthatmagneticfieldscannotexistinsidethesematerials.

Materialswith a relative permeability of just greater than one are defined as paramagnetic. In paramagneticmaterials, the magnetic moments due to electron orbit and spin are unequal, resulting in a small positive magnetic susceptibility.Magnetisation is not significant in paramagneticmaterials.Both diamagnetic andparamagneticmaterials are typically linear media.

Materialswitharelativepermeabilitymuchgreaterthanonearedefinedasferromagnetic.Ferromagneticmaterialsare always nonlinear. As such, these materials cannot be described by a single value of relative permeability. If a single number is given for the relative permeability of

any ferromagnetic material, this number represents an average value of µr

Ferromagnetic materials lose their ferromagnetic properties at very high temperatures (above a temperature known as the Curie temperature).

The characteristics of ferromagnetic materials are typically presented using the B-H curve, a plot of the magnetic fluxdensityBinthematerialduetoagivenappliedmagneticfieldH.

The B-H curve shows the initial magnetisation curve along with a curve known as a hysteresis loop. The initial magnetisationcurveshowsthemagneticfluxdensitythatwouldresultwhenanincreasingmagneticfieldisappliedtoaninitiallyunmagnetizedmaterial.AnunmagnetizedmaterialisdefinedbytheB=H=0pointontheB-Hcurve(nonetmagneticfluxgivennoappliedfield).Asthemagneticfieldincreases,atsomepoint,allofthemagneticmoments(currentloops)withinthematerialalignthemselveswiththeappliedfieldandthemagneticfluxdensitysaturates (Bm).Ifthemagneticfieldisthencycledbetweenthesaturationmagneticfieldvalueintheforwardandreverse directions (±Hm),thehysteresisloopresults.Theresponseofthematerialtoanyappliedfielddependsonthe initial state of the material magnetisation at that instant.


52/JNU OLE

Fig. 4.12 B-H curve(Source: http://www.ece.msstate.edu/~donohoe/ece3313notes8.pdf)

4.8 Magnetic Boundary ConditionsThefundamentalboundaryconditionsinvolvingmagneticfieldsrelatethetangentialcomponentsofmagneticfieldandthenormalcomponentsofmagneticfluxdensityoneithersideofthemediainterface.Thesametechniquesusedtodeterminetheelectricfieldboundaryconditionscanbeusedtodeterminethemagneticfieldboundaryconditions

Tangential magnetic field• Inordertodeterminetheboundaryconditiononthetangentialmagneticfieldatamediainterface,Ampere’slawisevaluated around a closed incremental path that extends into both regions as shown below. According to Ampere’s law,thelineintegralofthemagneticfieldaroundtheclosedloopyeildsthecurrentenclosed.

Fig. 4.13 Tangential magnetic field(Source: http://www.ece.msstate.edu/~donohoe/ece3313notes8.pdf)

If we take the limit of this integral as ∆y = 0, the integral contributions on the vertical paths goes to zero.

(µ1,Є1,σ1)

(µ2,Є2,σ2)

Δy

Δx

53/JNU OLE

Assuming themagneticfieldandsurfacecurrentcomponentsalong the incremental length )xareuniform, theintegrals above reduce to

Dividing this result by ∆x gives

Forthisexample,ifthesurfacecurrentflowsintheoppositedirection,weobtain

Or

Thus,thegeneralboundaryconditiononthetangentialmagneticfieldis

Thisisthetangentialcomponentsofmagneticfieldarediscontinuousacrossamediainterfacebyanamountequalto the surface current density on the interface.

Note that thepreviousboundaryconditionrelatesonlyscalarquantities.Thevectormagneticfieldandsurfacecurrentatthemediainterfacecanbeshowntosatisfyavectorboundaryconditiondefinedby

Theaboveequationindicatevectorboundaryconditionrelatingthemagneticfieldandsurfacecurrentatamediainterface.

Where n is a unit normal to the interface pointing into region 1


54/JNU OLE

Fig. 4.14 Tangential magnetic field(Source: http://www.ece.msstate.edu/~donohoe/ece3313notes8.pdf)

Givenasurfacecurrentontheinterface,thetangentialmagneticfieldcomponentsoneithersideoftheinterfacepoint in opposite directions.

Normal magnetic flux density• Inordertodeterminetheboundaryconditiononthenormalmagneticfluxdensityatamediainterface,weapplyGauss’slawformagneticfieldstoanincrementalvolumethatextendsintobothregionsasshownbelow

Fig. 4.15 Normal magnetic flux density(Source: http://www.ece.msstate.edu/~donohoe/ece3313notes8.pdf)

TheapplicationofGauss’slawformagneticfieldstotheclosedsurfaceabovegives

If we take the limit as the height of the volume )z approaches 0, the integral contributions on the four sides of the volume vanishes.

(µ1,Є1,σ1)

(µ2,Є2,σ2)

55/JNU OLE

The integrals over the upper and lower surfaces on either side of the interface reduce to

wherethemagneticfluxdensityisassumedtobeconstantovertheupperandlowerincrementalsurfaces.Evaluationof the surface integrals yields

Dividing by ∆x ∆y gives

suchthatthegeneralboundaryconditiononthenormalcomponentofmagneticfluxdensitybecomes

Thisisthenormalcomponentsofmagneticfluxdensityarecontinuousacrossamediainterface.

4.9 Inductors and InductanceAninductorisanenergystoragedevicethatstoresenergyinamagneticfield.Aninductortypicallyconsistsofsomeconfigurationofconductorcoils(anefficientwayofconcentratingthemagneticfield).Yet,evenstraightconductorscontaininductance.Theparametersthatdefineinductorsandinductancecanbedefinedasparallelquantitiestothose of capacitors and capacitance

Inductors CapacitorsStoresenergyinmagneticfield Storesenergyinanelectricfield

Inductance

L= Inductance (H)λ= Flux linkage (Wb)I=current

Capacitance

C=Capacitance(F)Q=Charge (C)V=Voltage (V)

Table 4.2 Difference between inductors and capacitors


56/JNU OLE

Thefluxlinkageofaninductordefinesthetotalmagneticfluxthatlinksthecurrent.Ifthemagneticfluxproducedbyagivencurrentlinksthatsamecurrent,theresultinginductanceisdefinedasaselfinductance.Ifthemagneticfluxproducedbyagivencurrentlinksthecurrentinanothercircuit,theresultinginductanceisdefinedasamutualinductance.

4.10 Mutual Inductance CalculationsThe mutual inductance between two distinct circuits can be determined by assuming a current in one circuit and determiningthefluxlinkagetotheoppositecircuit.

Example (Mutual inductance between coaxial loops)Determine the mutual inductance between two coaxial loops as shown below. Assume that the loop separation distance h is large relative to the radii of both loops (a, b). Also assume that loop (2) is much smaller than loop (1) (b<<a).

Fig. 4.16 Example mutual inductance between coaxial loops

Ifweassumethatacurrentflowsinloop(1),themagneticfluxdensityproducedbyloop(1)alongitsaxisis

Giventhat loop(2) ismuchsmaller thanloop(1)wemayassumethat thefluxproducedbyloop(1) isnearlyuniformovertheareaofloop(2)andisapproximatelyequaltothatattheloopcenter.Thefluxproducedbyloop(1)canalsobeassumedtobeapproximatelynormaltoloop(2)overitsarea.Thus,thetotalfluxproducedbyloop(1) linking loop (2) is approximately.

Fromthedefinitionofinductance,themutualinductancebetweentheloopsis

Note that the calculations required by assuming the current in loop (1) are much easier than those required by assuming the current in loop (2). The calculations are easier based on the approximations employed. If the current isassumedinloop(2),theresultingmagneticfluxdensityoverloop(1)isarathercomplicatedfunctionofposition

57/JNU OLE

which requires a complex integration to arrive at the same approximate

4.11 Internal And External InductanceIngeneral,acurrentcarryingconductorhasmagneticfluxinternalandexternaltotheconductor.Thus,themagneticfluxinsidetheconductorcanlinkportionsoftheconductorcurrentwhichproducesacomponentofinductancedesignated as internal inductance.Themagneticfluxoutside the conductor that links the conductor current isdesignated as external inductance

Themost efficient technique in determining the internal and external components of inductance is the energymethod. The energy method for determining inductance is based on the total magnetic energy expression for an inductor given by

Solving this equation for the inductance yields

The internal and external components of the inductance are found by integrating the internal and external magnetic fluxdensityexpressions,respectively.

4.12 Magnetic Forces on Magnetic MaterialsMagneticmaterialsexperienceaforcewhenplacedinanappliedmagneticfield.Thistypeofforceisseenwhenabarmagnetisplacedinamagneticfield,suchasthatofanotherbarmagnet.Thelikepolesrepeleachotherwhileunlike poles attract each other. Note that the bar magnets behave in the same way as current loops in an applied magneticfieldaseachmagnettendstoalignitsmagneticmomentinthedirectionoftheappliedfieldoftheoppositemagnet.

Fig.4.17 Magnetic forces on magnetic materials(Source: http://www.ece.msstate.edu/~donohoe/ece3313notes8.pdf)

The magnetised needle on a compass obeys the same principle as the compass needle aligns itself with the Earth’s magneticfield.ThemagneticfieldontheEarth’ssurfacehasahorizontalcomponentthatpointstowardthemagneticSouthPole(nearthegeographicNorthPole).ThehorizontalcomponentofmagneticfieldontheEarth’ssurfaceis


58/JNU OLE

maximum near the equator (approximately 35µT) and falls to zero at the magnetic poles.

Fig. 4.18 Magnetic field(Source: http://www.ece.msstate.edu/~donohoe/ece3313notes8.pdf)

An electromagnet can be used to lift ferromagnetic materials (such as scrap iron). The lifting force F of the electromagnet can be determined by considering how much energy is stored in the air gaps when the ferromagnetic materials are separated. The magnitude of the force necessary to separate the two pieces of ferromagnetic material equals the total amount of magnetic energy stored in the air gaps after separation. The total energy required to move the lower magnetic piece a distance l from the electromagnet is the product of the force magnitude F and the distance l. The total magnetic energy in the two air gaps is found by integrating the magnetic energy densities in the airgaps.Assumingshortairgaps,themagneticfieldintheairgapscanbeassumedtobeuniformandconfinedtothevolumebelowtheelectromagnetpoles(thefringingeffectofthemagneticfieldisnegligibleforsmallairgaps).Theuniformmagneticfieldintheairgapproducesauniformenergydensitysothatthetotalmagneticenergystoredin each air gap is a simply the product of the energy density and the volume of the air gap.

Solving this equation for the force magnitude gives

Theairgapmagneticfieldcanbedeterminedaccordingtotheboundaryconditionforthenormalcomponentofmagneticfluxacrosstheairgap.

Themagneticfluxdensityisnormaltotheinterfacesbetweentheairandtheferromagneticcore.Accordingtotheboundaryconditiononthenormalcomponentofmagneticfluxdensity,themagneticfluxdensitymustbecontinuousacross the air gap

Rewritingthisboundaryconditionintermsofthemagneticfieldsgives

Or

59/JNU OLE

Giveneitherthemagneticfieldorthemagneticfluxdensityinthecore,thecorrespondingquantityintheairgapcanbefoundaccordingtothepreviousequations.Notethatthemagneticfieldintheairgapismuchlargerthanthat in the core given the large relative permeability of the ferromagnetic core.

4.13 Magnetic CircuitsMagneticfieldproblemsinvolvingcomponentssuchascurrentcoils,ferromagneticcoresandairgapscanbesolvedas magnetic circuits according to the analogous behaviour of the magnetic quantities to the corresponding electric quantities in an electric circuit.

Fig. 4.19 Electric and magnetic circuits(Source: http://www.ece.msstate.edu/~donohoe/ece3313notes8.pdf)

Electric circuit Magnetic circuit

V=IRV=Electromotive Force

(V)[emf]I=total current (A)R=resistance(Ω)

F=magnetomotive force(A-turns)[mmf]

=totalmagneticflux(Wb)R=reluctance(H-1)

G=conductance (S) ℘=permeance (H)

Table 4.3 Difference between electric and magnetic circuits

Given that reluctance in a magnetic circuit is analogous to resistance in an electric circuit, and permeability in a magnetic circuit is analogous to conductivity in an electric circuit, we may interpret the permeability of a medium asameasureoftheresistanceofthematerialtomagneticflux.Justascurrentinanelectriccircuitfollowsthepathofleastresistance;themagneticfluxinamagneticcircuitfollowsthepathofleastreluctance


60/JNU OLE

SummaryElectricchargesmovinginamagneticfieldexperienceaforceduetothemagneticfield.Givenacharge.•The equivalence of the moving point charge and the differential length of line current yield the equivalent •magnetic force equation

(A-m) (A-m)

Thevectortorqueontheloopisdefinedintermsoftheforcemagnitude(IByl• 2), the torque moment arm distance (l1 /2), and the torque direction.Electricchargesmovinginamagneticfieldexperienceaforceduetothemagneticfield.•The charge-velocity product for a moving point charge can be related to an equivalent differential length of •line current.Thevectortorqueontheloopisdefinedintermsoftheforcemagnitude.•Magnetisationformagneticfieldsisthedualprocesstopolarisationforelectricfields.•An unmagnetized material can be characterized by a random distribution of the magnetic moments associated •with the electron orbits.Ifmostofthemagneticmomentsstayalignedaftertheappliedmagneticfieldisremoved,apermanentmagnet•is formed.The bar magnet can be viewed as the magnetic analogy to the electric dipole.•Magneticmaterialscanbeclassifiedbasedonthemagnitudeoftherelativepermeability.•Materialswitharelativepermeabilityofjustgreaterthanonearedefinedasparamagnetic.•Ferromagnetic materials lose their ferromagnetic properties at very high temperatures (above a temperature •known as the Curie temperature).

ReferencesBakshi, A.V., Bakshi, U.A., 2009. Field Theory. Technical Publications.http://www.ece.msstate.edu/~donohoe/•ece3313notes8.pdfBird, J., 2007. Electrical circuit theory and technology. Newnes.•

Recommended ReadingGoldman, A., 2010. Modern Ferrite Technology. Springer, 2 ed.•Cullity, B.D., Graham, C.D., 2008. Introduction to Magnetic Materials, Wiley-IEEE Press, 2 ed.•Yamaguchi, M., Tanimoto, Y., 2010. Magneto-Science: Magnetic Field Effects on Materials: Fundamentals •and Applications. Springer, 1 ed.

61/JNU OLE

Self Assessment

____________chargesmovinginamagneticfieldexperienceaforceduetothemagneticfield.1. Electromotivea. Electronicb. Electricc. Fixedd.

The charge-velocity product for a moving point charge can be related to an equivalent __________ length of 2. line current.

fixeda. differentialb. varyingc. time –variedd.

The_________torqueontheloopisdefinedintermsoftheforcemagnitude.3. pointa. lineb. vectorc. staticd.

____________formagneticfieldsisthedualprocesstopolarisationforelectricfields.4. Standarisationa. Fragmentationb. Divergencec. Magnetisationd.

A/an ____________ material can be characterized by a random distribution of the magnetic moments associated 5. with the electron orbits.

unmagnetiseda. magnetisedb. polarisedc. electrifiedd.

Ifmostofthemagneticmomentsstayalignedaftertheappliedmagneticfieldisremoved,a6. ____________ magnet is formed.

temporarya. permanentb. fixedc. stabled.

State which of the following is true.7. The bar magnet can be viewed as the magnetic analogy to the magnetic dipole.a. The pie magnet can be viewed as the magnetic analogy to the electric dipole.b. The magnet can be viewed as the magnetic analogy to the electric dipole.c. The ferro magnet can be viewed as the magnetic analogy to the electric dipole.d.


62/JNU OLE

State which of the following is true.8. Staticmaterialscanbeclassifiedbasedonthemagnitudeoftherelativepermeability.a. Magneticmaterialscanbeclassifiedbasedonthedisplacementoftherelativepermeability.b. Magneticmaterialscanbeclassifiedbasedonthemagnitudeoftherelativevelocity.c. Magneticmaterialscanbeclassifiedbasedonthemagnitudeoftherelativepermeability.d.

Materialswitharelativepermeabilityofjustgreaterthanonearedefinedaswhichofthefollowing?9. Bar magnetica. Ferromagneticb. Electromagneticc. Paramagneticd.

Ferromagnetic materials lose their ferromagnetic properties at which of the following?10. Low temperaturea. Very high temperaturesb. Room temperaturec. Cool temperatured.

63/JNU OLE

Chapter V

Time-Varying Fields And Maxwell’s Equations

Aim

The aim of the chapter is to:

analyse Faraday’s law•

explore displacement current•

highlight Maxwell’s equations•

Objectives


explain Faraday’s law•

explain Maxwell’s equations in point form•

examine• Maxwell’s equations in integral form

Learning outcome

At the end of this chapter students will be able to:

comprehend Faraday’s law•

determine • displacement current

understand • Maxwell’s equations in point form

formulate• Maxwell’s equations in integral form


64/JNU OLE

5.1 Faraday’s LawAfter Oersted demonstrated in 1820 that an electric current affected a compass needle, Faraday professed his beliefthatifacurrentcouldproduceamagneticfield,thenamagneticfieldshouldbeabletoproduceacurrent.Theconceptofthe"field"wasnotavailableatthattime,andFaraday’sgoalwastoshowthatacurrentcouldbeproduced by "magnetism".

Heworkedonthisproblemintermittentlyoveraperiodoftenyears,untilhewasfinallysuccessfulin1831.Hewound two separate windings on an iron toroid and placed a galvanometer in one circuit and a battery in the other. Uponclosingthebatterycircuit,henotedamomentarydeflectionofthegalvanometer;asimilardeflectionintheoppositedirectionoccurredwhenthebatterywasdisconnected.This,ofcourse,wasthefirstexperimenthemadeinvolvingachangingmagneticfield,andhefolloweditwithademonstrationthateitheramovingmagneticfieldoramovingcoilcouldalsoproduceagalvanometerdeflection.

In termsoffields,a time-varyingmagneticfieldproducesanelectromotive force (emf)whichmayestablishacurrent in a suitable closed circuit. An electromotive force is merely a voltage that arises from conductors moving inamagneticfieldorfromchangingmagneticfields,andweshalldefineitbelow.Faraday’slawiscustomarilystated as

(1)

Equation (1) implies a closed path, although not necessarily a closed conducting path; the closed path, for example, mightincludeacapacitor,oritmightbeapurelyimaginarylineinspace.Themagneticfluxisthatfluxwhichpasses through any and every surface whose perimeter is the closed path, and dΦ/dt is the time rate of change of thisflux.

A nonzero value of dΦ/dt may result from any of the following situations:Atime-changingfluxlinkingastationaryclosedpath•Relativemotionbetweenasteadyfluxandaclosedpath•A combination of the two•

Theminussignisanindicationthattheemfisinsuchadirectionastoproduceacurrentwhoseflux,ifaddedtotheoriginalflux,wouldreducethemagnitudeoftheemf.ThisstatementthattheinducedvoltageactstoproduceanopposingfluxisknownasLenz’slaw.

IftheclosedpathisthattakenbyanN-turnfilamentaryconductor,itisoftensufficientlyaccuratetoconsidertheturns as coincident and let

(2)

Where ΦisnowinterpretedasthefluxpassingthroughanyoneofNcoincidentpaths.Weneedtodefineemfasusedineq(1)oreq(2).Theemfisobviouslyascalar,and(perhapsnotsoobviously)adimensionalcheckshowsthatitismeasuredinvolts.Wedefinetheemfas

(3)

andnotethatitisthevoltageaboutaspecificclosedpath.Ifanypartofthepathischanged,generallytheemfchanges.Thedeparturefromstaticresultsisclearlyshownby(3),forelectricfieldintensityresultingfromastaticcharge distribution must lead to zero potential difference about a closed path. In electrostatics, the line integral leads toapotentialdifference;withtime-varyingfields,theresultisanemforavoltage.

Replacing Φ in eq (1) by the surface integral of B, we have

65/JNU OLE

(4)

Wherethefingersofourrighthandindicatethedirectionoftheclosedpath,andourthumbindicatesthedirectionofdS.AfluxdensityBinthedirectionofdSandincreasingwithtimethusproducesanaveragevalueofEwhichisopposite to the positive direction about the closed path. The right-handed relationship between the surface integral andtheclosedlineintegralin(4)shouldalwaysbekeptinmindduringfluxintegrationsandemfdeterminations.Letusdivideourinvestigationintotwopartsbyfirstfindingthecontributiontothetotalemfmadebyachangingfieldwithinastationarypath(transformeremf),andthenwewillconsideramovingpathwithinaconstant(motional,or generator, emf).

Wefirstconsiderastationarypath.Themagneticfluxistheonlytimevaryingquantityontherightsideof(4),anda partial derivative may be taken under the integral sign,

(5)

Before we apply this simple result to an example, let us obtain the point form of this integral equation. Applying Stokes’ theorem to the closed line integral, we have

where the surface integrals may be taken over identical surfaces. The surfaces are perfectly general and may be chosen as differentials,

(6)

Eq(5)istheintegralformofthisequationandisequivalenttoFaraday’slawasappliedtoafixedpath.IfBisnota function of time, (5) and (6) evidently reduce to the electrostatic equations,

(electrostatics)

and

(electrostatics)

5.2 Displacement CurrentFaraday’s experimental law has been used to obtain one of Maxwell’s equations in differential form,

(7)

whichshowsusthatatime-changingmagneticfieldproducesanelectricfield.Weseethatthiselectricfieldhasthespecialpropertyofcirculation;itslineintegralaboutageneralclosedpathisnotzero.Nowletusturnourattentiontothetime-changingelectricfield.

WeshouldfirstlookatthepointformofAmpere’scircuitallawasitappliestosteadymagneticfields,

(8)

and show its inadequacy for time-varying conditions by taking the divergence of each side,


66/JNU OLE

The divergence of the curl is identically zero, so ∇⋅J is also zero. However, the equation of continuity,

then shows us that (8) can be true only if . This is an unrealistic limitation, and (8) must be amended before wecanacceptitfortime-varyingfields.SupposeweaddanunknowntermGto(8),

Again taking the divergence, we have

Thus,

Replacing by ∇⋅D

from which we obtain the simplest solution for G,

Ampere’s circuital law in point form therefore becomes

(9)

Eq (9) has not been derived. It is merely a form we have obtained which does not disagree with the continuity equation. It is also consistent with all our other results, and we accept it as we did each experimental law and the equations derived from it.

WenowhaveasecondoneofMaxwell’sequationsandshallinvestigateitssignificance.Theadditionalterm has the dimensions of current density, amperes per square meter. Since it results from a time-varying electric

fluxdensity(ordisplacementdensity),Maxwelltermeditadisplacementcurrentdensity.Wesometimesdenoteitby Jd

This is the third type of current density we have met. Conduction current density,

is the motion of charge (usually electrons) in a region of zero net charge density, and convection current density,

is the motion of volume charge density. Both are represented by J in eq (9). Bound current density is, of course, included in H. In a non conducting medium in which no volume charge density is present, J = 0, and then

67/JNU OLE

(if J=0) (10)

Notice the symmetry between (10) and (7):

AgaintheanalogybetweentheintensityvectorsEandHandthefluxdensityvectorsDandBisapparent.Toomuchfaith cannot be placed in this analogy, however, for it fails when we investigate forces on particles. The force on a charge is related to E and to B, and some good arguments may be presented showing an analogy between E and B and between D and H. We shall omit them, however, and merely say that the concept of displacement current was probablysuggestedtoMaxwellbythesymmetryfirstmentionedabove.

The total displacement current crossing any given surface is expressed by the surface integral,

and we may obtain the time-varying version of Ampere’s circuital law by integrating eq (9) over the surface S,

and applying Stokes’ theorem

(11)

What is the nature of displacement current density? Let us study the simple circuit of Fig. 5.1, containing a filamentaryloopandaparallel-platecapacitor.Withintheloopamagneticfieldvaryingsinusoidalwithtimeisappliedtoproduceanemfabouttheclosedpath(thefilamentplusthedashedportionbetweenthecapacitorplates)which we shall take as

Using elementary circuit theory and assuming the loop has negligible resistance and inductance, we may obtain the current in the loop as

Where the quantities , S, and d pertain to the capacitor. Let us apply Ampere’s circuital law about the smaller closed circular path k and neglect displacement current for the moment:


68/JNU OLE

Fig. 5.1 A filamentary conductor forms a loop connecting the two plates of a parallel-plate capacitor(Source: http://www.ie.itcr.ac.cr/acotoc/Maestria_en_Computacion/Sistemas_de_Comunicacion_II/Material/


ThepathandthevalueofHalongthepatharebothdefinitequantities(althoughdifficulttodetermine),and∮k , H⋅dLisadefinitequantity.ThecurrentIk is that current through every surface whose perimeter is the path k. If we chooseasimplesurfacepuncturedbythefilament,suchastheplanecircularsurfacedefinedbythecircularpathk,the current is evidently the conduction current. Suppose now we consider the closed path k as the mouth of a paper bagwhosebottompassesbetweenthecapacitorplates.Thebagisnotpiercedbythefilament,andtheconductorcurrent is zero. Now we need to consider displacement current, for within the capacitor

and therefore

Thisisthesamevalueasthatoftheconductioncurrentinthefilamentaryloop.ThereforetheapplicationofAmpere’scircuitallawincludingdisplacementcurrenttothepathkleadstoadefinitevalueforthelineintegralofH.Thisvalue must be equal to the total current crossing the chosen surface. For some surfaces the current is almost entirely conduction current, but for those surfaces passing between the capacitor plates, the conduction current is zero, and it is the displacement current which is now equal to the closed line integral of H.

Physically,weshouldnotethatacapacitorstoreschargeandthattheelectricfieldbetweenthecapacitorplatesismuchgreaterthanthesmallleakagefieldsoutside.Wethereforeintroducelittleerrorwhenweneglectdisplacementcurrent on all those surfaces which do not pass between the plates. Displacement current is associated with time-varyingelectricfieldsandthereforeexistsinallimperfectconductorscarryingatime-varyingconductioncurrent.

5.3 Maxwell’s Equations in Point FormWehavealreadyobtainedtwoofMaxwell’sequationsfortime-varyingfields,

(12)

and (13)

The remaining two equations are unchanged from their non-time-varying form:

69/JNU OLE

(14)

(15)

Eq(14)essentiallystatesthatchargedensityisasource(orsink)ofelectricfluxlines.Notethatwecannolongersaythatallelectricfluxbeginsandterminatesoncharge,becausethepointformofFaraday’slaw(12)showsthatE,andhenceD,mayhavecirculationifachangingmagneticfieldispresent.

Thusthelinesofelectricfluxmayformclosedloops.However,theconverseisstilltrue,andeverycoulombofchargemusthaveonecoulombofelectricfluxdivergingfromit.

Equation(15)againacknowledgesthefactthat `magneticcharges,’’orpoles,arenotknowntoexist.Magneticfluxis always found in closed loops and never diverges from a point source.

These for equations form the basis of all electromagnetic theory. They are partial differential equations and relate the electricandmagneticfieldstoeachotherandtotheirsources,chargeandcurrentdensity.Theauxiliaryequationsrelating D and E are

(16)

relating B and H

(17)

definingconductioncurrentdensity,

(18)

anddefiningconvectioncurrentdensityintermsofthevolumechargedensity ,

(19)

arealsorequiredtodefineandrelatethequantitiesappearinginMaxwell’sequations.The potentials V and A have not been included above because they are not strictly necessary, although they are extremely useful.

5.4 Maxwell’s Equations in Integral FormThe integral forms of Maxwell’s equations are usually easier to recognize in terms of the experimental laws from which they have been obtained by a generalization process. Experiments must treat physical macroscopic quantities, and their results therefore are expressed in terms of integral relationships. A differential equation always represents a theory. Let us now collect the integral forms of Maxwell’s equations of the previous section.

Integrating eq (12) over a surface and applying Stokes’ theorem, we obtain Faraday’s law,

(20)

and the same process applied to (13) yields Ampere’s circuital law,

(21)

Gauss’slawsfortheelectricandmagneticfieldsareobtainedbyintegratingeq(14)and(15)throughoutavolumeand using the divergence theorem:


70/JNU OLE

(22)

(23)

ThesefourintegralequationsenableustofindtheboundaryconditionsonB,D,H,andEwhicharenecessarytoevaluate the constants obtained in solving Maxwell’s equations in partial differential form. These boundary conditions areingeneralunchangedfromtheirformsforstaticorsteadyfields,andthesamemethodsmaybeusedtoobtainthem. Between any two real physical media (where K must be zero on the boundary surface), (20) enables us to relatethetangentialE-fieldcomponents,

Et1=Et2 (24)

and from (21)

Ht1=Ht2 (25)

The surface integrals produce the boundary conditions on the normal components,

DN1‒DN2= (26)

and

BN1=BN2 (27)

It is often desirable to idealize a physical problem by assuming a perfect conductor for which σisinfinitebutJisfinite.FromOhm'slaw,inaperfectconductor,

E=0

and it follows from the point form of Faraday's law that

H=0

fortime-varyingfields.ThepointformofAmpere'scircuitallawthenshowsthatthefinitevalueofJis

J=0

and current must be carried on the conductor surface as a surface current K. Thus, if region 2 is a perfect conductor, (24) to (27) become, respectively,

Et1=0 (28)

Ht1=K (Ht1=k ×aN) (29)

DN1= (30)

BN1=0 (31)

Where aN is an outward normal at the conductor surface. Note that surface charge density is considered a physical possibility for dielectrics, perfect conductors, or imperfect conductors, but that surface current density is assumed only in conjunction with perfect conductors.

71/JNU OLE

The boundary conditions stated above are a very necessary part of Maxwell’s equations. All real physical problems have boundaries and require the solution of Maxwell’s equations in two or more regions and the matching of these solutions at the boundaries. In the case of perfect conductors, the solution of the equations within the conductor istrivial(alltime-varyingfieldsarezero),buttheapplicationoftheboundaryconditions(28)to(31)maybeverydifficult.

Certain fundamental properties of wave propagation are evident when Maxwell’s equations are solved for an unbounded region. It represents the simplest application of Maxwell’s equations, because it is the only problem which does not require the application of any boundary conditions.


72/JNU OLE

Summary

Faraday’s law is customarily stated as • .

Ampere’s circuital law in point form is given as • .

TheanalogybetweentheintensityvectorsEandHandthefluxdensityvectorsDandBisapparent•

TwoofMaxwell’sequationsfortime-varyingfieldsare• and .

Certain fundamental properties of wave propagation are evident when Maxwell’s equations are solved for an •unbounded region.

ReferencesTime –Varying Fields and Maxwell’s Equations• . Available at <http://www.ie.itcr.ac.cr/acotoc/Maestria_en_Computacion/Sistemas_de_Comunicacion_II/Material/Biblio2/chapt10.pdf>. [Accessed 11 February 2011].Time –Varying Fields and Maxwell’s Equations• . Available at <http://www.math.ethz.ch/~lanford/TP_2009/maxwell.pdf> [Accessed 11 February 2011].

Recommended ReadingFleisch, D., 2008. • A Student’s Guide to Maxwell’s Equations. Cambridge University Press, 1 ed.Huray, P.G., 2009. • Maxwell’s Equations. Wiley-IEEE Press.Karmel, P.R., Colef , G.D., Camisa, R.L., 1997. • Introduction to Electromagnetic and Microwave Engineering. Wiley-Interscience, 1 ed.

73/JNU OLE

Self Assessment

_____________professedhisbeliefthatifacurrentcouldproduceamagneticfield,thenamagneticfield1. should be able to produce a current.

Faradaya. Coulombb. Henryc. Jouled.

Intermsoffields,a__________magneticfieldproducesanelectromotiveforce(emf)whichmayestablisha2. current in a suitable closed circuit.

fixeda. time-varyingb. variablec. speed-varyingd.

A__________ismerelyavoltagethatarisesfromconductorsmovinginamagneticfieldorfromchanging3. magneticfields.

electromotive forcea. unit forceb. electric forcec. electronic forced.

Certain fundamental properties of wave propagation are evident when Maxwell’s equations are solved for a/4. an __________ region.

boundeda. fixedb. variablec. unboundedd.

_____________current is associatedwith time-varying electricfields and therefore exists in all imperfect5. conductors carrying a time-varying conduction current

Displacementa. Variableb. Varyingc. Fixedd.

in 1820, who demonstrated that an electric current affected a compass needle?6. Oersteda. Faradayb. Henryc. Jouled.


74/JNU OLE

Which of the following equation states Faraday’s law?7.

a.

b.

c.

d.

Ampere’s circuital law in point form is given by which of the following equation?8.

a.

b.

c.

d.

OneofMaxwell’sequationsfortime-varyingfieldsisgivenbywhichofthefollowingequation?9.

a.

b.

c.

d.

State which of the following is true.10. Magneticfluxisalwaysfoundinclosedloopsandneverdivergesfromapointsource.a. Magneticfluxisalwaysfoundinopenloopsandneverdivergesfromapointsource.b. Magneticfluxisalwaysfoundinclosedloopsanddivergesfromapointsource.c. Magneticfluxisnotfoundinclosedloopsandneverdivergesfromapointsource.d.

75/JNU OLE

Chapter VI

Antenna Fundamentals

Aim


analyse how antenna radiates•

explore radiation pattern of antenna •

classify• different types of antennas

Objectives


examine directivity of an antenna•

formulateantennaefficiency•

enlist various types of antenna •

Learning outcome


recall how antenna radiates•

identify • various performance parameters of antenna

categorise the• types of antenna


76/JNU OLE

6.1 Introduction Antennas are metallic structures designed for radiating and receiving electromagnetic energy. An antenna acts as a transitionalstructurebetweentheguidingdevices(e.g.,waveguide,transmissionline)andthefreespace.TheofficialIEEEdefinitionofanantennaasgivenbyStutzmanandThiele[4]followstheconcept:“Thatpartofatransmittingor receiving system that is designed to radiate or receive electromagnetic waves”.

6.2 How an Antenna RadiatesInordertoknowhowanantennaradiates,letusfirstconsiderhowradiationoccurs.Aconductingwireradiatesmainly because of time-varying current or an acceleration (or deceleration) of charge. If there is no motion of chargesinawire,noradiationtakesplace,sincenoflowofcurrentoccurs.Radiationwillnotoccurevenifchargesare moving with uniform velocity along a straight wire. However, charges moving with uniform velocity along a curved or bent wire will produce radiation. If the charge is oscillating with time, then radiation occurs even along a straight wire as explained by Balanis.

Theradiationfromanantennacanbeexplainedwiththehelpoffig.6.1whichshowsavoltagesourceconnectedto a two conductor transmission line. When a sinusoidal voltage is applied across the transmission line, an electric fieldiscreatedwhichissinusoidalinnatureandtheseresultsinthecreationofelectriclinesofforcewhicharetangentialtotheelectricfield.Themagnitudeoftheelectricfieldisindicatedbythebunchingoftheelectriclinesof force. The free electrons on the conductors are forcibly displaced by the electric lines of force and the movement ofthesechargescausestheflowofcurrentwhichinturnleadstothecreationofamagneticfield.

Fig. 6.1 Radiation from an antenna(Source: http://etd.lib.fsu.edu/theses/available/etd-04102004-143656/unrestricted/Chapter2.pdf)

77/JNU OLE

Duetothetimevaryingelectricandmagneticfields,electromagneticwavesarecreatedandthesetravelbetweentheconductors. As these waves approach open space, free space waves are formed by connecting the open ends of the electric lines. Since the sinusoidal source continuously creates the electric disturbance, electromagnetic waves are created continuously and these travel through the transmission line, through the antenna and are radiated into the free space. Inside the transmission line and the antenna, the electromagnetic waves are sustained due to the charges, but as soon as they enter the free space, they form closed loops and are radiated.

6.3 Near and Far Field RegionsThefieldpatterns,associatedwithanantenna,changewithdistanceandareassociatedwithtwotypesofenergy:-radiating energy and reactive energy. Hence, the space surrounding an antenna can be divided into three regions

Fig. 6.2 Field regions around an antenna(Source: http://etd.lib.fsu.edu/theses/available/etd-04102004-143656/unrestricted/Chapter2.pdf)

Thethreeregionsshowninfig6.2are:Reactive near-field region• :Inthisregion,thereactivefielddominates.Thereactiveenergyoscillatestowardsand away from the antenna, thus appearing as reactance. In this region, energy is only stored and no energy is dissipated. The outermost boundary for this region is at a distance where R1 is the distance fromtheantennasurface,Disthelargestdimensionoftheantennaandλisthewavelength.Radiating near-field region (also called Fresnel region• ): This is the region which lies between the reactive near-fieldregionandthefarfieldregion.Reactivefieldsaresmallerinthisfieldascomparedtothereactivenear-fieldregionandtheradiationfieldsdominate.Inthisregion,theangularfielddistributionisafunctionofthe distance from the antenna. The outermost boundary for this region is at a distance where R2 is the distance from the antenna surface.Far-field region (also called Fraunhofer region):• The region beyond isthefarfieldregion.Inthisregion,thereactivefieldsareabsentandonlytheradiationfieldsexist.Theangularfielddistributionisnotdependent on the distance from the antenna in this region and the power density varies as the inverse square of the radial distance in this region.

6.4 Antenna Performance ParametersThe performance of an antenna can be gauged from a number of parameters. Certain critical parameters are listed below:


78/JNU OLE

6.4.1 Radiation PatternTheradiationpatternofanantennaisaplotofthefar-fieldradiationpropertiesofanantennaasafunctionofthespatialco-ordinateswhicharespecifiedbytheelevationangleθandtheazimuthangleφ.Morespecificallyitisa plot of the power radiated from an antenna per unit solid angle which is nothing but the radiation intensity. Let us consider the case of an isotropic antenna. An isotropic antenna is one which radiates equally in all directions. If the total power radiated by the isotropic antenna is P, then the power is spread over a sphere of radius r, so that the power density S at this distance in any direction is given as

(1)

Then the radiation intensity for this isotropic antenna Ui can be written as

(2)

An isotropic antenna is not possible to realize in practice and is useful only for comparison purposes. A more practical type is the directional antenna which radiates more power in some directions and less power in other directions. A special case of the directional antenna is the omnidirectional antenna whose radiation pattern may be constant in one plane (e.g. E-plane) and varies in an orthogonal plane (e.g. H-plane). The radiation pattern plot of a generic directionalantennaisshowninfig.6.3

Fig. 6.3 Radiation pattern of a generic directional antenna(Source: http://etd.lib.fsu.edu/theses/available/etd-04102004-143656/unrestricted/Chapter2.pdf)

Figure 6.3 shows the followingHPBW• :Thehalfpowerbeamwidth(HPBW)canbedefinedastheanglesubtendedbythehalfpowerpointsof the main lobe.Main Lobe• : This is the radiation lobe containing the direction of maximum radiation.Minor Lobe• : All the lobes other then the main lobe are called the minor lobes. These lobes represent the radiation in undesired directions. The level of minor lobes is usually expressed as a ratio of the power density in the lobe in question to that of the major lobe. This ratio is called as the side lobe level (expressed in decibels).Back Lobe• : This is the minor lobe diametrically opposite the main lobe.Side Lobes• : These are the minor lobes adjacent to the main lobe and are separated by various nulls. Side lobes are generally the largest among the minor lobes

In most wireless systems, minor lobes are undesired. Hence a good antenna design should minimize the minor lobes.

79/JNU OLE

6.4.2 DirectivityThedirectivityofanantennaisdefinedas“theratiooftheradiationintensityinagivendirectionfromtheantennato the radiation intensity averaged over all directions”. In other words, the directivity of a nonisotropic source is equal to the ratio of its radiation intensity in a given direction, over that of an isotropic source,

(3)

Where,D is the directivity of the antennaU is the radiation intensity of the antennaUi is the radiation intensity of an isotropic sourceP is the total power radiated

Sometimes,thedirectionofthedirectivityisnotspecified.Inthiscase,thedirectionofthemaximumradiationintensity is implied and the maximum directivity is given by

(4)

WhereDmax = maximum directivityUmax = maximum radiation intensity.

Directivity is a dimensionless quantity, since it is the ratio of two radiation intensities.Hence, it is generally expressed in dBi. The directivity of an antenna can be easily estimated from the radiation pattern of the antenna. An antenna that has a narrow main lobe would have better directivity, then the one which has a broad main lobe, hence it is more directive.

6.4.3 Input ImpedanceTheinputimpedanceofanantennaisdefinedbyBalanisas“theimpedancepresentedbyanantennaatitsterminalsor the ratio of the voltage to the current at the pair of terminals or the ratio of the appropriate components of the electrictomagneticfieldsatapoint”.Hencetheimpedanceoftheantennacanbewrittenas

(5)

Where Zin = antenna impedance at the terminalsRin = antenna resistance at the terminalsXin = antenna reactance at the terminals

Theimaginarypart,Xinoftheinputimpedancerepresentsthepowerstoredinthenearfieldoftheantenna.Theresistivepart, Rin of the input impedance consists of two components, the radiation resistance Rr and the loss resistance RL. The power associated with the radiation resistance is the power actually radiated by the antenna, while the power dissipated in the loss resistance is lost as heat in the antenna itself due to dielectric or conducting losses.


80/JNU OLE

6.4.4 Voltage Standing Wave Ratio (VSWR)

Fig. 6.4 Equivalent circuit of transmitting antenna(Source: http://etd.lib.fsu.edu/theses/available/etd-04102004-143656/unrestricted/Chapter2.pdf)

Inorderfortheantennatooperateefficiently,maximumtransferofpowermusttakeplacebetweenthetransmitterand the antenna. Maximum power transfer can take place only when the impedance of the antenna (Zin ) is matched to that of the transmitter (ZS). According to the maximum power transfer theorem, maximum power can be transferred only if the impedance of the transmitter is a complex conjugate of the impedance of the antenna under consideration and vice-versa. Thus, the condition for matching is

(6)

Where

Zin= Rin +jXinZS=RS+jXS

Iftheconditionformatchingisnotsatisfied,thensomeofthepowermaybereflectedbackandthisleadstothecreation of standing waves, which can be characterized by a parameter called as the Voltage Standing Wave Ratio (VSWR). The VSWR is given by Makarov as

(7)

(8)

WhereΓ iscalledthereflectioncoefficientVr istheamplitudeofthereflectedwaveVi is the amplitude of the incident wave

81/JNU OLE

The VSWR is basically a measure of the impedance mismatch between the transmitter and the antenna. The higher the VSWR, the greater is the mismatch. The minimum VSWR which corresponds to a perfect match is unity. A practicalantennadesignshouldhaveaninputimpedanceofeither50Ωor75Ωsincemostradioequipmentisbuiltfor this impedance

6.4.5 Return Loss (RL)The Return Loss (RL) is a parameter which indicates the amount of power that is “lost” to the load and does not returnasareflection.Asexplainedintheprecedingsection,wavesarereflectedleadingtotheformationofstandingwaves, when the transmitter and antenna impedance do not match. Hence the RL is a parameter similar to the VSWR to indicate how well the matching between the transmitter and antenna has taken place. The RL is given as by Makarov as

(dB) (9)

Forperfectmatchingbetweenthetransmitterandtheantenna,Γ=0andRL=∞whichmeansnopowerwouldbereflectedback,whereasaΓ=1hasaRL=0dB,whichimpliesthatallincidentpowerisreflected.Forpracticalapplications, a VSWR of 2 is acceptable, since this corresponds to a RL of -9.54 dB.

6.4.6 Antenna EfficiencyTheantennaefficiencyisaparameterwhichtakesintoaccounttheamountoflossesattheterminalsoftheantennaand within the structure of the antenna. These losses are given as

Reflectionsbecauseofmismatchbetweenthetransmitterandtheantenna•I• 2R losses (conduction and dielectric)

Hencethetotalantennaefficiencycanbewrittenas

(10)

Where et=totalantennaefficiency

er=(1− )=reflection(mismatch)efficiency

ec=conductionefficiencyed =dielectricefficiency

Since ec and edaredifficulttoseparate,theyarelumpedtogethertoformtheecdefficiencywhichisgivenas

(11)

ecd iscalledastheantennaradiationefficiencyandisdefinedastheratioofthepowerdeliveredtotheradiationresistance Rr, to the power delivered to Rr and RL.

6.4.7 Antenna GainAntenna gain is a parameter which is closely related to the directivity of the antenna. We know that the directivity is how much an antenna concentrates energy in one direction in preference to radiation in other directions. Hence, iftheantennais100%efficient,thenthedirectivitywouldbeequaltotheantennagainandtheantennawouldbean isotropic radiator. Since all antennas will radiate more in some direction that in others, therefore the gain is the amount of power that can be achieved in one direction at the expense of the power lost in the others as explained byUlaby[7].Thegainisalwaysrelatedtothemainlobeandisspecifiedinthedirectionofmaximumradiationunless indicated. It is given as


82/JNU OLE

(dBi) (12)

6.4.8 PolarisationPolarisationofaradiatedwaveisdefinedbyBalarisas“thatpropertyofanelectromagneticwavedescribingthetimevaryingdirectionandrelativemagnitudeoftheelectricfieldvector”.Thepolarisationofanantennareferstothepolarisationoftheelectricfieldvectoroftheradiatedwave.Inotherwords,thepositionanddirectionoftheelectricfieldwithreferencetotheearth’ssurfaceorgrounddeterminesthewavepolarisation.Themostcommontypes of polarisation include the linear (horizontal or vertical) and circular (right hand polarisation or the left hand polarisation).

Fig. 6.5 A linearly (vertically) polarised wave(Source: http://etd.lib.fsu.edu/theses/available/etd-04102004-143656/unrestricted/Chapter2.pdf)

Ifthepathoftheelectricfieldvectorisbackandforthalongaline,itissaidtobelinearlypolarised.Fig.6.5showsalinearlypolarisedwave.Inacircularlypolarisedwave,theelectricfieldvectorremainsconstantinlengthbutrotatesaround in a circular path. A left hand circular polarised wave is one in which the wave rotates counter clockwise whereasrighthandcircularpolarisedwaveexhibitsclockwisemotionasshowninfig6.6

Fig. 6.6 Commonly used polarisation schemes(Source: http://etd.lib.fsu.edu/theses/available/etd-04102004-143656/unrestricted/Chapter2.pdf)

83/JNU OLE

6.4.9 BandwidthThebandwidthofanantennaisdefinedas“therangeofusablefrequencieswithinwhichtheperformanceoftheantenna,withrespecttosomecharacteristic,conformstoaspecifiedstandard.”Thebandwidthcanbetherangeoffrequencies on either side of the center frequency where the antenna characteristics like input impedance, radiation pattern, beam width, polarisation, side lobe level or gain, are close to those values which have been obtained at the centerfrequency.Thebandwidthofabroadbandantennacanbedefinedastheratiooftheuppertolowerfrequenciesofacceptableoperation.ThebandwidthofanarrowbandantennacanbedefinedasthepercentageofthefrequencydifferenceoverthecenterfrequencyBalaris.Thesedefinitionscanbewrittenintermsofequationsasfollows

(13)

(14)

Where fH =upper frequencyfL =lower frequencyfC =center frequencyAn antenna is said to be broadband if .OnemethodofjudginghowefficientlyanantennaisoperatingovertherequiredrangeoffrequenciesisbymeasuringitsVSWR.AVSWR≤2(RL≥−9.5dB)ensuresgoodperformance

Fig. 6.7 Measuring bandwidth from the plot of the reflection coefficient(Source: http://etd.lib.fsu.edu/theses/available/etd-04102004-143656/unrestricted/Chapter2.pdf)

6.5 Types of AntennasAntennas come in different shapes and sizes to suit different types of wireless applications. The characteristics of an antenna are very much determined by its shape, size and the type of material that it is made of. Some of the commonlyusedantennasarebrieflydescribedbelow.

6.5.1 Half Wave Dipole


84/JNU OLE

The length of this antenna is equal to half of its wavelength as the name itself suggests.Dipoles can be shorter or longer than half the wavelength, but a trade off exists in the performance and hence the half wavelength dipole is widely used

Fig. 6.8 Half wave dipole(Source: http://etd.lib.fsu.edu/theses/available/etd-04102004-143656/unrestricted/Chapter2.pdf)

The dipole antenna is fed by a two wire transmission line, where the two currents in the conductors are of sinusoidal distribution and equal in amplitude, but opposite in direction. Hence, due to cancelling effects, no radiation occurs fromthetransmissionline.Asshowninfig.6.8thecurrentsinthearmsofthedipoleareinthesamedirectionandthey produce radiation in the horizontal direction. Thus, for a vertical orientation, the dipole radiates in the horizontal direction.Thetypicalgainofthedipoleis2dBandithasabandwidthofabout10%.Thehalfpowerbeamwidthisabout78degreesintheEplaneanditsdirectivityis1.64(2.15dB)witharadiationresistanceof73Ω[4].Fig.6.9shows the radiation pattern for the half wave dipole

Fig. 6.9 Radiation pattern for half wave dipole(Source: http://etd.lib.fsu.edu/theses/available/etd-04102004-143656/unrestricted/Chapter2.pdf)

85/JNU OLE

6.5.2 Monopole AntennaThemonopoleantenna,showninfig.6.10,resultsfromapplyingtheimagetheorytothedipole.Accordingtothistheory, if a conducting plane is placed below a single element of length L/2 carrying a current, then the combination of the element and its image acts identically to a dipole of length L except that the radiation occurs only in the space above the plane as discussed by Saunders [8].

Fig. 6.10 Monopole antenna(Source: http://etd.lib.fsu.edu/theses/available/etd-04102004-143656/unrestricted/Chapter2.pdf)

For this type of antenna, the directivity is doubled and the radiation resistance is halved when compared to the dipole.Thus,ahalfwavedipolecanbeapproximatedbyaquarterwavemonopole(L/2=λ/4).Themonopoleisvery useful in mobile antennas where the conducting plane can be the car body or the handset case. The typical gain forthequarterwavelengthmonopoleis2-6dBandithasabandwidthofabout10%.Itsradiationresistanceis36.5Ωanditsdirectivityis3.28(5.16dB)[4].Theradiationpatternforthemonopoleisshownbelowinfig7.11.

Fig. 6.11 Radiation pattern for the monopole antenna(Source: http://etd.lib.fsu.edu/theses/available/etd-04102004-143656/unrestricted/Chapter2.pdf)

6.5.3 Loop AntennaThe loop antenna is a conductor bent into the shape of a closed curve such as a circle or a square with a gap in theconductortoformtheterminalsasshowninfig6.12.Therearetwotypesofloopantennas-electricallysmallloop antennas and electrically large loop antennas. If the total loop circumference is very small as compared to thewavelength(L<<<λ),thentheloopantennaissaidtobeelectricallysmall.Anelectricallylargeloopantennatypicallyhasitscircumferenceclosetoawavelength.Thefar-fieldradiationpatternsofthesmallloopantennaareinsensitive to shape [4].


86/JNU OLE

Fig. 6.12 Loop antenna(Source: http://etd.lib.fsu.edu/theses/available/etd-04102004-143656/unrestricted/Chapter2.pdf)

Asshowninfig.6.12,theradiationpatternsareidenticaltothatofadipoledespitethefactthatthedipoleisverticallypolarised whereas the small circular loop is horizontally polarised.

Fig. 6.13 Radiation pattern of small and large loop antenna(Source: http://etd.lib.fsu.edu/theses/available/etd-04102004-143656/unrestricted/Chapter2.pdf)

Theperformanceoftheloopantennacanbeincreasedbyfillingthecorewithferrite.Thishelpsinincreasingtheradiation resistance. When the perimeter or circumference of the loop antenna is close to a wavelength, then the antenna is said to be a large loop antenna. The radiation pattern of the large loop antenna is different then that of the small loop antenna. For a one wavelength square loop antenna, radiation is maximum normal to the plane of the loop (along the z axis). In the plane of the loop, there is a null in the direction parallel to the side containing the feed (along the x axis), and there is a lobe in a direction perpendicular to the side containing the feed (along the y axis). Loopantennasgenerallyhaveagainfrom-2dBto3dBandabandwidthofaround10%..Thesmallloopantennais very popular as a receiving antenna. Single turn loop antennas are used in pagers and multiturn loop antennas are used in AM broadcast receivers.

87/JNU OLE

6.5.4 Helical AntennasA helical antenna or helix is one in which a conductor connected to a ground plane, is wound into a helical shape. Fig 6.14 illustrates a helix antenna. The antenna can operate in a number of modes, however the two principal modes are thenormalmode(broadsideradiation)andtheaxialmode(endfireradiation).Whenthehelixdiameterisverysmallas compared to the wavelength, then the antenna operates in the normal mode. However, when the circumference of the helix is of the order of a wavelength, then the helical antenna is said to be operating in the axial mode

Fig. 6.14 Helix antenna(Source: http://etd.lib.fsu.edu/theses/available/etd-04102004-143656/unrestricted/Chapter2.pdf)

Inthenormalmodeofoperation,theantennafieldismaximuminaplanenormaltothehelixaxisandminimumalong its axis. This mode provides low bandwidth and is generally used for hand-portable mobile applications

Fig. 6.15 Radiation pattern of helix antenna(Source: http://etd.lib.fsu.edu/theses/available/etd-04102004-143656/unrestricted/Chapter2.pdf)


88/JNU OLE

Intheaxialmodeofoperation,theantennaradiatesasanendfireradiatorwithasinglebeamalongthehelixaxis.This mode provides better gain (upto 15dB) [4] and high bandwidth ratio (1.78:1) as compared to the normal mode of operation. For this mode of operation, the beam becomes narrower as the number of turns on the helix is increased. Due to its broadband nature of operation, the antenna in the axial mode is used mainly for satellite communications. Fig.7.15 above shows the radiation patterns for the normal mode as well as the axial mode of operations.

6.5.5 Horn AntennasHorn antennas are used typically in the microwave region (gigahertz range) where waveguides are the standard feedmethod,sincehornantennasessentiallyconsistofawaveguidewhoseendwallsareflaredoutwardstoforma megaphone like structure

Fig. 6.16 Types of horn antenna(Source: http://etd.lib.fsu.edu/theses/available/etd-04102004-143656/unrestricted/Chapter2.pdf)

Horns provide high gain, low VSWR, relatively wide bandwidth, low weight, and are easy to construct [4]. The aperture of the horn can be rectangular, circular or elliptical. However, rectangular horns are widely used. The three basictypesofhornantennasthatutilizearectangulargeometryareshowninfig.6.16.Thesehornsarefedbyarectangularwaveguidewhichhaveabroadhorizontalwallasshowninthefigure.Fordominantwaveguidemodeexcitation,theE-planeisverticalandH-planehorizontal.Ifthebroadwalldimensionofthehornisflaredwiththenarrow wall of the waveguide being left as it is, then it is called an H-plane sectoral horn antenna as shown in the figure.IftheflaringoccursonlyintheE-planedimension,itiscalledanE-planesectoralhornantenna.Apyramidalhornantennaisobtainedwhenflaringoccursalongboththedimensions.Thehornbasicallyactsasatransitionfromthewaveguidemodetothefree-spacemodeandthistransitionreducesthereflectedwavesandemphasizesthe travelling waves which lead to low VSWR and wide bandwidth [4]. The horn is widely used as a feed element for large radio astronomy, satellite tracking, and communication dishes.

89/JNU OLE

SummaryAntennas are metallic structures designed for radiating and receiving electromagnetic energy.•A conducting wire radiates mainly because of time-varying current or an acceleration (or deceleration) of •charge.Duetothetimevaryingelectricandmagneticfields,electromagneticwavesarecreatedandthesetravelbetween•the conductors.Thefieldpatterns,associatedwithanantenna,changewithdistanceandareassociatedwithtwotypesofenergy:•- radiating energy and reactive energy.Theradiationpatternofanantennaisaplotofthefar-fieldradiationpropertiesofanantennaasafunctionof•thespatialco-ordinateswhicharespecifiedbytheelevationangleθandtheazimuthangleφ.An isotropic antenna is not possible to realize in practice and is useful only for comparison purposes.•In most wireless systems, minor lobes are undesired. Hence a good antenna design should minimize the minor •lobes.Thedirectivityofanantennaisdefinedas“theratiooftheradiationintensityinagivendirectionfromthe•antenna to the radiation intensity averaged over all directions”.The Return Loss (RL) is a parameter which indicates the amount of power that is “lost” to the load and does •notreturnasareflection.Theantennaefficiencyisaparameterwhichtakesintoaccounttheamountoflossesattheterminalsofthe•antenna and within the structure of the antenna.Antenna gain is a parameter which is closely related to the directivity of the antenna.•Polarisationofaradiatedwaveisdefinedbyas“thatpropertyofanelectromagneticwavedescribingthetime•varyingdirectionandrelativemagnitudeoftheelectricfieldvector”.Antennas come in different shapes and sizes to suit different types of wireless applications.•The characteristics of an antenna are very much determined by its shape, size and the type of material that it •is made of.The loop antenna is a conductor bent into the shape of a closed curve such as a circle or a square with a gap in •the conductor to form the terminals.A helical antenna or helix is one in which a conductor connected to a ground plane, is wound into a helical •shape.Horn antennas are used typically in the microwave region (gigahertz range) where waveguides are the standard •feed method.

ReferencesAntenna Basics• . Available at: <http://wireless.ictp.it/handbook/C4.pdf> [Accessed 17 February 2011].Antenna Fundamentals• . Available at: <http://etd.lib.fsu.edu/theses/available/etd-04102004-143656/unrestricted/Chapter2.pdf> [Accessed 17 February 2011].

Recommended ReadingBalanis, C.A., 2005. • Antenna Theory: Analysis and Design, Wiley-Interscience, 3 ed.Minin, I.V., Minin, O.V., 2010. • Basic Principles of Fresnel Antenna Arrays. Springer, 1 ed.Kraus, J.D., Marhefka, R.J., 2001. • Antennas for All Applications. McGraw-Hill Science/Engineering/Math, 3 ed.


90/JNU OLE

Self assessment

Antennas are ________ structures designed for radiating and receiving electromagnetic energy.1. metallica. plasticb. oxidec. silverd.

A conducting wire radiates mainly because of time-varying current or a/an ___________ (or deceleration) of 2. charge.

accelerationa. speedb. displacementc. volumed.

An ____________ antenna is not possible to realize in practice and is useful only for comparison purposes.3. bidirectionala. rhombicb. isotropicc. unidirectionald.

The ___________ is a parameter which indicates the amount of power that is “lost” to the load and does not 4. returnasareflection.

load lossa. reflectionlossb. loss factorc. return lossd.

The antenna __________ is a parameter which takes into account the amount of losses at the terminals of the 5. antenna and within the structure of the antenna.

gaina. resolutionb. directivityc. efficiencyd.

State which of the following statement is true.6. Antennaefficiencyisaparameterwhichiscloselyrelatedtothedirectivityoftheantenna.a. Antenna gain is a parameter which is closely related to the directivity of the antenna.b. Antenna loss is a parameter which is closely related to the directivity of the antenna.c. Antenna power factor is a parameter which is closely related to the directivity of the antenna.d.

State which of the following statement is true.7. Antennas do not come in different shapes and sizes to suit different types of wireless applications.a. Antennas come only in rectangular shape to suit different types of wireless applications.b. Antennas come in different shapes and sizes to suit different types of wireless applications.c. Antennas come in different shapes and sizes to suit all types of non-wireless applications.d.

91/JNU OLE

State which of the following statement is false.8. The characteristics of an antenna are very much determined by its shape, size and the type of material that a. it is made of.Duetothetimevaryingelectricandmagneticfields,electromagneticwavesarecreatedandthesetravelb. between the conductors.Horn antennas are used typically in the microwave region (gigahertz range) where waveguides are the c. standard feed method.The loop antennas are used typically in the microwave region (gigahertz range) where waveguides are the d. standard feed method.

Which of the following antenna is the one in which a conductor connected to a ground plane, is wound into a 9. helical shape?

Helical antennaa. Loop antennab. Rhombic antennac. Isotropic antennad.

Which of the following antennas are used typically in the microwave region (gigahertz range) where waveguides 10. are the standard feed method?

Helical antennaa. Horn antennab. Rhombicc. Pie antennad.


92/JNU OLE

Application I

Helical Antenna for Medical Applications

IntroductionAn increased interest for thermotherapy using microwaves has been observed in the last decade. A large number of devices have been designed and tested in order to produce therapeutic heating for medical applications and more particularly microwave hyperthermia (for the treatment of tumors having different sizes and located in various places of the human body). Among these devices, there is large interest in the study of interstitial coaxial applicators and more particularly of endocavitary applicators. They are generally used in urology for the heating of tumors or for the improvement of medical treatments like radiotherapy or chemotherapy. In this paper, we present the theoretical studyandtheexperimentalverificationsconcerningagenerationofapplicatorsofhelicaltype.Theyhavebeendesigned as to reduce the heating zone along the cable in order to avoid possible thermal necrosis.

Materials and method usedThemicrowaveantennaisrealizedfromaflexiblecoaxialcableof50Wcharacteristicimpedance.Previousantennaswere realized by removing the outer conductor of the cable on a length h. The improvement consists in making a doublehelicalantenna,thefirsthelixistheinnerconductorwhichisrolleduparoundtheteflonsheath.Thesecondhelixissolderedattheouterconductorandisrolleduparoundthecable(figure1).Thethermotherapysystemconsists of a microwave generator (heating frequency 915 MHz and maximum power 100 W) and a microwave radiometer centered around 3 GHz for the measurement of the temperatures.

In order to take into account the heterogeneousness of the volume surrounding the antenna, but also the exact shape of tissues and applicator, a complete 3D model based on the well known FDTD method [3] has been developed. With this model, it is possible to know how the electromagnetic energy is deposited inside lossy media and, so to obtain thespecificabsorptionrate(SAR).Wecanalsodeterminethematchingoftheapplicatorinsidethesurroundingmedia at the heating frequency, but also in the radiometric frequency bandwidth. The heating pattern is then deduced from the resolution of the bio-heat transfer equation.

As to verify the theoretical results, experimental measurements have been carried out on phantom model of human tissues (polyacrylamide gel). First, the return loss (S11 parameter) has been measured as a function of frequency by means of a network analyzer HP 8510 in order to obtain the level of adaptation of the applicator at the heating frequency and in the radiometric bandwidth. The next part of the experiment consists in the determination of the energy distribution. The method is based on the temperature increase in a polyacrylamide gel, induced by microwave power for a short time (about one minute) in order to avoid thermal conduction phenomena inside the gel. The thermal performances of the applicator are obtained from temperatures measurement on a polyacrylamide gel after aheatingsessionofaboutfortyfiveminutesusinganautomaticexperimentalsystem

Results and discussionThe comparison between theoretical results and experimental measurements concerning the S11 parameter as a functionoffrequencyisshownonfigure2:wecanobservethatthematchingisquitegood.Thereflectioncoefficientisbelow-10dBattheheatingfrequency,thatistosaythatatleast90%oftheincidentpowerisdeliveredtothesurrounding media.

We can observe a maximum of power behind the junction plane of the two helix (the junction plane is the plane wherethetwohelixbegin).The40%isopowerlinespreadsonalengthnearlyequaltothetotalantennalength.Asuccession of power peaks appears in the vicinity of each metallic element corresponding to the helix. If we compare these results to the ones obtained with the previous urethral antenna, we can observe that the maximum of the SAR extends in the front of the junction plane of the applicator. So, we can conclude that the power deposition spreads on a less extensive zone for the helical applicator.

93/JNU OLE

ConclusionWe have studied a new kind of endocavitary applicator of helical type. The theoretical results (obtained from the FDTDmethod)andconfirmedbyexperimentalmeasurementsshowclearlyanimprovementofthepowerdepositionalong the coaxial cable, which will make possible to avoid potential burns near the bladder neck.

Fig. 1 Scheme of helical applicator

Fig. 2 Comparison between experimental measurements (dotted line) and theoreticalresults (full line) for the reflection coefficient (S11 parameter) as a function of frequency

obtained for the helical applicator dived in a polyacrylamid gel

Questions:Which material is used to realize helical antenna?1. AnswerThehelicalantennaisrealizedfromaflexiblecoaxialcablewith50Wcharacteristicimpedance.

According to the above study what improvements have been made in previous antenna?2. AnswerPrevious antennas were realized by removing the outer conductor of the cable on a length h. The improvement consistsinmakingadoublehelicalantenna,thefirsthelixistheinnerconductorwhichisrolleduparoundtheteflonsheath.Thesecondhelixissolderedattheouterconductorandisrolleduparoundthecable.

The thermotherapy system consists of which devices?3. AnswerThe thermotherapy system consists of a microwave generator (heating frequency 915 MHz and maximum power 100 W) and a microwave radiometer centered around 3 GHz for the measurement of the temperatures.


94/JNU OLE

Application II

Design And Analysis of Uwb Tem Horn Antenna for Ground Penetrating Radar Applications

IntroductionA TEM horn antenna is usually applied to the air-launching GPR system. It is made from two tapered metal plates, including exponentially tapered or linearly tapered. There are a narrow feed point and a wide open end at both sides of TEM horn antennas. Traditionally, the variation of characteristic impedance of a TEM horn antenna is usually set to range from 50 Ω (characteristic impedance of a coaxial cable) to 376.7 Ω (free space wave impedance). However, a difference regularly exists between transmission-line characteristic impedance and free space wave impedance.Sincetherewillbealargereflectionfromtheapertureoftheantenna,theconceptofmatchingtheimpedance at the antenna aperture to that of free space was proved not the best case. Therefore, in this paper, we designed several TEM horn antennas with different aperture impedances to better understand which impedance is better to be used in antenna aperture. From the simulated result, it shows that the performance of the designed TEM horn antenna matching with impedance of 200 Ω is better than that with free space impedance of 376.7Ω, but there is no huge difference of performance with the different aperture impedance from 200 Ω to 400 Ω of the antenna. It demonstrated that the matching impedance at the antenna aperture is not a critical factor to manipulate the TEM horn antenna performance.

Design approachBecauseTEMhornantennaiscomposedoftwotaperedmetalplates,thecurrentflowsonthesetwoplatesandtheTEMwavepropagatesbetweenthesetwoplatessimultaneously.ThecurrentflowingonthetwoplatesleadstothegenerationofthemagneticfieldsofTEMmodewave.ThevoltagedifferencebetweentwoplatesleadstothegenerationoftheelectricfieldsofTEMmodewave.ATEMhornantennacanbeconsideredtobeatransformerfromthe impedance of a transmission line to the impedance of the free space and the variation of characteristic impedance is usually designed to be between 50 ohms and 376.7 ohms. Since the characteristic impedance variation can be adjusted with the difference of the width of the plates and distance between two plates, the variation of characteristic impedancebetweentwoplateshastobecalculatedcarefullyinordertomakereflectioncoefficientassmallaspossible over a large frequency range. In general, there are four main steps in designing a TEM horn antenna.

Antenna geometryFollowing this design procedure, we can obtain the shape of the conductor plate and make the plates to be a linear TEM horn antenna. Since difference between the transmission-line wave characteristic impedance and the free space wave characteristic impedance usually exists, such difference should be taken into consideration in the design of TEM horn antenna. In order to determine the best matching impedance between the antenna aperture and the free space, different shapes of TEM horn antennas are designed based on different characteristic impedance at the antenna aperture.Thesevirtualantennamodelsareruninthefinitedifferencetimedomain(FDTD)methodbasedsoftwareXFDTD®.Theantennaaperturecharacteristicimpedanceisdesignedseparatelytobe200Ω, 250Ω, 300Ω, and up to 400Ω. There may be a little difference in the lengths of the antennas because of the requirements of matching section. Generally speaking, when matching 50 Ω to 100Ω, we get the shortest antenna length; when matching 50Ω to 400Ω,wegetthelongestantennalength.Theflareanglesbetweentwoconductorplatesareallfixedat20°.

FDTD simulated resultsThemostimportantfactortothepulseantennaiswidebandwidthwithlowreflectioncoefficient(S11). As for a usable GPR antenna, S11 should be at least less than -10 dB over a wide frequency range. Large S11 usually causes by the mismatch of the impedance between the feed cable and the antenna or the antenna and free space. The FDTD simulation of the TEM horn antennas can help us to understand better what the best design aperture impedance is.Fig.1showthesimulatedresultsofthereflectioncoefficient(S11) with different aperture impedances are from 200 Ω to 400Ω.WefindthatthereisnoobviousdifferenceofperformancesoftheTEMhornantennasasapertureimpedances ranging from 200Ω to 400Ω. Setting the antenna aperture impedances between 200Ω and 300Ω will result

95/JNU OLE

in better performance. Their S11 can be less than -15 dB from 1.2 GHz to 3 GHz. It seems that the best performance happens when aperture impedance is set to be 200Ω. If aperture impedance is set over than 300Ω, we can see that the performance of S11 becomes worse. With impedances ranging over 300Ω as shown in Fig. 1, the S11 is greater than -15dB; and the performance of low frequency band (below 1 GHz) is worse than that with impedances ranging from 200Ω to 300Ω It is obvious that the best matching impedance between the free space and the antenna aperture is not the free space wave impedance. To get the best performances of S11, the aperture impedance should be set within the range from 200Ω to 300Ω. For aperture impedance of 200Ω, the S11 is less than -15 dB from 1.0 GHz to 3 GHz, even close to -20 dB from 1.2 GHz up to 3 GHz.

ConclusionFrom this research, it demonstrates that for generalTEMhorn antennas, there is no significant difference ofperformance with the different characteristic impedance of the antenna aperture. By comparing the simulated results withdifferentapertureimpedances,wecanfindoutthatmatchingtheapertureimpedancetofreespaceimpedancedoes not guarantee the best matching performance. The antenna can obtain the best performance with aperture impedance of 200Ω..

Questions:

What is the antenna geometry used in the given application1. What is the design approach used in the above application?2. State the applications of TEM horn antenna.3.


96/JNU OLE

Application III

Uses of Electromagnetism in Life

IntroductionElectromagnetismhascreatedarevolutionnotonlyinthefieldofengineering,butalsoinvariousotherfieldslikemedicine, space, construction etc. Read here to know about various uses of electromagnetism in everyday life from household appliances to research labs.

Uses of Electromagnetism in lifeWhatever powered devices we use, from table clocks to microwave ovens, have some form of electromagnetic principleinvolvedintheirfunctioning.Itiselectromagnetismwhichhasgiventheflexibilityforswitchingof/onelectricity as required.

Electromagnets are created by having an iron core wound with a conductor carrying current. The strength of the electromagnet depends upon the amount of current passing through the conductor. Also the current can be easily stopped and started to form an electromagnet and de-energize respectively as per the need of the work to be performed. This is the principle used for moving heavy objects in the scrap yard. Electricity is connected to the circuit to power the electromagnets when they are energized. Thus the magnets start to attract scrap metal (junk cars), and carry them to the designated area. After locating them in a particular location, the electricity is disconnected from the circuit, thus de-energizing the electromagnet, making the scrap metal detach from the magnet.

Uses in home appliancesMany of our electrical home appliances use electromagnetism as a basic principle of working. If we take an example of an electric fan, the motor works on the principle of electromagnetic induction, which keeps it rotating on and on and thus making the blade hub of the fan to rotate, blowing air. Not restricting to fan, many other appliances use electromagnetism as a basic principle. Electric door bell works on this principle too. When the door bell button is put on, the coil gets energized, and due to the electromagnetic forces, the bell sounds. The working of an electric bell is discussed in detailed manner in one of our articles. The loudspeaker which we use for public announcements in meetings, or to transmit message over a long distance, is a perfect example for an electromagnetic appliance. The movement of the coil under the electromagnetic force produces sound which is heard over a very long distance.

Also the modern way of locking the door or a bank safe is to have a magnetic locking device. Either they may be having a number secret code or a magnetic card which when swiped opens the door. The number keys are stored in the magnetic tape on the back of the card, interacts with the magnetic card reader in the door. When the data stored on the card and the memory matches, the door opens. Similar principle is used in the bank’s safe lockers.

Power circuits and communication devicesThe telephones and mobiles we use to make a call over huge distances could have not taken shape with out electromagnetism. The interaction of the signals and the electromagnetic pulses, make the telephones and mobiles very handy. In power circuits, we use a device called relays, which has the potential to cut down a large current to the load, with the application of small amount of current. A small magnetic coil, which when energized, makes or brakes contact, thus doing a greater amount of work on the other end. Not to forget the usage of electromagnetism in medicalfield.EveryonemusthaveheardofMRIscans.MRI-istheacronymforMagneticResonanceImaging.Thissophisticated equipment can scan any minute details in the human body on the principle of electromagnetism

Thus it is evident that the usage of electromagnetism is wide and everywhere. Everyday more and more equipments take birth in the market due to the development in the magnetism and electromagnetic studies. The above applications are only a few of many uses of electromagnetism. It plays a very vital role in our day-to-day life.

97/JNU OLE

Questions:

What is Electromagnetism?1. State the uses of Electromagnetism in Home Appliances.2. State the uses of Electromagnetism power circuits and communication devices3.

.


98/JNU OLE

Bibliography

References

Antenna Basics• . Available at: <http://wireless.ictp.it/handbook/C4.pdf> [Accessed 17 February 2011].Antenna Fundamentals• . Available at: <http://etd.lib.fsu.edu/theses/available/etd-04102004-143656/unrestricted/Chapter2.pdf> [Accessed 17 February 2011].Bakshi, A.V., Bakshi, U.A., 2009. • Field Theory. Technical Publications.Bakshi, U.A., Bakshi, A.V., 2009. E• lectromagnetic Field Theory, Technical Publications.Bird, J., 2007. • Electrical circuit theory and technology. Newnes.Coulomb’s Law. Available at: <http://web.mit.edu/8.02t/www/802TEAL3D/visualizations/coursenotes/modules/•guide02.pdf)> [Accessed on 2 February 2011].Electrostatics• .Availableat:<http://www.nu.edu.sa/userfiles/iaalyafrosi/chapter%2004.pdf)>[Accessedon2

February 2011].Energy and Potential• . Available at: <http://www.ie.itcr.ac.cr/acotoc/Maestria_en_Computacion/Sistemas_de_Comunicacion_II/Material/Biblio2/chapt04.pdf>. [Accessed 13 February 2011].Laplace’s and Poisson’s Equations• . Available at: <http://www.ifm.liu.se/courses/TFYY67/Lect4.pdf> [Accessed 14 February 2010].Magnetic Force on a Moving Charge• . Available at: <http://www.ece.msstate.edu/~donohoe/ece3313notes8.pdf> [Accessed 15 February 2011].Potential Energy• . Available at: <http://www.scar.utoronto.ca/~pat/fun/NEWT3D/PDF/ENERGY3D.PDF>. [Accessed 9 February 2011].Time –Varying Fields and Maxwell’s Equations• . Available at: <http://www.ie.itcr.ac.cr/acotoc/Maestria_en_Computacion/Sistemas_de_Comunicacion_II/Material/Biblio2/chapt10.pdf> [Accessed 11 February 2011].Unit 3 - Electric Flux Density, Gauss’s Law and Divergence• . Available at: <http://www.engr.mun.ca/~egill/index_files/5812_w10/5812_unit3_2011.pdf>[Accessedon2 February 2011].

Recommended Reading

Balanis, C.A, 1989. • Advanced Engineering Electromagnetics. Wiley, Solution Manual edition.Balanis, C.A., 2005. • Antenna Theory: Analysis and Design, Wiley-Interscience, 3 ed.Cheng, D.K., 1992. • Fundamentals of Engineering Electromagnetics. Prentice Hall, 1st ed.Cullity, B.D., Graham, C.D., 2008. • Introduction to Magnetic Materials, Wiley-IEEE Press, 2 ed.Edminister, J., 2010. • Schaum’s Outline of Electromagnetics, McGraw-Hill. 3rd ed.Fleisch, D., 2008. • A Student’s Guide to Maxwell’s Equations. Cambridge University Press, 1 ed.Goldman, A., 2010. • Modern Ferrite Technology. Springer; 2 ed.Huray, P.G., 2009. • Maxwell’s Equations. Wiley-IEEE Press.Ida, N., 2004. • Engineering Electromagnetics. Springer, 2nd ed.Karmel, P.R., Colef , G.D., Camisa, R.L., 1997. • Introduction to Electromagnetic and Microwave Engineering. Wiley-Interscience, 1 ed.Kraus, J.D., Fleisch, D., 1999. • Electromagnetics, McGraw Hill Higher Education, 5th ed.Kraus, J.D., Marhefka, R.J., 2001. • Antennas for All Applications. McGraw-Hill Science/Engineering/Math, 3

ed.Minin, I.V., Minin, O.V., 2010. • Basic Principles of Fresnel Antenna Arrays. Springer, 1 ed.Rao, N.N., 2004. • Elements of Engineering Electromagnetics. Prentice Hall, 6th ed.

99/JNU OLE

Ryaben’kii, V.S., Kulman, N.K., (Translator), 2001. • Method of Difference Potentials and its Applications. Springer, 1st ed.Ulaby, F.T., 2006. • Fundamentals of Applied Electromagnetics. Prentice Hall, 5th ed.Yamaguchi, M., Tanimoto, Y., 2010. • Magneto-Science: Magnetic Field Effects on Materials: Fundamentals and Applications. Springer, 1st ed.


100/JNU OLE

Self Assessment Answers

Chapter Ia1. b2. d3. d4. a5. a6. c7. d8. a9. b10.

Chapter IId1. b2. c3. a4. b5. b6. c7. d8. a9. b10.

Chapter IIIa1. a2. c3. d4. a5. b6. c7. d8. a9. b10.

Chapter IVc1. b2. c3. d4. a5. b6. c7. d8. d9. b10.

101/JNU OLE

Chapter Va1. b2. a3. d4. a5. a6. b7. c8. d9. a10.

Chapter VIa1. a2. c3. d4. d5. b6. c7. d8. a9. b10.

electromagnetics and radiationjnujprdistance.com/assets/lms/lms jnu/mba/mba - telecom mana… ·...

Documents