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Unit I Static Electric Fields

Electromagnetic field

A changing magnetic field always produces an electric field, and conversely, a changingelectric field always produces a magnetic field. This interaction of electric and magneticforces gives rise to a condition in space known as an electromagnetic field. Thecharacteristics of an electromagnetic field are expressed mathematically by Maxwell'sequation.

Vector

A directed line segment. As such, vectors have magnitude and direction. Many physicaquantities, for example, velocity, acceleration, and force, are vectors.

Cross product

theCross Product is a binary operation on two vectors in a three dimensional!uclideanspace that results in another vector which is perpendicular to the two input vectors. "ycontrast, the dot product produces ascalar result. #n many engineering and physics problems, it is handy to be able to construct a perpendicular vector from two existingvectors, and the cross product provides a means for doing so. The cross product is alsoknown as thevector product , or Gibbs vector product .

The cross product is not defined except in three dimensions $and thealgebradefined bythe cross product is notassociative%. &ike the dot product, it depends on themetric of !uclidean space. nlike thedot product, it also depends on the choice oforientation or (handedness(. )ertain features of the cross product can be generali*ed to other situations.+or arbitrary choices of orientation, the cross product must be regarded not as a vector, but as a pseudovector . +or arbitrary choices of metric, and in arbitrary dimensions, thecross product can be generali*ed by theexterior product of vectors, defining atwo forminstead of a vector.

+ig . #llustration of the cross product in respect to a right handed coordinate system.
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+ig .- +inding the direction of the cross product by theright hand rule.

The cross product of two vectorsa andb is denoted bya b . #n a three dimensional!uclidean space, with a usualright handed coordinate system, it is defined as a vectorcthat is perpendicularto botha andb , with a direction given by theright hand rule and amagnitude equal to the area of the parallelogram that the vectors span.

The cross product is given by the formula

where is the measure of theangle betweena andb $/0 1 1 2/0%,a andb are themagnitudesof vectorsa and b , and is aunit vector perpendicular to the planecontaininga andb . #f the vectorsa andb are collinear $i.e., the angle between them iseither /0 or 2/0%, by the above formula, the cross product ofa andb is the *ero vector0.

The direction of the vector is given by the right hand rule, where one simply points the

forefinger of the right hand in the direction ofa and the middle finger in the direction of b . Then, the vector is coming out of the thumb $see the picture on the right%.

sing the cross product requires the handedness of the coordinate system to be taken intoaccount $as explicit in the definition above%. #f aleft handed coordinate system is used,the direction of the vector is given by the left hand rule and points in the oppositedirection.

Dot product

Thedot product , also known as thescalar product , is an operation which takes two

vectors over thereal numbers R and returns a real valued scalar quantity.
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where

3a3 and 3b3 denote thelength$magnitude% ofa andb 4 is theangle between them.

5ince 3a3cos$4% is thescalar pro6ection of a ontob , the dot product can be understoodgeometrically as the product of this pro6ection with the length ofb .

3a37cos$4% is thescalar pro6ection ofa ontob

Coordinate S stem

A coordinate system is a mathematical language that is used to describe geometrical

ob6ects analyticallyA cartesian coordinate systemis one of the simplest and most useful systems of coordinates. #t is constructed by choosing a pointO designated as the origin. Through itthree intersecting directed linesOX, OY, OZ , the coordinate axes, are constructed. Thecoordinates of a point P are x, the distance of P from the planeYOZ measured parallel toOX , and y and z , which are determined similarly $+ig. %. sually the three axes are takento be mutually perpendicular , in which case the system is a rectangularcartesian one.8bviously a similar construction can be made in the plane, in which case a point has twocoordinates $ x,y%.

fig .9 Cartesian coordinate system.
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C lindrical Coordinate S stem

The c lindrical coordinate s stem is a three dimensionalcoordinate systemwhichessentially extends circular polar coordinates by adding a third coordinate $usuallydenotedh% which measures the height of a point above the plane.

A point : is given as $r ,4,h%. #n terms of the )artesian coordinate system;

r is the distance from 8 to :', the orthogonal pro6ection of the point : onto the


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and and lose significance when / and loses significance when sin$% / $at / and 2/0%.

To plot a point from its spherical coordinates, go units from the origin along the positive * axis, rotate about the y axis in the direction of the positive x axis and rotate

about the * axis in the direction of the positive y axis.Coordinate s stem conversions

As the spherical coordinate system is only one of many three dimensional coordinatesystems, there exist equations for converting coordinates between the sphericalcoordinate system and others.

Cartesian coordinate s stem

The three spherical coordinates are obtained from )artesian coordinates by;

Bote that the arctangent must be defined suitably so as to take account of the correctquadrant of y C x. Theatan- or equivalent function accomplishes this for computational purposes.

)onversely, )artesian coordinates may be retrieved from spherical coordinates by;

Divergence of a Vector Field:

In study of vector fields, directed line segments, also called flux lines or streamlines,represent field variations graphically. The intensity of the field is proportional to the density of lines. For example, the number of flux lines passing through a unit surface S normal to thevector measures the vector field strength.
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Fig 1.5: Flux Lines

We have already defined flux of a vector field as

....................................................(1.1)

For a volume enclosed by a surface,

.........................................................................................(1. )

We define the divergence of a vector field at a point P as the net out!ard flux from avolume enclosing P , as the volume shrin"s to #ero.

.................................................................(1.$)

%ere is the volume that encloses P and S is the corresponding closed surface.

&et us consider a differential volume centered on point :$u, ,! % in a vector field . The fluxthrough an elementary area normal to u is given by ,

........................................(1.')


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et out!ard flux along u can be calculated considering the t!o elementary surfaces perpendicular to u .

.......................................(1. )

*onsidering the contribution from all six surfaces that enclose the volume, !e can !rite

.......................................(1.+)

%ence for the *artesian, cylindrical and spherical polar coordinate system, the expressions for divergence can b!ritten as

In Cartesian coordinates:

................................(1.-)

Fig 1.6 Evaluation of divergence in curvilinear coordinate

In cylindrical coordinates

....................................................................(1. )


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and in spherical polar coordinates

......................................(1./)

In connection !ith the divergence of a vector field, the follo!ing can be noted

0ivergence of a vector field gives a scalar.

..............................................................................(1.1 )

Divergence theorem :

0ivergence theorem states that the volume integral of the divergence of vector field is e2ualto the net out!ard flux of the vector through the closed surface that bounds the volume.

3athematically,

roof:

&et us consider a volume " enclosed by a surface S . &et us subdivide the volume in large

number of cells. &et the # th cell has a volume and the corresponding surface is denotedby S k . Interior to the volume, cells have common surfaces. 4ut!ard flux through thesecommon surfaces from one cell becomes the in!ard flux for the neighboring cells. Therefore

!hen the total flux from these cells are considered, !e actually get the net out!ard fluxthrough the surface surrounding the volume. %ence !e can !rite

......................................(1.11)

In the limit, that is !hen and the right hand of the expression can be

!ritten as .

%ence !e get , !hich is the divergence theorem.


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Curl of a vector field:

We have defined the circulation of a vector field $ around a closed path as .

Curl of a vector field is a measure of the vector field5s tendency to rotate about a point. *url

, also !ritten as is defined as a vector !hose magnitude is maximum of the netcirculation per unit area !hen the area tends to #ero and its direction is the normal directionto the area !hen the area is oriented in such a !ay so as to ma"e the circulation maximum.

Therefore, !e can !rite

......................................(1.1 )

To derive the expression for curl in generali#ed curvilinear coordinate system, !e first

compute and to do so let us consider the figure 1.-

Fig 1.! Curl of a Vector

If C 1 represents the boundary of , then !e can !rite

......................................(1.1$)

The integrals on the 6%7 can be evaluated as follo!s

.................................(1.1')

................................................(1.1 )

The negative sign is because of the fact that the direction of traversal reverses. 7imilarly,


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..................................................(1.1+)

............................................................................(1.1-)

8dding the contribution from all components, !e can !rite

........................................................................(1.1 )

Therefore, ......................................................(1.1/)

In the same manner if !e compute for and !e can !rite,

.......(1

This can be !ritten as,

......................................................(1. 1)

In *artesian coordinates .......................................(1. )


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In *ylindrical coordinates, ....................................(1. $)

In 7pherical polar coordinates, ..............(1. ')

*url operation exhibits the follo!ing properties

..............(1. )

"to#e$s theorem :

It states that the circulation of a vector field around a closed path is e2ual to the integral of

over the surface bounded by this path. It may be noted that this e2uality holds

provided and are continuous on the surface.

i.e,

..............(1. +)

roof: &et us consider an area 7 that is subdivided into large number of cells as sho!n in the

figure 1.


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Fig 1.% "to#es theorem

&et k t hcell has surface area and is bounded path %" !hile the total area isbounded by path %. 8s seen from the figure that if !e evaluate the sum of the lineintegrals around the elementary areas, there is cancellation along every interiorpath and !e are left the line integral along path %. Therefore !e can !rite,

..............(1. -)

As

. .............(1. )

!hich is the sto"e5s theorem.

Coulomb"s #a$

Coulomb"s #a$ may be stated as follows;&'he ma(nitude of the e)ectrostatic force bet!een t!o *oint char(es is direct)y

*ro*ortiona) to the ma(nitudes of each char(e and in erse)y *ro*ortiona) to the s+uareof the distance bet!een the char(es.&

)oulomb's law states that the electrical force between two charged ob6ects is directly proportional to the product of the quantity of charge on the ob6ects and inversely


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proportional to the square of the separation distance between the two ob6ects. #n equatioform, )oulomb's law can be stated as

DDDDDDDDDD$ .-E%

where%& represents the quantity of charge on ob6ect $in )oulombs%,%' represents thequantity of charge on ob6ect - $in )oulombs%, andd represents the distance of separation between the two ob6ects $in meters%. The symbol( is a proportionality constant known asthe )oulomb's law constant. The value of this constant is dependent upon the mediumthat the charged ob6ects are immersed in.

3athematically, ,!here # is the proportionality constant.

In 7I units, Q 1 and Q are expressed in *oulombs(*) and R is in meters.

Force F is in e!tons ( N ) and , is called the permittivity of free space.

(We are assuming the charges are in free space. If the charges are any other dielectric

medium, !e !ill use instead !here is called the relative permittivity or thedielectric constant of the medium).

Therefore .......................(1.$ )

8s sho!n in the Figure .1 let the position vectors of the point charges Q1and Q are given

by and . &et represent the force on Q 1 due to charge Q .

Fig 1.&: Coulom'$s La(


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The charges are separated by a distance of . We define the unit vectorsas

and ..................................(1.$1)

can be defined as

. 9..(1.$ )

7imilarly the force on due to charge - can be calculated and if represents this force

then !e can !rite

When !e have a number of point charges, to determine the force on a particular charge dueto all other charges, !e apply principle of superposition. If !e have N number of charges

Q 1,Q ,......... Q located respectively at the points represented by the position vectors ,

,...... , the force experienced by a charge Q located at is given by,

.................................(1.$$)

Electric Field

The electric field intensity or the electric field strength at a point is defined as the force perunit charge. That is

or, .......................................(1.$')

The electric field intensity E at a point r (observation point) due a point charge Q located at

(source point) is given by

..........................................(1.$ )


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For a collection of - point charges , - ,......... B located at , ,...... , the electric field intensity at point obtained as

........................................(1.$+)

The expression ( .+) can be modified suitably tocompute the electric filed due to a continuousdistribution of charges.

In figure 1.1 !e consider a continuous volumedistribution of charge t/ in the region denoted as thesource region.

For an elementary charge , i.e.considering this charge as point charge, !e can !rite

the field expression as

.............( .-)

Fig1.1): Continuous Volume Distri'ution

When this expression is integrated over the source region, !e get the electric field at thepoint P due to this distribution of charges. Thus the expression for the electric field at P canbe !ritten as

..........................................(1.$-)

7imilar techni2ue can be adopted !hen the charge distribution is in the form of a line chargedensity or a surface charge density.

........................................(1.$ )

........................................(1.$/)


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Electric field strengt!

!lectric field strength isa vector quantityF it has both magnitude and direction. Themagnitude of the electric field strength is defined in terms of how it is measured. &et'ssuppose that an electric charge can be denoted by the

symbol%. This electric charge creates an electric fieldFsince% is the source of the electric field, we will refer toit as the source c!arge . The strength of the sourcecharge's electric field could be measured by any other charge placed somewhere in its surroundings. The chargethat is used to measure the electric field strength is referred to as atest c!arge since it isused totest the field strength. The test charge has a quantity of charge denoted by thesymbol) . Ghen placed within the electric field, the test charge will experience anelectric force either attractive or repulsive. As is usually the case, this force will bedenoted by the symbolF . The magnitude of the electric field is simply defined as theforce per charge on the test charge.

#f the electric field strength is denoted by the symbolE , then the equation can berewritten in symbolic form as

.

The standard metric units on electric field strength arise from its definition. 5ince electricfield is defined as a force per charge, its units would be force units divided by chargeunits. #n this case, the standard metric units are BewtonC)oulomb or BC).

Electric Field #ines

The magnitude or strength of an electric field in the space surrounding a source charge isrelated directly to the quantity of charge on the source charge and inversely to thedistance from the source charge. The direction of the electric field is always directed inthe direction that a positive test charge would be pushed or pulled if placed in the spacesurrounding the source charge. 5ince electric field is a vector quantity, it can berepresented by a vector arrow. +or any given location, the arrows point in the direction ofthe electric field and their length is proportional to the strength of the electric field at thatlocation. 5uch vector arrows are shown in the diagram below. Bote that the length of thearrows are longer when closer to the source charge and shorter when further from thesource charge.
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Electric Fields Inside of C!arged Conductors

)harged conductors which have reached electrostatic equilibrium share a variety of unusual characteristics. 8ne characteristic of a conductor at electrostatic equilibrium isthat the electric field anywhere beneath the surface of a charged conductor is *ero. #f an

electric field did exist beneath the surface of a conductor $and inside of it%, then thelectric field would exert a force on all electrons that were present there. This net forcewould begin to accelerate and move these electrons. "ut ob6ects at electrostaticequilibrium have no further motion of charge about the surface. 5o if this were to occur,then the original claim that the ob6ect was at electrostatic equilibrium would be a falseclaim. #f the electrons within a conductor have assumed an equilibrium state, then the neforce upon those electrons is *ero. The electric field lines either begin or end upon acharge and in the case of a conductor, the charge exists solely upon its outer surface. Thelines extend from this surface outward, not inward. This of course presumes that ourconductor does not surround a region of space where there was another charge.

To illustrate this characteristic, let's consider the space between and inside of twoconcentric, conducting cylinders of different radii as shown in the diagram at the right.The outer cylinder is charged positively. The inner cylinder ischarged negatively. The electric field about the inner cylinder is directed towards the negatively charged cylinder. 5ince thiscylinder does not surround a region of space where there isanother charge, it can be concluded that the excess chargeresides solely upon the outer surface of this inner cylinder. Theelectric field inside the inner cylinder would be *ero. Ghendrawing electric field lines, the lines would be drawn from theinner surface of the outer cylinder to the outer surface of the

inner cylinder. +or the excess charge on the outer cylinder, there is more to consider thanmerely the repulsive forces between charges on its surface. Ghile the excess charge onthe outer cylinder seeks to reduce repulsive forces between its excess charge, it must balance this with the tendency to be attracted to the negative charges on the inner cylinder. 5ince the outer cylinder surrounds a region which is charged, the characteristicof charge residing on the outer surface of the conductor does not apply.

This concept of the electric field being *ero inside of a closed conducting surface wasfirst demonstrated by Michael +araday, a Eth century physicist who promoted the fieldtheory of electricity. +araday constructed a room within a room, covering the inner roomwith a metal foil. Ie sat inside the inner room with an electroscope and charged thesurfaces of the outer and inner room using an electrostatic generator. Ghile sparks wereseen flying between the walls of the two rooms, there was no detection of an electric fieldwithin the inner room. The excess charge on the walls of the inner room resided entirelyupon the outer surface of the room.

The inner room with the conducting frame which protected +araday from the staticcharge is now referred to as aFarada "s cage . The cage serves to shield whomever andwhatever is on the inside from the influence of electric fields. Any closed, conductingsurface can serve as a +araday's cage, shielding whatever it surrounds from the


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Electric Fields and Surface Curvature

A third characteristic of conducting ob6ects at electrostatic equilibrium is that the electrifields are strongest at locations along the surface where the ob6ect is most curved. Thecurvature of a surface can range from absolute flatness on one extreme to being curved to

a b)unt point on the other extreme.

A flat location has no curvature and is characteri*ed by relatively weak electric fields. 8nthe other hand, ab)unt *oint has a high degree of curvature and is characteri*ed by

relatively strong electric fields. A sphere is uniformly shaped with the same curvature atevery location along its surface. As such, the electric field strength on the surface of asphere is everywhere the same.

To understand the rationale for this third characteristic, we will consider an irregularlyshaped ob6ect which is negatively charged. 5uch an ob6ect has an excess of electronsThese electrons would distribute themselves in such a manner as to reduce the affect oftheir repulsive forces. 5ince electrostatic forces vary inversely with thesquare of the distance, these electrons would tend to position themselvesso as to increase their distance from one another. 8n a regularly shapedsphere, the ultimate distance between every neighboring electron would

be the same. "ut on an irregularly shaped ob6ect, excess electrons wouldtend to accumulate in greater density along locations of greatestcurvature. )onsider the diagram at the right. !lectrons A and " arelocated along a flatter section of the surface. &ike all well behavedelectrons, they repel each other. The repulsive forces are directed along aline connecting charge to charge, making the repulsive force primarily parallel to thesurface. 8n the other hand, electrons ) and J are located along a section of the surfacewith a sharper curvature. These excess electrons also repel each other with a forcedirected along a line connecting charge to charge. "ut now the force is directed at asharper angle to the surface. The components of these forces parallel to the surface areconsiderably less. A ma6ority of the repulsive force between electrons ) and J is directed

perpendicular to the surface.The parallel components of these repulsive forces is what causes excess electrons tomove along the surface of the conductor. The electrons will move and distributethemselves until electrostatic equilibrium is reached. 8nce reached, the resultant of all parallel components on any given excess electron $and on all excess electrons% will adup to *ero. All the parallel components of force on each of the electrons must be *erosince the net force parallel to the surface of the conductor is always *ero $thesecondcharacteristic discussed above%. +or the same separation distance, the parallel component
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of force is greatest in the case of electrons A and ". 5o to acquire this balance of parallelforces, electrons A and " must distance themselves further from each other than electrons) and J. !lectrons ) and J on the other hand can crowd closer together at their locationsince that the parallel component of repulsive forces is less. #n the end, a relatively largequantity of charge accumulates on the locations of greatest curvature. This larger quantity

of charge combined with the fact that their repulsive forces are primarily directed perpendicular to the surface results in a considerably stronger electric field at suchlocations of increased curvature.

The fact that surfaces which are sharply curved to a blunt edge create strong electricfields is the underlying principle for the use of lightning rods.

Electric scalar PotentialIn the previous sections !e have seen ho! the electric fieldintensity due to a charge or a charge distribution can be foundusing *oulomb5s la! or :auss5s la!. 7ince a charge placed in thevicinity of another charge (or in other !ords in the field of othercharge) experiences a force, the movement of the chargerepresents energy exchange. ;lectrostatic potential is related to the!or" done in carrying a charge from one point to the other in thepresence of an electric field.

&et us suppose that !e !ish to move a positive test chargefrom a point P to another point as sho!n in the Fig. 1.11

The force at any point along its path !ould cause the particle toaccelerate and move it out of the region if unconstrained. 7ince !eare dealing !ith an electrostatic case, a force e2ual to the negative

of that acting on the charge is to be applied !hile moves from P to . The !or" done by this external agent in moving the charge by

a distance is given by

Fig 1.11 *ovement of +est Charge in ElectricField


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.............................(1.' )

The negative sign accounts for the fact that !or" is done on the system by the externalagent.

.....................................(1.'1)

The potential difference bet!een t!o points P and , " P , is defined as the !or" done perunit charge, i.e.

...............................(1.' )

It may be noted that in moving a charge from the initial point to the final point if the potentialdifference is positive, there is a gain in potential energy in the movement, external agentperforms the !or" against the field. If the sign of the potential difference is negative, !or" isdone by the field.

We !ill see that the electrostatic system is conservative in that no net energy is exchanged if the test charge is moved about a closed path, i.e. returning to its initial position. Further, thepotential difference bet!een t!o points in an electrostatic field is a point function< it isindependent of the path ta"en. The potential difference is measured in =oules>*oulomb!hich is referred to as Volts .

&et us consider a point charge as sho!n in the Fig. 1.1

Fig 1.1, Electrostatic otential calculation for a -oint charge


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Further consider the t!o points $ and 0 as sho!n in the Fig.1.1 . *onsidering the movementof a unit positive test charge from 0 to $ , !e can !rite an expression for the potentialdifference as

..................................(1.'$)

It is customary to choose the potential to be #ero at infinity. Thus potential at any point ( r $ ?r ) due to a point charge @ can be !ritten as the amount of !or" done in bringing a unitpositive charge from infinity to that point (i.e. r 0 ? ).

..................................(1.'')

4r, in other !ords,

..................................(1.' )

&et us no! consider a situation !here the point charge is not located at the origin assho!n in Fig. 1.1$.

Fig 1.1 : Electrostatic otential due a Dis-laced Charge

The potential at a point P becomes

..................................(1.'+)

7o far !e have considered the potential due to point charges only. 8s any other type ofcharge distribution can be considered to be consisting of point charges, the same basicideas no! can be extended to other types of charge distribution also.


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&et us first consider N point charges Q 1, Q 2 ,.....Q N located at points !ith position vectors ,

,...... . The potential at a point having position vector can be !ritten as

..................................(1.'-)

or , ...........................................................(1.' )

For continuous charge distribution, !e replace point charges Q n by corresponding charge

elements or or depending on !hether the charge distribution is linear,surface or a volume charge distribution and the summation is replaced by an integral. With

these modifications !e can !rite

For line charge, ..................................(1.'/)

For surface charge, .................................(1. )

For volume charge, .................................(1. 1)

It may be noted here that the primed coordinates represent the source coordinates and theunprimed coordinates represent field point.

Further, in our discussion so far !e have used the reference or #ero potential at infinity. Ifany other point is chosen as reference, !e can !rite

.................................(1. )

!here C is a constant. In the same manner !hen potential is computed from a "no!nelectric field !e can !rite

.................................(1. $)

The potential difference is ho!ever independent of the choice of reference.


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.......................(1. ')

We have mentioned that electrostatic field is a conservative field< the !or" done in moving acharge from one point to the other is independent of the path. &et us consider moving acharge from point P to P - in one path and then from point P bac" to P 1 over a different path.If the !or" done on the t!o paths !ere different, a net positive or negative amount of !or"!ould have been done !hen the body returns to its original position P 1. In a conservativefield there is no mechanism for dissipating energy corresponding to any positive !or" neither any source is present from !hich energy could be absorbed in the case of negative !or".%ence the 2uestion of different !or"s in t!o paths is untenable, the !or" must have to beindependent of path and depends on the initial and final positions.

7ince the potential difference is independent of the paths ta"en, " $0 ? A " 0$ , and over aclosed path,

.................................(1. )

8pplying 7to"es5s theorem, !e can !rite

............................(1. +)

from !hich it follo!s that for electrostatic field,

........................................(1. -)

8ny vector field that satisfies is called an irrotational field.

From our definition of potential, !e can !rite

.................................(1. )

from !hich !e obtain,

..........................................(1. /)


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From the foregoing discussions !e observe that the electric field strength at any point is the negative of the pote

gradient at any point, negative sign sho!s that is directed from higher to lo!er values of . This gives us anomethod of computing the electric field, i. e. if !e "no! the potential function, the electric field may be computed.

may note here that that one scalar function contain all the information that three components of carry, the s

possible because of the fact that three components of are interrelated by the relation .Exam-le: Electric Di-ole

8n electric dipole consists oft!o point charges of e2ualmagnitude but of opposite signand separated by a smalldistance.

8s sho!n in figure 1.1', thedipole is formed by the t!opoint charges and - separated by a distance d , thecharges being placedsymmetrically about the origin.&et us consider a point P at adistance r , !here !e areinterested to find the field.

Fig 1.1/ : Electric Di-ole

The potential at B due to the dipole can be !ritten as

..........................(1.+ )

When r and r CCd , !e can !rite and .

Therefore,

....................................................(1.+ )

We can !rite,

...............................................(1.+$)

The 2uantity is called the di-ole moment of the electric dipole.

%ence the expression for the electric potential can no! be !ritten as

................................(1.+')


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It may be noted that !hile potential of an isolated charge varies !ith distance as 1> r that ofan electric dipole varies as 1> r !ith distance.

If the dipole is not centered at the origin, but the dipole center lies at , the expression forthe potential can be !ritten as

........................(1.+ )

The electric field for the dipole centered at the origin can be computed as

........................(1.++)

is the magnitude of the dipole moment. 4nce again !e note that the electric field ofelectric dipole varies as 1> r $ !here as that of a point charge varies as 1> r .

Electric flux densit0:

8s stated earlier electric field intensity or simply D;lectric field5 gives the strength of the fieldat a particular point. The electric field depends on the material media in !hich the field isbeing considered. The flux density vector is defined to be independent of the material media(as !e5ll see that it relates to the charge that is producing it).For a linear

isotropic medium under consideration< the flux density vector is defined as

................................................(1.+-)

We define the electric flux as

.....................................(1.+ )

auss$s La(: :auss5s la! is one of the fundamental la!s of electromagnetism and it statesthat the total electric flux through a closed surface is e2ual to the total charge enclosed bythe surface.


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Fig 1.15 auss$s La(

&et us consider a point charge located in an isotropic homogeneous medium of dielectricconstant . The flux density at a distance r on a surface enclosing the charge is given by

...............................................(1.+/)

If !e consider an elementary area d s , the amount of flux passing through the elementaryarea is given by

.....................................(1.- )

Eut , is the elementary solid angle subtended by the area at the location of

. Therefore !e can !rite

For a closed surface enclosing the charge, !e can !rite

!hich can seen to be same as !hat !e have stated in the definition of :auss5s &a!.

2--lication of auss$s La(

:auss5s la! is particularly useful in computing or !here the charge distribution hassome symmetry. We shall illustrate the application of :auss5s &a! !ith some examples.

1.2n infinite line charge

8s the first example of illustration of use of :auss5s la!, let consider the problem ofdetermination of the electric field produced by an infinite line charge of density &*>m. &et usconsider a line charge positioned along the z Aaxis as sho!n in Fig.1.1+(a) (next slide). 7incethe line charge is assumed to be infinitely long, the electric field !ill be of the form as sho!nin Fig. .'(b) (next slide).


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If !e consider a close cylindrical surface as sho!n in Fig.1.1+(a), using :auss5s theorm !ecan !rite,

.....................................(1.-1)

*onsidering the fact that the unit normal vector to areas S 1 and S $ are perpendicular to theelectric field, the surface integrals for the top and bottom surfaces evaluates to #ero. %ence

!e can !rite,

Fig 1.16 Infinite Line Charge

.....................................(1.- )


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,. Infinite "heet of Charge

8s a second example of application of :auss5stheorem, !e consider an infinite charged sheetcovering the x1z plane as sho!n in figure 1.1-

8ssuming a surface charge density of for theinfinite surface charge, if !e consider a cylindricalvolume having sides placed symmetrically , !ecan !rite

..............(1.-$)Fig1.1!: Infinite "heet of ChargeIt may be noted that the electric field strength is independent of distance. This is true for the infinite plane of

charge< electric lines of force on either side of the charge !ill be perpendicular to the sheet and extend to

infinity as parallel lines. 8s number of lines of force per unit area gives the strength of the field, the field

becomes independent of distance. For a finite charge sheet, the field !ill be a function of distance.

. 3niforml0 Charged "-here

&et us consider a sphere of radius r / having auniform volume charge density of v *>m $. To

determine every!here, inside and outside thesphere, !e construct :aussian surfaces of radius r 2r and r 3 r as sho!n in Fig. 1.1 (a) and Fig.1.1 (b).

For the region < the total enclosed charge !illbe

.........................(1.-') Fig 1.1% 3niforml0 Charged "-here

Ey applying :auss5s theorem,

...............(1.- )


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Therefore

...............................................(1.-+)

For the region < the total enclosed charge !ill be

....................................................................(1.--)

Ey applying :auss5s theorem,

.....................................................(1.- )

Unit II Static *agnetic Field

Introduction :


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In previous chapters !e have seen that an electrostatic field is produced by static orstationary charges. The relationship of the steady magnetic field to its sources is much morecomplicated.

The source of steady magnetic field may be a permanent magnet, a direct current or anelectric field changing !ith time. In this chapter !e shall mainly consider the magnetic field

produced by a direct current. The magnetic field produced due to time varying electric field!ill be discussed later. %istorically, the lin" bet!een the electric and magnetic field !asestablished 4ersted in 1 . 8mpere and others extended the investigation of magneticeffect of electricity . There are t!o ma or la!s governing the magnetostatic fields are

EiotA7avart &a! 8mpere5s &a!

Gsually, the magnetic field intensity is represented by the vector . It is customary torepresent the direction of the magnetic field intensity (or current) by a small circle !ith a dotor cross sign depending on !hether the field (or current) is out of or into the page as sho!nin Fig. .1.

(or l ) out of the page (or l ) into the page

Fig. ,.1: 4e-resentation of magnetic field or current

7iot8 "avart La(

This la! relates the magnetic field intensity dH produced at a point due to a differential

current element as sho!n in Fig. . .

Fig. ,.,: *agnetic field intensit0 due to a current element


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The magnetic field intensity at B can be !ritten as,

............................( .1a)

..............................................( .1b)

!here is the distance of the current element from the point B.

7imilar to different charge distributions, !e can have different current distribution such as linecurrent, surface current and volume current. These different types of current densities aresho!n in Fig. .$.

&ine *urrent 7urface *urrent Holume *urrent

Fig. ,. : Different t0-es of current distri'utions


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Ey denoting the surface current density as (in amp>m) and volume current density as = (inamp>m ) !e can !rite

......................................( . )

( It may be noted that )

;mploying EiotA7avart &a!, !e can no! express the magnetic field intensity %. In terms of these current distributions.

............................. for line current. ...........................( .$a)

........................ for surface current ....................( .$b)

....................... for volume current ......................( .$c)To illustrate the application of Eiot A 7avart5s &a!, !e consider the follo!ing example.

Exam-le ,.1: We consider a finite length of a conductor carrying a current placed along #Aaxis as sho!n in the Fig .'. We determine the magnetic field at point B due to this currentcarrying conductor.

Fig. ,./: Field at a -oint due to a finite length current carr0ing conductor

With reference to Fig. .', !e find that

..........................................( .')

8pplying Eiot A 7avart5s la! for the current element


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!e can !rite,

..............................................( . )

7ubstituting !e can !rite,

..............( .+)

We find that, for an infinitely long conductor carrying a current I , and

Therefore, .........................................................................................( .-)

The value of the constant of proportionality ' K ' depends upon a property called permeability of the medium around the conductor. :ermeability is represented bysymbol"m" and the constant ' K 4 is expressed in terms of"m" as

Magnetic field '+" is a vector and unless we give the direction of 'd+ ', its description isnot complete. #ts direction is found to be perpendicular to the plane of 'r ' and 'dl '.

#f we assign the direction of the current 'I ' to the length element 'dl ', the vector productdl x r has magnituder dl sin) and direction perpendicular to 'r ' and 'dl '.

Ience, "iotK5avart law can be stated in vector form to give both the magnitude as wellas direction of magnetic field due to a current element as

Lalue of permeability changes from medium to medium. +or ferromagnetic materials itis much higher than that for other materials. The permeability of free space $vacuum% denoted by the symbol 'm 0' and its value is,p - &0 ./ GbCAm


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2m-ere$s Circuital La(:

8mpere5s circuital la! states that the line integral of the magnetic field (circulation of H )around a closed path is the net current enclosed by this path. 3athematically,

......................................( '. )

The total current I enc can be !ritten as,

......................................( './)Ey applying 7to"e5s theorem, !e can !rite

......................................( .1 )!hich is the 8mpere5s la! in the point form.

2--lications of 2m-ere$s la(:

We illustrate the application of 8mpere5s &a! !ith some examples.

Exam-le,.,: We compute magnetic field due to an infinitely long thin current carrying

conductor as sho!n in Fig. . . Gsing 8mpere5s &a!, !e consider the close path to be acircle of radius as sho!n in the Fig. '. .

If !e consider a small current element , is perpendicular to the plane

containing both and . Therefore only component of that !ill be present is

,i.e., .

Ey applying 8mpere5s la! !e can !rite,

......................................( .11)

Therefore, !hich is same as e2uation ( .-)


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Fig. ,.5: *agnetic field due to an infinite thin current carr0ing conductor

Exam-le ,. : We consider the cross section of an infinitely long coaxial conductor, theinner conductor carrying a current I and outer conductor carrying current A I as sho!n in

figure .+. We compute the magnetic field as a function of as follo!s

In the region

......................................( .1 )

............................( .1$)

In the region

......................................( .1')

Fig. ,.6: Coaxial conductor carr0ing e9ual and o--osite currents


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In the region

......................................( .1 )

........................................( .1+)

In the region

......................................( .1-)#orent force

A charged particle at rest will not interact with a static magnetic field. "ut if the charged particle is moving in a magnetic field, the magnetic character of a charge in motion becomes evident. #t experiences a deflecting force. The force is greatest when the particle moves in a direction perpendicular to the magnetic field lines. At other angles,the force is less and becomes *ero when the particles move parallel to the field lines. #nany case, the direction of the force is always perpendicular to the magnetic field linesand to the velocity of the charged particle.

*agnetic Flu- Densit

The amount of magnetic flux through a unit area taken perpendicular to the direction ofthe magnetic flux. Also calledma(netic induction .

Definition 1f 2mpere

Ghen two current carrying conductors are placed next to each other, we notice that eachinduces a force on the other. !ach conductor produces a magnetic field around itself $"iotK5avart law% and the second experiences a force that is given by the &orent* force


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which is a very powerful tool for calculating electric fields.

2pplication of 2mpere"s la$3

Ampere's law can be used to calculate '+ ' for various current carrying conductor configurations.

Gauss"s #a$

Gauss"s la$ for magnetic field This law deals with magnetic flux inside a closedsurface and is equivalent to auss's law for electric field discussed in !lectric )hargeand !lectric +ield, connected electric flux 6 ! and electric charge.

And 4 E E . 2

5imilarly, magnetic fluxf + can be defined as the number of lines of force crossing a unitarea.

Magnetic fluxf + +52

5ince there are no free magnetic charges, the magnetic flux crossing a closed surfacewill always be *ero. Thus auss's law of magnetic field says that the net magneticfluxf + out of any closed surface is *ero.

or+52 6 0

#en "s la$

5oon after +araday proposed his law of electromagnetic induction, &en* gave the lawdetermining the direction of the induced emf.

&en*'s law may be stated as follows;

The direction of the induced current is such as to oppose the cause producing it.

&en*'s law can be compared with the Bewton's third law N every action has equal andopposite reaction.

Ghen an emf is generated by a change in magnetic flux according to the +araday's law,the polarity of the induced emf is such that it produces a current whose magnetic fieldopposes the change that produces it. The induced magnetic field inside any loop of wirealways acts to keep the magnetic flux in the loop constant.

#n the examples below, if the '"' field is increasing, the induced field acts in oppositionto it. #f it is decreasing, the induced field acts in the direction of the applied field to tryto keep it constant.


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*agnetic Flu-

+araday understood that the magnitude of the induced current in a loop was due to the(amount of magnetic field( passing through the loop.

To visuali*e this (amount of magnetic field(, which is now called the magnetic flux, heintroduced a mental picture of magnetic field as lines of force. This is exactly analogousto electric flux.

Magnetic flux is the product of the '"' times the perpendicular area that it penetrates.

The contribution to 6" for a given area is equal to the area times the component of magnetic field perpendicular to the area.

+or a closed surface, the sum of magnetic flux is always equal to *ero $This is alsoknown as auss's law for magnetic field%.

The standard unit for magnetic flux is a weber $Gb%, it is the number of magnetic lineof force $Tesla% crossing a unit area $m-%.

*agnetic Flux Densit0:

In simple matter, the magnetic flux density related to the magnetic field intensity as!here called the permeability. In particular !hen !e consider the free space

!here %>m is the permeability of the free space. 3agnetic fluxdensity is measured in terms of Wb>m .

The magnetic flux density through a surface is given by


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Wb ......................................( .1 )

In the case of electrostatic field, !e have seen that if the surface is a closed surface, the net

flux passing through the surface is e2ual to the charge enclosed by the surface. In case ofmagnetic field isolated magnetic charge (i. e. pole) does not exist. 3agnetic poles al!aysoccur in pair (as A7). For example, if !e desire to have an isolated magnetic pole bydividing the magnetic bar successively into t!o, !e end up !ith pieces each having north ( )and south (7) pole as sho!n in Fig. .- (a). This process could be continued until themagnets are of atomic dimensions< still !e !ill have A7 pair occurring together. This meansthat the magnetic poles cannot be isolated.

Fig. ,.!: a "u'division of a magnet ' *agnetic field flux lines of a straight currentcarr0ing conductor

7imilarly if !e consider the field>flux lines of a current carrying conductor as sho!n in Fig. .-(b), !e find that these lines are closed lines, that is, if !e consider a closed surface, thenumber of flux lines that !ould leave the surface !ould be same as the number of flux linesthat !ould enter the surface.

From our discussions above, it is evident that for magnetic field,

......................................( .1/)

!hich is the :auss5s la! for the magnetic field.

Ey applying divergence theorem, !e can !rite


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%ence, ......................................( . )

!hich is the :auss5s la! for the magnetic field in point form.

*agnetic "calar and Vector otentials:

In studying electric field problems, !e introduced the concept of electric potential that simplified the computation ofelectric fields for certain types of problems. In the same manner let us relate the magnetic field intensity to a scalamagnetic -otential and !rite

...................................( . 1)

From 8mpere5s la! , !e "no! that

......................................( . )

Therefore, ............................( . $)

Eut using vector identity, !e find that is valid only !here . Thus the scalar magnetic

potential is defined only in the region !here . 3oreover, V m in general is not a single valued function of position.

This point can be illustrated as follo!s. &et us consider the cross section of a coaxial line assho!n in fig . .

In the region , and

Fig. ,.%: Cross "ection of a Coaxial Line


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If V m is the magnetic potential then,

If !e set V m ? at then c? and

We observe that as !e ma"e a complete lap around the current carrying conductor , !e

reach again but V m this time becomes

We observe that value of V m "eeps changing as !e complete additional laps to pass throughthe same point. We introduced V m analogous to electostatic potential V . Eut for static electric

fields, and , !hereas for steady magnetic field !herever

but even if along the path of integration.

We no! introduce the vector magnetic -otential !hich can be used in regions !herecurrent density may be #ero or non#ero and the same can be easily extended to time varyingcases. The use of vector magnetic potential provides elegant !ays of solving ;3 fieldproblems.

7ince and !e have the vector identity that for any vector , , !e can

!rite .

%ere, the vector field is called the vector magnetic potential. Its 7I unit is Wb>m. Thus ifcan find of a given current distribution, can be found from through a curl operation.

We have introduced the vector function and related its curl to . 8 vector function is

defined fully in terms of its curl as !ell as divergence. The choice of is made as follo!s.

...........................................( . ')


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Ey using vector identity, .................................................( . )

.........................................( . +)

:reat deal of simplification can be achieved if !e choose .

Butting , !e get !hich is vector poisson e2uation.In *artesian coordinates, the above e2uation can be !ritten in terms of the components as

......................................( . -a)

......................................( . -b)

......................................( . -c)

The form of all the above e2uation is same as that of

..........................................( . )

for !hich the solution is

..................( . /)

In case of time varying fields !e shall see that , !hich is "no!n as &orent#condition, V being the electric potential. %ere !e are dealing !ith static magnetic field, so

.

Ey comparison, !e can !rite the solution for 8 x as

...................................( .$ )

*omputing similar solutions for other t!o components of the vector potential, the vectorpotential can be !ritten as

.......................................( .$1)


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This e2uation enables us to find the vector potential at a given point because of a volume

current density . 7imilarly for line or surface current density !e can !rite

...................................................( .$ )

respectively. ..............................( .$$)

The magnetic flux through a given area 7 is given by

.............................................( .$')

7ubstituting

.........................................( .$ )

Hector potential thus have the physical significance that its integral around any closed path ise2ual to the magnetic flux passing through that path.

*agnetic *oment In 2 *agnetic Field

Themagnetic moment of an ob6ect is avector relating the aligningtorque in amagnetic

field experienced by the ob6ect to the field vector itself. The relationship is given by

where

is the torque, measured innewton meters,is the magnetic moment, measured in ampere meters squared, andis the magnetic field, measured in teslas or, equivalently in newtons per

$ampere meter%.

*agnetic Scalar Potential

Themagnetic scalar potential is another useful tool in describing the magnetic fieldaround a current source. #t is only defined in regions of space in absence of $but could bnear% currents.

The magnetic scalar potential is defined by the equation;
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Applying Ampere's &aw to the above definition we get;

5ince in any continuous field, the curl of a gradient is *ero, this would suggest thatmagnetic scalar potential fields cannot support any sources. #n fact, sources can besupported by applying discontinuities to the potential field $thus the same point can havtwo values for points along the disconuity%. These discontinuities are also known a(cuts(. Ghen solvingmagnetostatics problems using magnetic scalar potential, the sourcecurrents must be applied at the discontinuity.

The magnetic scalar potential is suited to use around linesCloops of currents, but not region of space with finite current density. The use of magnetic potential reduces thethree components of the magnetic field to one component , making computationsand algebraic manipulations easier. #t is often used in magnetostatics, but rarely used inother applications.

*agnetic Vector Potential

The magnetic vector potential is a three dimensionalvector field whosecurl is themagnetic field in the theory of electromagnetism;

5ince the magnetic field isdivergence free $i.e. %, always exists.
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#aplace"s e)uation

#n three dimensions, the problem is to find twicedifferentiable real valued functions,of real variables, x, y, and z , such that

This is often written as

or

where div is thedivergence, and grad is thegradient, or

where P is the&aplace operator .

5olutions of &aplace's equation are calledharmonic functions.

#f the right hand side is specified as a given function, f $ x, y, z %, i.e.

then the equation is called (:oisson's equation.( &aplace's equation and :oisson'sequation are the simplest examples ofelliptic partial differential equations. The partialdifferential operator, , or P, $which may be defined in any number of dimensions% icalled the&aplace operator , or 6ust the &aplacian

For electrostatic field, !e have seen that

..........................................................................................($.1)

Form the above t!o e2uations !e can !rite

..................................................................($. )

Gsing vector identity !e can !rite, ................($.$)
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For a simple homogeneous medium, is constant and . Therefore,

................($.')

This e2uation is "no!n as oisson;s e9uation . %ere !e have introduced a ne! operator,

( del s2uare), called the &aplacian operator. In *artesian coordinates,

...............($.')

Therefore, in *artesian coordinates, Boisson e2uation can be !ritten as

...............($. )

In cylindrical coordinates,

...............($.+)

In spherical polar coordinate system,

...............($.-)

8t points in simple media, !here no free charge is present, BoissonJs e2uation reduces to

...................................($. )

!hich is "no!n as &aplaceJs e2uation.

&aplaceJs and BoissonJs e2uation are very useful for solving many practical electrostatic fieldproblems !here only the electrostatic conditions (potential and charge) at some boundariesare "no!n and solution of electric field and potential is to be found throughout the volume.We shall consider such applications in the section !here !e deal !ith boundary value

problems.

Polari ation densit in *a-$ell"s e)uations

The behavior of electric fields $E , D%,magnetic fields $+ , 7 %,charge density $O% andcurrent density $8% are described byMaxwell's equations. The role of the polari*ationdensityP is described below.
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Relations bet$een E9 D and P

The polari*ation densityP defines theelectric displacement field D as

which is convenient for various calculations.A relation betweenP andE exists in manymaterials, as described later in the article.

+ound c!arge

!lectric polari*ation corresponds to a rearrangement of the boundelectrons in thematerial, which creates an additionalcharge density, known as thebound c!arge densitO b;so that the total charge density that enters Maxwell's equations is given bywhere Of isthe free c!arge densit $describing charges brought from outside%.At the surface of the polari*ed material, the bound charge appears as asurface charge densitywhere is thenormal vector . #fP is uniform inside the material, this surface charge is the only boundcharge.

Ghen the polari*ation density changes with time, the time dependent bound chargedensity creates acurrent density of so that the total current density that enters Maxwell'sequations is given by where8f is the free charge current density, and the second term is acontribution from themagneti*ation $when it exists%.

Ca-acitance and Ca-acitors

Capacitance is a measure of the amount ofelectric charge stored $or separated% for agiven electric potential. The most common form of charge storage device is a two platecapacitor . #f the charges on the plates are QR and R, and L gives the voltage difference between the plates, then the capacitance is given by

The5# unit of capacitance is the faradF farad coulomb pervolt.

Capacitors

The capacitance of the ma6ority of capacitors used in electronic circuits is several orderof magnitude smaller than the farad. The most common subunits of capacitance in usetoday are themillifarad $m+%,microfarad $S+%, thenanofarad $n+% and the picofarad $p+%

The capacitance can be calculated if the geometry of the conductors and the dielectric properties of the insulator between the conductors are known. +or example, thecapacitance of a *ara))e)1*)ate capacitor constructed of two parallel plates of area $separated by a distanced is approximately equal to the following;
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or

where

C is the capacitance infarads, +s is the static permittivity of the insulator used $or/ for a vacuum%

$ is the area of each plate, measured in square metres r is the relative static permittivity $sometimes called the dielectric constant% of

the material between the plates, $vacuum %d is the separation between the plates, measured inmetres

The equation is a good approximation ifd is small compared to the other dimensions of the plates.

We have already stated that a conductor in an electrostatic field is an ;2uipotentialbody and any charge given to such conductor !ill distribute themselves in such a mannerthat electric field inside the conductor vanishes. If an additional amount of charge is suppliedto an isolated conductor at a given potential, this additional charge !ill increase the surface

charge density . 7ince the potential of the conductor is given by , the

potential of the conductor !ill also increase maintaining the ratio same. Thus !e can !rite

!here the constant of proportionality C is called the capacitance of the isolatedconductor. 7I unit of capacitance is *oulomb> Holt also called Farad denoted by 5 . It can Itcan be seen that if " ?1, C ? . Thus capacity of an isolated conductor can also be definedas the amount of charge in *oulomb re2uired to raise the potential of the conductor by 1Holt.

4f considerable interest in practice is a capacitor that consists of t!o (or more) conductorscarrying e2ual and opposite charges and separated by some dielectric media or free space.The conductors may have arbitrary shapes. 8 t!oAconductor capacitor is sho!n in figure $.1
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Fig .1: Ca-acitance and Ca-acitorsWhen a dAc voltage source is connected bet!een the conductors, a charge transfer occurs !hich results into a positivecharge on one conductor and negative charge on the other conductor. The conductors are e2uipotential surfaces andthe field lines are perpendicular to the conductor surface. If " is the mean potential difference bet!een the conductors,

the capacitance is given by . *apacitance of a capacitor depends on the geometry of the conductor and thepermittivity of the medium bet!een them and does not depend on the charge or potential difference bet!eenconductors. The capacitance can be computed by assuming (at the same time A on the other conductor), first

determining using :aussJs theorem and then determining . We illustrate this procedure by ta"ing theexample of a parallel plate capacitor Exam-le: arallel -late ca-acitor

Fig .,: arallel late Ca-acitor For the parallel plate capacitor sho!n in the figure $. , let each plate has area 8 and a distance h separates the plates.

8 dielectric of permittivity fills the region bet!een the plates. The electric field lines are confined bet!een the plates.

We ignore the flux fringing at the edges of the plates and charges are assumed to be uniformly distributed over the

conducting plates !ith densities and A , .

Ey :aussJs theorem !e can !rite, .......................($./)


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8s !e have assumed to be uniform and fringing of field is neglected, !e see that ; is

constant in the region bet!een the plates and therefore, !e can !rite . Thus,

for a parallel plate capacitor !e have,........................($.1 )

"eries and -arallel Connection of ca-acitors

*apacitors are connected in various manners in electrical circuits< series and parallelconnections are the t!o basic !ays of connecting capacitors. We compute the e2uivalentcapacitance for such connections.

"eries Case: 7eries connection of t!o capacitors is sho!n in the figure $.$. For this case!e can !rite,

Fig . : "eries Connection of Ca-acitors Fig ./: arallel Connection of Ca-

The same approach may be extended to more than t!o capacitors connected in series.

arallel Case: For the parallel case, the voltages across the capacitors are the same.

The total charge


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.......................($.11)

Therefore, .......................($.1 )

Electrostatic Energ0 and Energ0 Densit0

We have stated that the electric potential at a point in an electric field is the amount of !or"re2uired to bring a unit positive charge from infinity (reference of #ero potential) to that point.To determine the energy that is present in an assembly of charges, let us first determine the

amount of !or" re2uired to assemble them. &et us consider a number of discrete chargesQ 1, Q ,......., Q are brought from infinity to their present position one by one. 7ince initiallythere is no field present, the amount of !or" done in bring @ 1 is #ero. @ is brought in thepresence of the field of @ 1, the !or" done 6 1? " 1 !here " 1 is the potential at the locationof @ due to @ 1. Broceeding in this manner, !e can !rite, the total !or" done

.................................................($.1$)

%ad the charges been brought in the reverse order,

.................($.1')

Therefore,

................($.1 )

%ere " IJ represent voltage at the 7 th charge location due to 8 th charge. Therefore,

4r, ................($.1+)


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If instead of discrete charges, !e no! have a distribution of charges over a volume then !ecan !rite,

................($.1-)

!here is the volume charge density and " represents the potential function.

7ince, , !e can !rite

.......................................($.1 )

Gsing the vector identity,

, !e can !rite

................($.1/)

In the expression , for point charges, since " varies as and 0 varies as ,

the term " varies as !hile the area varies as r . %ence the integral term varies at least

as and the as surface becomes large (i.e. ) the integral term tends to #ero

Thus the e2uation for 6 reduces to

................($. )

, is called the energy density in the electrostatic field.

7oundar0 conditions for Electrostatic fields

In our discussions so far we have considered the existence of electric field in the homogeneousmedium. :ractical electromagnetic problems often involve media with different physical properties.


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Jetermination of electric field for such problems requires the knowledge of the relations of fieldquantities at an interface between two media. The conditions that the fields must satisfy at theinterface of two different media are referred to asboundary conditions .

In order to discuss the boundary conditions, !e first consider the field behavior in somecommon material media.

In general, based on the electric properties, materials can be classified into three categoriesconductors, semiconductors and insulators (dielectrics). In conductor , electrons in theoutermost shells of the atoms are very loosely held and they migrate easily from one atom tothe other. 3ost metals belong to this group. The electrons in the atoms of insulators ordi l ctrics remain confined to their orbits and under normal circumstances they are notliberated under the influence of an externally applied field. The electrical properties ofs miconductors fall bet!een those of conductors and insulators since semiconductors havevery fe! numbers of free charges.

The parameter conducti!it" is used characteri#es the macroscopic electrical property of amaterial medium. The notion of conductivity is more important in dealing !ith the current flo!

and hence the same !ill be considered in detail later on.

If some free charge is introduced inside a conductor, the charges !ill experience a force dueto mutual repulsion and o!ing to the fact that they are free to move, the charges !ill appearon the surface. The charges !ill redistribute themselves in such a manner that the field

!ithin the conductor is #ero. Therefore, under steady condition, inside a conductor .

From :auss5s theorem it follo!s that

? .......................($. 1)

The surface charge distribution on a conductor depends on the shape of the conductor. The charges on the surface ofthe conductor !ill not be in e2uilibrium if there is a tangential component of the electric field is present, !hich !ouldproduce movement of the charges. %ence under static field conditions, tangential component of the electric field on theconductor surface is #ero. +he electric field on the surface of the conductor is normal ever0(here to the surface

7ince the tangential component of electric field is #ero, the conductor surface is an e9ui-otential surface . 8s ? inside the conductor, the conductor as a !hole has the same potential. We may further note that charges re2uire a finite

time to redistribute in a conductor. %o!ever, this time is very small sec for good conductor li"e copper.

&et us no! consider an interfacebet!een a conductor and freespace as sho!n in the figure $. .

Fig .5: 7oundar0 Conditions for at the surface of a Conductor


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net dipole moment #ero. %o!ever, in the absence of an external field, the molecules arrangethemselves in a random manner so that net dipole moment over a volume becomes #ero.Gnder the influence of an applied electric field, these dipoles tend to align themselves alongthe field. There are some materials that can exhibit net permanent dipole moment even inthe absence of applied field. These materials are called l ctr ts that made by heatingcertain !axes or plastics in the presence of electric field. The applied field aligns the

polari#ed molecules !hen the material is in the heated state and they are fro#en to their ne!position !hen after the temperature is brought do!n to its normal temperatures. Bermanentpolari#ation remains !ithout an externally applied field.

8s a measure of intensity of polari#ation, polari#ation vector (in *>m ) is defined as

.......................($. /)

n being the number of molecules per unit volume i.e. is the dipole moment per unit volume. &et us no! consider a

dielectric material having polari#ation and compute the potential at an external point 4 due to an elementary dipoled 4 .

Fig .6: otential at an External oint due to an

Elementar0 Di-ole dv' .

With reference to the figure .1+, !e can !rite..........................................($.$ )

Therefore,

....................($.$1))

........($.$ )!here x,y,# represent the coordinates of the external point 4and x5,y5,#5 are the coordinates of the source point.

From the expression of R , !e can verify that

.............................................($.$$)

.........................................($.$')

Gsing the vector identity, ,!here $ is a scalar 2uantity , !e have,


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.......................($.$ )

*onverting the first volume integral of the above expression to surface integral, !e can !rite

.................($.$+)

!here is the out!ard normal from the surface element ds4 of the dielectric. From theabove expression !e find that the electric potential of a polari#ed dielectric may be foundfrom the contribution of volume and surface charge distributions having densities

......................................................................($.$-)

......................($.$ )

These are referred to as polarisation or bound charge densities. Therefore !e may replace apolari#ed dielectric by an e2uivalent polari#ation surface charge density and a polari#ationvolume charge density. We recall that bound charges are those charges that are not free tomove !ithin the dielectric material, such charges are result of displacement that occurs on amolecular scale during polari#ation. The total bound charge on the surface is

......................($.$/)

The charge that remains inside the surface is

......................($.' )

The total charge in the dielectric material is #ero as

......................($.'1)

If !e no! consider that the dielectric region containing charge density the total volumecharge density becomes

....................($.' )

7ince !e have ta"en into account the effect of the bound charge density, !e can !rite


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....................($.'$)

Gsing the definition of !e have

....................($.'')

Therefore the electric flux density

When the dielectric properties of the medium are linear and isotropic, polarisation is directlyproportional to the applied field strength and

........................($.' )

is the electric susceptibility of the dielectric. Therefore,

.......................($.'+)

is called relative permeability or the dielectric constant of the medium. is calledthe absolute permittivity.

8 dielectric medium is said to be linear !hen is independent of and the medium is

homogeneous if is also independent of space coordinates. 8 linear homogeneous andisotropic medium is called a sim-le medium and for such medium the relative permittivity isa constant.

0ielectric constant may be a function of space coordinates. For anistropic materials, thedielectric constant is different in different directions of the electric field, 0 and ; are relatedby a permittivity tensor !hich may be !ritten as

.......................($.'-)

For crystals, the reference coordinates can be chosen along the principal axes, !hich ma"eoff diagonal elements of the permittivity matrix #ero. Therefore, !e have

.......................($.' )


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3edia exhibiting such characteristics are called 'iaxial . Further, if then the medium is

called uniaxial . It may be noted that for isotropic media, .

&ossy dielectric materials are represented by a complex dielectric constant, the imaginarypart of !hich provides the po!er loss in the medium and this is in general dependant onfre2uency.

8nother phenomenon is of importance is dielectric 'rea#do(n . We observed that theapplied electric field causes small displacement of bound charges in a dielectric material thatresults into polari#ation. 7trong field can pull electrons completely out of the molecules.These electrons being accelerated under influence of electric field !ill collide !ith molecularlattice structure causing damage or distortion of material. For very strong fields, avalanchebrea"do!n may also occur. The dielectric under such condition !ill become conducting.

The maximum electric field intensity a dielectric can !ithstand !ithout brea"do!n is referredto as the dielectric strength of the material.

7oundar0 Conditions for Electrostatic Fields:

&et us consider the relationship among the field components that exist at the interfacebet!een t!o dielectrics as sho!n in the figure $.-. The permittivity of the medium 1 and

medium are and respectively and the interface may also have a net charge density

*oulomb>m.

Fig .!: 7oundar0 Conditions at the interface 'et(een t(o dielectrics

We can express the electric field in terms of the tangential and normal components

..........($.'/)

!here 9 t and 9 n are the tangential and normal components of the electric field respectively.

&et us assume that the closed path is very small so that over the elemental path length thevariation of ; can be neglected. 3oreover very near to the interface, . Therefore


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.......................($. )

Thus, !e have,

or i.e. the tangential com-onent of an electric field is continuousacross the interface .

For relating the flux density vectors on t!o sides of the interface !e apply :aussJs la! to asmall pillbox volume as sho!n in the figure. 4nce again as , !e can !rite

..................($. 1a)

i.e., .................................................($. 1b)

i.e., .......................($. 1c)

Thus !e find that the normal com-onent of the flux densit0 vector D is discontinuousacross an interface '0 an amount of discontinuit0 e9ual to the surface charge densit0at the interface.

Exam-le

T!o further illustrate these points< let us consider an example, !hich involves the refractionof 0 or ; at a charge free dielectric interface as sho!n in the figure $. .

Gsing the relationships !e have ust derived, !e can !rite

.......................($. a)

.......................($. b)

In terms of flux density vectors,

.......................($. $a)

.......................($. $b)

Therefore, .......................($. ')


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Fig .%: 4efraction of D or E at a Charge Free Dielectric Interface

Energ

Theenergy $measured in 6oules% stored in a capacitor is equal to the!or# done to chargeit. )onsider a capacitanceC , holding a charge


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9% 5olve for the normal components like this;The normal components depend on the surface charge densitys $)Cm-% .J n K J-n s $)Cm-% 8H ! n -! -n s

Special Cases3

Perfect Dielectrics :conductivit 6 0;5urface charge density can only exist on a conductive surface, so if both materialsare perfect dielectrics $have no conductivity%, thens /.

Perfect Conductors :conductivit is infinite; :metals;

emf material

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