EMF Notes of Lesson 1

8/7/2019 EMF Notes of Lesson 1

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DEPARTMENT OF ECE

SEMESTER - IV

NOTES OF LESSON FOR

ELECTROMAGNETICFIELDS (SUBJECTCODE: EC 43)

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


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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 abinary operationon twovectorsin a three-dimensionalEuclidean spacethat results in another vector which isperpendicular to the two input vectors. Bycontrast, thedot productproduces ascalar result. In many engineering and physicsproblems, 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). Like the dot product, it depends on themetricof Euclidean space. Unlike thedot product,it also depends on the choice of orientationor "handedness". Certain features of the cross product can be generalized to other situations

For arbitrary choices of orientation, the cross product must be regarded not as a vector,but as apseudovector . For arbitrary choices of metric, and in arbitrary dimensions, thecross product can be generalized by theexterior productof vectors, defining atwo-form instead of a vector.

Fig 1.1 Illustration of the cross-product in respect to a right-handed coordinate system.

Fig 1.2 Finding the direction of the cross product by theright-hand rule.
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The cross product of two vectorsa and b is denoted bya b . In a three-dimensionalEuclidean space, with a usualright-handed coordinate system,it is defined as a vector cthat isperpendicular to botha andb , with a direction given by theright-hand ruleand amagnitude equal to the area of theparallelogramthat the vectors span.

The cross product is given by the formula

where is the measure of theanglebetweena and b (0 180),a and b are themagnitudesof vectorsa and b , and is aunit vector perpendicular to the planecontaininga andb . If the vectorsa andb are collinear (i.e., the angle between them iseither 0 or 180), by the above formula, the cross product of a andb is the zero vector 0.

The direction of the vector is given by the right-hand rule, where one simply points theforefinger of the right hand in the direction of a and the middle finger in the direction of b . Then, the vector is coming out of the thumb (see the picture on the right).Using the cross product requires the handedness of the coordinate system to be taken intoaccount (as explicit in the definition above). If aleft-handed coordinate systemis used,the direction of the vector is given by the left-hand rule and points in the oppositedirection.

Dot product

The dot product , also known as thescalar product , is an operation which takes twovectorsover thereal numbers R and returns a real-valuedscalar quantity.

where

|a | and |b | denote thelength(magnitude) of a andb is theanglebetween them.

Since |a |cos() is thescalar projectionof a onto b , the dot product can be understoodgeometrically as the product of this projection with the length of b .
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|a |cos() is thescalar projectionof a ontob

Coordinate System

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

objects analyticallyA cartesian coordinate systemis one of the simplest and most useful systems of coordinates. It 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 pointP arex, the distance of P from the planeYOZ measured parallel toOX , andy andz , which are determined similarly (Fig. 1). Usually the three axes are takento be mutuallyperpendicular , in which case the system is a rectangular cartesianone.Obviously a similar construction can be made in the plane, in which case a point has twocoordinates (x,y).

fig 1.3Cartesian coordinate system.

Cylindrical Coordinate System

The cylindrical coordinate system 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 P is given as (r ,,h). In terms of the Cartesian coordinate system:

r is the distance from O to P', the orthogonal projection of the point P onto the XYplane. This is the same as the distance of P to the z-axis.

is the angle between the positive x-axis and the line OP', measuredcounterclockwise.

h is the same asz .
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Thus, the conversion functionf from cylindrical coordinates to Cartesiancoordinates is .

Fig 1.4 :A point plotted with cylindrical coordinates

Spherical Coordinates

The three coordinates (, , ) are defined as:

0 is the distance from the origin to a given pointP . 0 2 is the angle between the positive x-axis and the line from the origin to

theP projected onto the xy-plane. 0 is the angle between the positive z-axis and the line formed between the

origin andP .

is referred to as the azimuth, while is referred to as the zenith, colatitude or polar angle.

and and lose significance when = 0 and loses significance when sin() = 0 (at =0 and = 180).

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

Coordinate system 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.


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Cartesian coordinate system

The three spherical coordinates are obtained fromCartesian coordinatesby:

Note that the arctangent must be defined suitably so as to take account of the correctquadrant of y / x. Theatan2or equivalent function accomplishes this for computationalpurposes.

Conversely, Cartesian 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.

Fig 1.5: Flux Lines

We have already defined flux of a vector field as

....................................................(1.1)
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For a volume enclosed by a surface,

.........................................................................................(1.2)

We define the divergence of a vector field at a point P as the net outward flux from avolume enclosing P , as the volume shrinks to zero.

.................................................................(1.3)

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

Let us consider a differential volume centered on point P(u,v,w ) in a vector field . The fluxthrough an elementary area normal to u is given by ,

........................................(1.4)


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Fig 1.6 Evaluation of divergence in curvilinear coordinate

In cylindrical coordinates:

....................................................................(1.8)

Net outward flux along u can be calculated considering the two elementary surfaces perpendicular to u .

.......................................(1.5)Considering the contribution from all six surfaces that enclose the volume, we can write

.......................................(1.6)

Hence for the Cartesian, cylindrical and spherical polar coordinate system, the expressions for divergence can bewritten as:

In Cartesian coordinates:

................................(1.7)


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and in spherical polar coordinates:

......................................(1.9)

In connection with the divergence of a vector field, the following can be noted

Divergence of a vector field gives a scalar.

..............................................................................(1.10)

Divergence theorem :

Divergence theorem states that the volume integral of the divergence of vector field is equalto the net outward flux of the vector through the closed surface that bounds the volume.

Mathematically,

Proof:

Let us consider a volume V enclosed by a surface S . Let us subdivide the volume in large

number of cells. Let the k th cell has a volume and the corresponding surface is denotedby S k . Interior to the volume, cells have common surfaces. Outward flux through thesecommon surfaces from one cell becomes the inward flux for the neighboring cells. Therefore

when the total flux from these cells are considered, we actually get the net outward fluxthrough the surface surrounding the volume. Hence we can write:

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

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

written as .

Hence we get , which is the divergence theorem.


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Curl of a vector field:

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

Curl of a vector field is a measure of the vector field's tendency to rotate about a point. Curl

, also written as is defined as a vector whose magnitude is maximum of the netcirculation per unit area when the area tends to zero and its direction is the normal directionto the area when the area is oriented in such a way so as to make the circulation maximum.

Therefore, we can write:

......................................(1.12)

To derive the expression for curl in generalized curvilinear coordinate system, we first

compute and to do so let us consider the figure 1.7:

Fig 1.7 Curl of a Vector

If C 1 represents the boundary of , then we can write

......................................(1.13)The integrals on the RHS can be evaluated as follows:

.................................(1.14)

................................................(1.15)

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


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..................................................(1.16)

............................................................................(1.17)

Adding the contribution from all components, we can write:

........................................................................(1.18)

Therefore, ......................................................(1.19)

In the same manner if we compute for and we can write,

.......(1

This can be written as,

......................................................(1.21)

In Cartesian coordinates: .......................................(1.22)


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In Cylindrical coordinates, ....................................(1.23)

In Spherical polar coordinates, ..............(1.24)

Curl operation exhibits the following properties:

..............(1.25)

Stoke's theorem :

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

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

provided and are continuous on the surface.

i.e,

..............(1.26)

Proof: Let us consider an area S that is subdivided into large number of cells as shown in the

figure 1.8


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Fig 1.8 Stokes theorem

Let k t hcell has surface area and is bounded path Lk while the total area isbounded by path L. As seen from the figure that if we evaluate the sum of the lineintegrals around the elementary areas, there is cancellation along every interior path and we are left the line integral along path L. Therefore we can write,

..............(1.27)

As 0

. .............(1.28)

which is the stoke's theorem.

Coulomb's Law

Coulomb's Law may be stated as follows:"The magnitude of the electrostatic force between two point charges is directlyproportional to the magnitudes of each charge and inversely proportional to the squareof the distance between the charges."

Coulomb's law states that the electrical force between two charged objects is directlyproportional to the product of the quantity of charge on the objects and inversely


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proportional to the square of the separation distance between the two objects. In equationform, Coulomb's law can be stated as

(1.29)

whereQ 1 represents the quantity of charge on object 1 (in Coulombs),Q 2 represents thequantity of charge on object 2 (in Coulombs), andd represents the distance of separationbetween the two objects (in meters). The symbolk is a proportionality constant known asthe Coulomb's law constant. The value of this constant is dependent upon the mediumthat the charged objects are immersed in.

Mathematically, ,where k is the proportionality constant.

In SI units, Q 1 and Q 2 are expressed in Coulombs(C) and R is in meters.

Force F is in Newtons ( 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, we will use instead where is called the relative permittivity or thedielectric constant of the medium).

Therefore .......................(1.30)

As shown in the Figure 2.1 let the position vectors of the point charges Q1and Q 2 are given

by and . Let represent the force on Q 1 due to charge Q 2.

Fig 1.9: Coulomb's Law


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

and ..................................(1.31)

can be defined as

. ..(1.32)

Similarly the force on Q1 due to charge Q2 can be calculated and if represents this force

then we can write

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

Q 1,Q 2,......... Q N located respectively at the points represented by the position vectors ,

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

.................................(1.33)

Electric Field

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

or, .......................................(1.34)

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

(source point) is given by:

..........................................(1.35)


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For a collection of N point charges Q1 ,Q2 ,.........Q N located at , ,...... , the electric field intensity at point obtained as

........................................(1.36)

The expression (2.6) can be modified suitably tocompute the electric filed due to a continuousdistribution of charges.

In figure 1.10we 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, we can writethe field expression as:

.............(2.7)

Fig1.10: Continuous Volume Distribution

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

..........................................(1.37)

Similar technique can be adopted when the charge distribution is in the form of a line chargedensity or a surface charge density.

........................................(1.38)

........................................(1.39)


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Electric field strength

Electric field strength isa vector quantity; it has both magnitude and direction. The

magnitude of the electric field strength is defined in terms of how it is measured. Let'ssuppose that an electric charge can be denoted by thesymbolQ . This electric charge creates an electric field;sinceQ is the source of the electric field, we will refer toit as the source charge . 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 charge since it isused totest the field strength. The test charge has a quantity of charge denoted by thesymbolq . When placed within the electric field, the test charge will experience anelectric force - either attractive or repulsive. As is usually the case, this force will be

denoted by the symbolF . The magnitude of the electric field is simply defined as theforce per charge on the test charge.

If 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. Since electricfield is defined as a force per charge, its units would be force units divided by chargeunits. In this case, the standard metric units are Newton/Coulomb or N/C.

Electric Field Lines

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. Since electric field is a vector quantity, it can berepresented by a vector arrow. For 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. Such vector arrows are shown in the diagram below. Note that the length of thearrows are longer when closer to the source charge and shorter when further from thesource charge.
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A more useful means of visually representing the vector nature of an electric field isthrough the use of electric field lines of force. Rather than draw countless vector arrowin the space surrounding a source charge, it is perhaps more useful to draw a pattern ofseveral lines which extend betweeninfinity and the source charge. These pattern of lines,sometimes referred to aselectric field lines , point in the direction which a positive testcharge would accelerate if placed upon the line. As such, the lines are directed away frompositively charged source charges and toward negatively charged source charges. To

communicate information about the direction of the field, each line must include anarrowhead which points in the appropriate direction. An electric field line pattern couldinclude an infinite number of lines. Because drawing such large quantities of lines tendsto decrease the readability of the patterns, the number of lines are usually limited. Thepresence of a few lines around a charge is typically sufficient to convey the nature of theelectric field in the space surrounding the lines.

Electric Fields and Conductors

Electrostatic equilibrium is the condition established by charged conductors in whichthe excess charge has optimally distanced itself so as to reduce the total amount of repulsive forces. Once a charged conductor has reached the state of electrostaticequilibrium, there is no further motion of charge about the surface.


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Electric Fields Inside of Charged Conductors

Charged conductors which have reached electrostatic equilibrium share a variety ofunusual characteristics. One characteristic of a conductor at electrostatic equilibrium isthat the electric field anywhere beneath the surface of a charged conductor is zero. If an

electric field did exist beneath the surface of a conductor (and inside of it), then theelectric field would exert a force on all electrons that were present there. This net forcewould begin to accelerate and move these electrons. But objects at electrostaticequilibrium have no further motion of charge about the surface. So if this were to occurthen the original claim that the object was at electrostatic equilibrium would be a falseclaim. If the electrons within a conductor have assumed an equilibrium state, then the neforce upon those electrons is zero. 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 rightThe outer cylinder is charged positively. The inner cylinder ischarged negatively. The electric field about the inner cylinder is directed towards the negatively charged cylinder. Since 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 zero. Whendrawing electric field lines, the lines would be drawn from theinner surface of the outer cylinder to the outer surface of the

inner cylinder. For the excess charge on the outer cylinder, there is more to consider thanmerely the repulsive forces between charges on its surface. While the excess charge onthe outer cylinder seeks to reduce repulsive forces between its excess charge, it mustbalance this with the tendency to be attracted to the negative charges on the inner cylinder. Since 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 zero inside of a closed conducting surface wasfirst demonstrated by Michael Faraday, a 19th century physicist who promoted the fieldtheory of electricity. Faraday constructed a room within a room, covering the inner roomwith a metal foil. He sat inside the inner room with an electroscope and charged thesurfaces of the outer and inner room using an electrostatic generator. While 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 Faraday from the staticcharge is now referred to as aFaraday'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 Faraday's cage, shielding whatever it surrounds from the


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potentially damaging affects of electric fields. This principle of shielding is commonlyutilized today as we protect delicate electrical equipment by enclosing them in metalcases. Even delicate computer chips and other components are shipped inside ofconducting plastic packaging which shields the chips from potentially damaging affectsof electric fields.

Electric Fields are Perpendicular to Charged Surfaces

A second characteristic of conductors at electrostatic equilibrium is that the electric fieldupon the surface of the conductor is directed entirely perpendicular to the surface. Therecannot be a component of electric field (or electric force) that is parallel to the surface. Ithe conducting object is spherical, then this means that the perpendicular electric fieldvector are aligned with the center of the sphere. If the object is irregularly shaped, thenthe electric field vector at any location is perpendicular to a tangent line drawn to thesurface at that location.

Understanding why this characteristic is true demands an understanding of vectors, forcand motion. The motion of electrons, like any physical object, isgoverned by Newton's laws. One outcome of Newton's laws was thatunbalanced forces cause objects to accelerate in the direction of the unbalanced force and a balance of forces cause objects to remain at equilibrium. This truth provides the foundation for the rationale behind whyelectric fields must be directed perpendicular to the surface of conducting objects. If therewere a component of electric field directed parallel to the surface, then the excess chargeon the surface would be forced into accelerated motion by this component. If a charge isset into motion, then the object upon which it is on is not in a state of electrostaticequilibrium. Therefore, the electric field must be entirely perpendicular to the conductingsurface for objects which are at electrostatic equilibrium. Certainly a conducting object

which has recently acquired an excess charge has a component of electric field (andelectric force) parallel to the surface; it is this component which acts upon the newlyacquired excess charge to distribute the excess charge over the surface and establishelectrostatic equilibrium. But once reached, there is no longer any parallel component ofelectric field and no longer any motion of excess charge.
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Electric Fields and Surface Curvature

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

a blunt point on the other extreme.

A flat location has no curvature and is characterized by relatively weak electric fields. Onthe other hand, ablunt point has a high degree of curvature and is characterized 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 object which is negatively charged. Such an object has an excess of electrons.These electrons would distribute themselves in such a manner as to reduce the affect oftheir repulsive forces. Since electrostatic forces vary inversely with thesquare of the distance, these electrons would tend to position themselvesso as to increase their distance from one another. On a regularly shapedsphere, the ultimate distance between every neighboring electron would

be the same. But on an irregularly shaped object, excess electrons wouldtend to accumulate in greater density along locations of greatestcurvature. Consider the diagram at the right. Electrons A and B arelocated along a flatter section of the surface. Like all well-behavedelectrons, they repel each other. The repulsive forces are directed alonga line connecting charge to charge, making the repulsive force primarily parallel to thesurface. On the other hand, electrons C and D 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. But now the force is directed at asharper angle to the surface. The components of these forces parallel to the surface areconsiderably less. A majority of the repulsive force between electrons C and D 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. Once reached, the resultant of allparallel components on any given excess electron (and on all excess electrons) will addup to zero. All the parallel components of force on each of the electrons must be zerosince the net force parallel to the surface of the conductor is always zero (thesecond characteristicdiscussed above). For the same separation distance, the parallel component
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.............................(1.40)

The negative sign accounts for the fact that work is done on the system by the externalagent.

.....................................(1.41)

The potential difference between two points P and Q , V PQ, is defined as the work done per unit charge, i.e.

...............................(1.42)

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 work against the field. If the sign of the potential difference is negative, work isdone by the field.

We will 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 between two points in an electrostatic field is a point function; it isindependent of the path taken. The potential difference is measured in Joules/Coulombwhich is referred to as Volts .

Let us consider a point charge Q as shown in the Fig. 1.12

Fig 1.12 Electrostatic Potential calculation for a point charge


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Further consider the two points A and B as shown in the Fig.1.12. Considering the movementof a unit positive test charge from B to A , we can write an expression for the potentialdifference as:

..................................(1.43)

It is customary to choose the potential to be zero at infinity. Thus potential at any point ( r A =r ) due to a point charge Q can be written as the amount of work done in bringing a unitpositive charge from infinity to that point (i.e. r B = 0).

..................................(1.44)

Or, in other words,

..................................(1.45)

Let us now consider a situation where the point charge Q is not located at the origin asshown in Fig. 1.13.

Fig 1.13: Electrostatic Potential due a Displaced Charge

The potential at a point P becomes

..................................(1.46)

So far we have considered the potential due to point charges only. As any other type of charge distribution can be considered to be consisting of point charges, the same basicideas now can be extended to other types of charge distribution also.


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Let us first consider N point charges Q 1, Q 2 ,.....Q N located at points with position vectors ,

,...... . The potential at a point having position vector can be written as:

..................................(1.47)

or , ...........................................................(1.48)

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

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

these modifications we can write:

For line charge, ..................................(1.49)

For surface charge, .................................(1.50)

For volume charge, .................................(1.51)

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 we have used the reference or zero potential at infinity. If any other point is chosen as reference, we can write:

.................................(1.52)

where C is a constant. In the same manner when potential is computed from a knownelectric field we can write:

.................................(1.53)

The potential difference is however independent of the choice of reference.


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.......................(1.54)

We have mentioned that electrostatic field is a conservative field; the work done in moving acharge from one point to the other is independent of the path. Let us consider moving acharge from point P 1 to P 2 in one path and then from point P 2 back to P 1 over a different path.If the work done on the two paths were different, a net positive or negative amount of workwould have been done when the body returns to its original position P 1. In a conservativefield there is no mechanism for dissipating energy corresponding to any positive work neither any source is present from which energy could be absorbed in the case of negative work.Hence the question of different works in two paths is untenable, the work must have to beindependent of path and depends on the initial and final positions.

Since the potential difference is independent of the paths taken, V AB = - V BA , and over aclosed path,

.................................(1.55)

Applying Stokes's theorem, we can write:

............................(1.56)

from which it follows that for electrostatic field,

........................................(1.57)

Any vector field that satisfies is called an irrotational field.

From our definition of potential, we can write

.................................(1.58)

from which we obtain,

..........................................(1.59)


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

gradient at any point, negative sign shows that is directed from higher to lower values of . This gives us anomethod of computing the electric field, i. e. if we know 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 .Example: Electric Dipole

An electric dipole consists of two point charges of equalmagnitude but of opposite signand separated by a smalldistance.

As shown in figure 1.14, thedipole is formed by the twopoint charges Q and -Qseparated by a distance d , thecharges being placedsymmetrically about the origin.Let us consider a point P at adistance r , where we areinterested to find the field. Fig 1.14 : Electric DipoleThe potential at P due to the dipole can be written as:

..........................(1.60)

When r 1 and r 2>>d , we can write and .

Therefore,

....................................................(1.62)

We can write,

...............................................(1.63)

The quantity is called the dipole moment of the electric dipole.

Hence the expression for the electric potential can now be written as:

................................(1.64)


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It may be noted that while potential of an isolated charge varies with distance as 1/ r that of an electric dipole varies as 1/ r 2 with distance.

If the dipole is not centered at the origin, but the dipole center lies at , the expression for the potential can be written as:

........................(1.65)

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

........................(1.66)

is the magnitude of the dipole moment. Once again we note that the electric field of electric dipole varies as 1/ r 3 where as that of a point charge varies as 1/ r 2.

Electric flux density:

As stated earlier electric field intensity or simply Electric field' gives the strength of the fieldat a particular point. The electric field depends on the material media in which the field isbeing considered. The flux density vector is defined to be independent of the material media(as we'll 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:

................................................(1.67)

We define the electric flux as

.....................................(1.68)

Gauss's Law: Gauss's law is one of the fundamental laws of electromagnetism and it statesthat the total electric flux through a closed surface is equal to the total charge enclosed bythe surface.


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Fig 1.15 Gauss's Law

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

...............................................(1.69)

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

.....................................(1.70)

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

Q. Therefore we can write

For a closed surface enclosing the charge, we can write

which can seen to be same as what we have stated in the definition of Gauss's Law.

Application of Gauss's Law

Gauss's law is particularly useful in computing or where the charge distribution hassome symmetry. We shall illustrate the application of Gauss's Law with some examples.

1.An infinite line charge

As the first example of illustration of use of Gauss's law, let consider the problem of determination of the electric field produced by an infinite line charge of density LC/m. Letus consider a line charge positioned along the z -axis as shown in Fig.1.16(a) (next slide).Since the line charge is assumed to be infinitely long, the electric field will be of the form asshown in Fig. 2.4(b) (next slide).


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If we consider a close cylindrical surface as shown in Fig.1.16(a), using Gauss's theorm wecan write,

.....................................(1.71)

Considering the fact that the unit normal vector to areas S 1 and S 3 are perpendicular to theelectric field, the surface integrals for the top and bottom surfaces evaluates to zero. Hence

we can write,

Fig 1.16 Infinite Line Charge

.....................................(1.72)


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2. Infinite Sheet of Charge

As a second example of application of Gauss'stheorem, we consider an infinite charged sheetcovering the x-z plane as shown in figure 1.17

Assuming a surface charge density of for theinfinite surface charge, if we consider a cylindricalvolume having sides placed symmetrically , wecan write:

..............(1.73)

Fig1.17: Infinite Sheet 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 will be perpendicular to the sheet and extend to

infinity as parallel lines. As 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 will be a function of distance.

3. Uniformly Charged Sphere

Let us consider a sphere of radius r 0 having a uniform

volume charge density of v C/m 3. To determineeverywhere, inside and outside the sphere, weconstruct Gaussian surfaces of radius r < r 0 and r >r 0 as shown in Fig. 1.18(a) and Fig.1.18(b).

For the region ; the total enclosed charge willbe

.........................(1.74) Fig 1.18 Uniformly Charged Sphere

By applying Gauss's theorem,


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...............(1.75)

Therefore

...............................................(1.76)

For the region ; the total enclosed charge will be

....................................................................(1.77)

By applying Gauss's theorem,

.....................................................(1.78)


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Unit II Static Magnetic Field

Introduction :

In previous chapters we have seen that an electrostatic field is produced by static or

stationary 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 with time. In this chapter we shall mainly consider the magnetic fieldproduced by a direct current. The magnetic field produced due to time varying electric fieldwill be discussed later. Historically, the link between the electric and magnetic field wasestablished Oersted in 1820. Ampere and others extended the investigation of magneticeffect of electricity . There are two major laws governing the magnetostatic fields are:

Biot-Savart Law Ampere's Law

Usually, 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 with a dotor cross sign depending on whether the field (or current) is out of or into the page as shownin Fig. 2.1.

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

Fig. 2.1: Representation of magnetic field (or current)

Biot- Savart Law

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

current element as shown in Fig. 2.2.


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Fig. 2.2: Magnetic field intensity due to a current element

The magnetic field intensity at P can be written as,

............................(2.1a)

..............................................(2.1b)

where is the distance of the current element from the point P.

Similar to different charge distributions, we can have different current distribution such as line

current, surface current and volume current. These different types of current densities areshown in Fig. 2.3.

Line Current Surface Current Volume Current

Fig. 2.3: Different types of current distributions


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By denoting the surface current density as K (in amp/m) and volume current density as J (inamp/m 2) we can write:

......................................(2.2)

( It may be noted that )

Employing Biot-Savart Law, we can now express the magnetic field intensity H. In terms of these current distributions.

............................. for line current. ...........................(2.3a)

........................ for surface current ....................(2.3b)

....................... for volume current ......................(2.3c)To illustrate the application of Biot - Savart's Law, we consider the following example.

Example 2.1: We consider a finite length of a conductor carrying a current placed along z-axis as shown in the Fig 2.4. We determine the magnetic field at point P due to this currentcarrying conductor.

Fig. 2.4: Field at a point P due to a finite length current carrying conductor

With reference to Fig. 2.4, we find that

..........................................(2.4)

Applying Biot - Savart's law for the current element


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we can write,

..............................................(2.5)

Substituting we can write,

..............(2.6)

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

Therefore, .........................................................................................(2.7)

The value of the constant of proportionality 'K ' depends upon a property calledpermeability of the medium around the conductor. Permeability is represented bysymbol'm' and the constant 'K ' is expressed in terms of 'm' as

Magnetic field 'B' is a vector and unless we give the direction of 'dB ', its description isnot complete. Its direction is found to be perpendicular to the plane of 'r ' and 'dl '.

If we assign the direction of the current 'I ' to the length element 'dl ', the vector productdl x r has magnituder dl sinq and direction perpendicular to 'r ' and 'dl '.

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

Value of permeability changes from medium to medium. For ferromagnetic materials itis much higher than that for other materials. The permeability of free space (vacuum) isdenoted by the symbol 'm 0' and its value is4p x 10 7 Wb/Am


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Ampere's Circuital Law:

Ampere's circuital law states that the line integral of the magnetic field (circulation of H )around a closed path is the net current enclosed by this path. Mathematically,

......................................(24.8)

The total current I enc can be written as,

......................................(24.9)By applying Stoke's theorem, we can write

......................................(2.10)which is the Ampere's law in the point form.

Applications of Ampere's law:

We illustrate the application of Ampere's Law with some examples.

Example2.2: We compute magnetic field due to an infinitely long thin current carrying

conductor as shown in Fig. 2.5. Using Ampere's Law, we consider the close path to be acircle of radius as shown in the Fig. 4.5.

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

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

,i.e., .

By applying Ampere's law we can write,

......................................(2.11)

Therefore, which is same as equation (2.7)


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Fig. 2.5: Magnetic field due to an infinite thin current carrying conductor

Example 2.3: We consider the cross section of an infinitely long coaxial conductor, theinner conductor carrying a current I and outer conductor carrying current - I as shown in

figure 2.6. We compute the magnetic field as a function of as follows:

In the region

......................................(2.12)

............................(2.13)

In the region

......................................(2.14)

Fig. 2.6: Coaxial conductor carrying equal and opposite currents


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In the region

......................................(2.15)

........................................(2.16)

In the region

......................................(2.17)Lorentz force

A charged particle at rest will not interact with a static magnetic field. But if the chargedparticle is moving in a magnetic field, the magnetic character of a charge in motionbecomes evident. It experiences a deflecting force. The force is greatest when theparticle moves in a direction perpendicular to the magnetic field lines. At other angles,the force is less and becomes zero when the particles move parallel to the field lines. Inany case, the direction of the force is always perpendicular to the magnetic field linesand to the velocity of the charged particle.

Magnetic Flux Density

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

Definition Of Ampere

When two current carrying conductors are placed next to each other, we notice that eachinduces a force on the other. Each conductor produces a magnetic field around itself (BiotSavart law) and the second experiences a force that is given by the Lorentz force.


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Ampere's LawThe magnetic field in space around an electric current is proportional to the electriccurrent which serves as its source, just as the electric field in space is proportionalto the charge which serves as its source.

Ampere's law states that for any closed loop path, the sum of the length elementstimes the magnetic field in the direction of the length element is equal to thepermeability times the electric current enclosed in the loop.

Or

In the electric case, the relation of field to source is quantified in Gauss's Law


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which is a very powerful tool for calculating electric fields.

Application of Ampere's law:

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

Gauss's Law

Gauss's law for magnetic field This law deals with magnetic flux inside a closedsurface and is equivalent to Gauss's law for electric field discussed in Electric Chargeand Electric Field, connected electric flux jE and electric charge.

Andj E = E . A

Similarly, magnetic fluxf B can be defined as the number of lines of force crossing a unitarea.

Magnetic fluxf B = B.A

Since there are no free magnetic charges, the magnetic flux crossing a closed surfacewill always be zero. Thus Gauss's law of magnetic field says that the net magneticfluxf B out of any closed surface is zero.

or B.A = 0

Lenz's law

Soon after Faraday proposed his law of electromagnetic induction, Lenz gave the lawdetermining the direction of the induced emf.

Lenz's law may be stated as follows:

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

Lenz's law can be compared with the Newton's third law ? every action has equal andopposite reaction.

When an emf is generated by a change in magnetic flux according to the Faraday'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.

In the examples below, if the 'B' field is increasing, the induced field acts in oppositionto it. If it is decreasing, the induced field acts in the direction of the applied field to tryto keep it constant.


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Magnetic Flux

Faraday understood that the magnitude of the induced current in a loop was due to the"amount of magnetic field" passing through the loop.

To visualize 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 'B' times the perpendicular area that it penetrates.

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

For a closed surface, the sum of magnetic flux is always equal to zero (This is alsoknown as Gauss's law for magnetic field).

The standard unit for magnetic flux is a weber (Wb), it is the number of magnetic linesof force (Tesla) crossing a unit area (m2).

Magnetic Flux Density:

In simple matter, the magnetic flux density related to the magnetic field intensity aswhere called the permeability. In particular when we consider the free space

where H/m is the permeability of the free space. Magnetic fluxdensity is measured in terms of Wb/m 2 .

The magnetic flux density through a surface is given by:


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Wb ......................................(2.18)

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

flux passing through the surface is equal to the charge enclosed by the surface. In case of magnetic field isolated magnetic charge (i. e. pole) does not exist. Magnetic poles alwaysoccur in pair (as N-S). For example, if we desire to have an isolated magnetic pole bydividing the magnetic bar successively into two, we end up with pieces each having north (N)and south (S) pole as shown in Fig. 2.7 (a). This process could be continued until themagnets are of atomic dimensions; still we will have N-S pair occurring together. This meansthat the magnetic poles cannot be isolated.

Fig. 2.7: (a) Subdivision of a magnet (b) Magnetic field/ flux lines of a straight currentcarrying conductor

Similarly if we consider the field/flux lines of a current carrying conductor as shown in Fig. 2.7(b), we find that these lines are closed lines, that is, if we consider a closed surface, thenumber of flux lines that would leave the surface would be same as the number of flux linesthat would enter the surface.

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

......................................(2.19)

which is the Gauss's law for the magnetic field.

By applying divergence theorem, we can write:


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Hence, ......................................(2.20)

which is the Gauss's law for the magnetic field in point form.

Magnetic Scalar and Vector Potentials:

In studying electric field problems, we introduced the concept of electric potential that simplified the computation of electric fields for certain types of problems. In the same manner let us relate the magnetic field intensity to a scalamagnetic potential and write:

...................................(2.21)

From Ampere's law , we know that

......................................(2.22)

Therefore, ............................(2.23)

But using vector identity, we find that is valid only where . Thus the scalar magnetic

potential is defined only in the region where . Moreover, V m in general is not a single valued function of position.

This point can be illustrated as follows. Let us consider the cross section of a coaxial line asshown in fig 2.8.

In the region , and

Fig. 2.8: Cross Section of a Coaxial Line


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If V m is the magnetic potential then,

If we set V m = 0 at then c=0 and

We observe that as we make a complete lap around the current carrying conductor , we

reach again but V m this time becomes

We observe that value of V m keeps changing as we complete additional laps to pass throughthe same point. We introduced V m analogous to electostatic potential V . But for static electric

fields, and , whereas for steady magnetic field wherever

but even if along the path of integration.

We now introduce the vector magnetic potential which can be used in regions wherecurrent density may be zero or nonzero and the same can be easily extended to time varyingcases. The use of vector magnetic potential provides elegant ways of solving EM fieldproblems.

Since and we have the vector identity that for any vector , , we can

write .

Here, the vector field is called the vector magnetic potential. Its SI unit is Wb/m. Thus if can 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 . A vector function is

defined fully in terms of its curl as well as divergence. The choice of is made as follows.

...........................................(2.24)


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By using vector identity, .................................................(2.25)

.........................................(2.26)

Great deal of simplification can be achieved if we choose .

Putting , we get which is vector poisson equation.In Cartesian coordinates, the above equation can be written in terms of the components as

......................................(2.27a)

......................................(2.27b)

......................................(2.27c)

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

..........................................(2.28)

for which the solution is

..................(2.29)

In case of time varying fields we shall see that , which is known as Lorentzcondition, V being the electric potential. Here we are dealing with static magnetic field, so

.

By comparison, we can write the solution for A x as

...................................(2.30)

Computing similar solutions for other two components of the vector potential, the vector potential can be written as

.......................................(2.31)


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

current density . Similarly for line or surface current density we can write

...................................................(2.32)

respectively. ..............................(2.33)

The magnetic flux through a given area S is given by

.............................................(2.34)

Substituting

.........................................(2.35)

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

Magnetic Moment In A Magnetic Field

Themagnetic moment of an object is avector relating the aligningtorquein amagnetic

fieldexperienced by the object to the field vector itself. The relationship is given by

where

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

(ampere-meter).

Magnetic Scalar Potential

The magnetic scalar potential is another useful tool in describing the magnetic fieldaround a current source. It is only defined in regions of space in absence of (but could benear) currents.

The magnetic scalar potential is defined by the equation:
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Applying Ampere's Law to the above definition we get:

Since in any continuous field, the curl of a gradient is zero, this would suggest thatmagnetic scalar potential fields cannot support any sources. In fact, sources can besupported by applying discontinuities to the potential field (thus the same point can havetwo values for points along the disconuity). These discontinuities are also known as"cuts". When solvingmagnetostaticsproblems using magnetic scalar potential, the sourcecurrents must be applied at the discontinuity.

The magnetic scalar potential is suited to use around lines/loops of currents, but not aregion 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. It is often used in magnetostatics, but rarely used inother applications.

Magnetic Vector Potential

The magnetic vector potential is a three-dimensionalvector field whosecurlis themagnetic field in the theory of electromagnetism:

Since the magnetic field isdivergencefree (i.e. ), always exists.
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Unit III Electric And Magnetic Fiels In Materials

Poisson's equation

The derivation of Poisson's equation in electrostatics follows. SI units are used andEuclidean space is assumed.

Starting withGauss' lawfor electricity (also part of Maxwell's equations) in a differentialcontrol volume, we have:

means to take the divergence. is the electric displacement field.

is thecharge density.

Assuming the medium is linear, isotropic, and homogeneous then:

is thepermittivityof the the medium.is theelectric field.is the vacuum permittivity. is the relative permittivity of the medium.

By substitution and division, we have:

Since thecurlof the electric field is zero, it is defined by a scalar electric potential field,

Eliminating by substitution, we have a form of the Poisson equation:

Solving Poisson's equation for the potential requires knowing the charge densitydistribution. If the charge density is zero, thenLaplace's equationresults.
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Laplace's equation

In three dimensions, the problem is to find twice-differentiablereal-valued functions,of real variables,x, y, andz , such that

This is often written as

or

where div is thedivergence, and grad is thegradient, or

where is theLaplace operator .

Solutions of Laplace's equation are calledharmonic functions.

If the right-hand side is specified as a given function,f (x, y, z ), i.e.

then the equation is called "Poisson's equation." Laplace's equation and Poisson'sequation are the simplest examples of elliptic partial differential equations. The partialdifferential operator, , or , (which may be defined in any number of dimensions) iscalled theLaplace operator , or just the Laplacian

For electrostatic field, we have seen that

..........................................................................................(3.1)

Form the above two equations we can write

..................................................................(3.2)

Using vector identity we can write, ................(3.3)
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For a simple homogeneous medium, is constant and . Therefore,

................(3.4)

This equation is known as Poissons equation . Here we have introduced a new operator,

( del square), called the Laplacian operator. In Cartesian coordinates,

...............(3.4)

Therefore, in Cartesian coordinates, Poisson equation can be written as:

...............(3.5)

In cylindrical coordinates,

...............(3.6)

In spherical polar coordinate system,

...............(3.7)

At points in simple media, where no free charge is present, Poissons equation reduces to

...................................(3.8)

which is known as Laplaces equation.

Laplaces and Poissons equation are very useful for solving many practical electrostatic fieldproblems where only the electrostatic conditions (potential and charge) at some boundariesare known and solution of electric field and potential is to be found throughout the volume.We shall consider such applications in the section where we deal with boundary value

problems.

Polarization density in Maxwell's equations

The behavior of electric fields(E , D),magnetic fields(B, H ), charge density() andcurrent density(J ) are described byMaxwell's equations. The role of the polarizationdensityP is described below.
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Relations between E, D and P

The polarization densityP defines theelectric displacement field D as

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

Bound charge

Electric polarization corresponds to a rearrangement of the boundelectronsin thematerial, which creates an additional charge density, known as thebound charge densityb:so that the total charge density that enters Maxwell's equations is given bywhere f isthe free charge density (describing charges brought from outside).At the surface of thepolarized material, the bound charge appears as asurface chargedensitywhere is thenormal vector . If P is uniform inside the material, this surface charge is the only boundcharge.

When the polarization density changes with time, the time-dependent bound-chargedensity creates acurrent densityof so that the total current density that enters Maxwell'sequations is given by whereJ f is the free-charge current density, and the second term is acontribution from themagnetization(when it exists).

Capacitance and Capacitors

Capacitance is a measure of the amount of electric chargestored (or separated) for agiven electric potential. The most common form of charge storage device is a two-platecapacitor . If the charges on the plates are +Q and -Q, and V gives the voltage differencebetween the plates, then the capacitance is given by

TheSIunit of capacitance is the farad; 1 farad = 1coulombper volt.

Capacitors

The capacitance of the majority of capacitors used in electronic circuits is several orderof magnitude smaller than the farad. The most common subunits of capacitance in usetoday are themillifarad (mF),microfarad(F), thenanofarad (nF) and thepicofarad(pF)

The capacitance can be calculated if the geometry of the conductors and the dielectricproperties of the insulator between the conductors are known. For example, thecapacitance of aparallel-plate capacitor constructed of two parallel plates of areaAseparated by a distanced is approximately equal to the following:
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or

where

C is the capacitance infarads, F s is the staticpermittivityof the insulator used (or 0 for a vacuum)A 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 =1)d is the separation between the plates, measured inmetres

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

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

charge density . Since the potential of the conductor is given by , the

potential of the conductor will also increase maintaining the ratio same. Thus we can write

where the constant of proportionality C is called the capacitance of the isolatedconductor. SI unit of capacitance is Coulomb/ Volt also called Farad denoted by F . It can Itcan be seen that if V =1, C = Q. Thus capacity of an isolated conductor can also be definedas the amount of charge in Coulomb required to raise the potential of the conductor by 1Volt.

Of considerable interest in practice is a capacitor that consists of two (or more) conductorscarrying equal and opposite charges and separated by some dielectric media or free space.The conductors may have arbitrary shapes. A two-conductor capacitor is shown in figure 3.1
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Fig 3.1: Capacitance and CapacitorsWhen a d-c voltage source is connected between the conductors, a charge transfer occurs which results into a positivecharge on one conductor and negative charge on the other conductor. The conductors are equipotential surfaces andthe field lines are perpendicular to the conductor surface. If V is the mean potential difference between the conductors,

the capacitance is given by . Capacitance of a capacitor depends on the geometry of the conductor and thepermittivity of the medium between them and does not depend on the charge or potential difference betweenconductors. The capacitance can be computed by assuming Q(at the same time - Q on the other conductor), first

determining using Gausss theorem and then determining . We illustrate this procedure by taking theexample of a parallel plate capacitor Example: Parallel plate capacitor

Fig 3.2: Parallel Plate Capacitor For the parallel plate capacitor shown in the figure 3.2, let each plate has area A and a distance h separates the plates.

A dielectric of permittivity fills the region between the plates. The electric field lines are confined between 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 with densities and - , .

By Gausss theorem we can write, .......................(3.9)


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As we have assumed to be uniform and fringing of field is neglected, we see that E is

constant in the region between the plates and therefore, we can write . Thus,

for a parallel plate capacitor we have,........................(3.10)

Series and parallel Connection of capacitors

Capacitors are connected in various manners in electrical circuits; series and parallelconnections are the two basic ways of connecting capacitors. We compute the equivalentcapacitance for such connections.

Series Case: Series connection of two capacitors is shown in the figure 3.3. For this casewe can write,

.......................(3.11)

Therefore, .......................(3.12)

Fig 3.3: Series Connection of Capacitors Fig 3.4: Parallel Connection of Capacitors

The same approach may be extended to more than two capacitors connected in series.

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

The total charge


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Electrostatic Energy and Energy Density

We have stated that the electric potential at a point in an electric field is the amount of workrequired to bring a unit positive charge from infinity (reference of zero potential) to that point.To determine the energy that is present in an assembly of charges, let us first determine theamount of work required to assemble them. Let us consider a number of discrete charges

Q 1, Q2,......., Q N are brought from infinity to their present position one by one. Since initiallythere is no field present, the amount of work done in bring Q 1 is zero. Q 2 is brought in thepresence of the field of Q 1, the work done W 1= Q2V 21 where V 21 is the potential at the locationof Q 2 due to Q 1. Proceeding in this manner, we can write, the total work done

.................................................(3.13)

Had the charges been brought in the reverse order,

.................(3.14)

Therefore,

................(3.15)

Here V IJ represent voltage at the I th charge location due to J th charge. Therefore,

Or, ................(3.16)

If instead of discrete charges, we now have a distribution of charges over a volume v thenwe can write,

................(3.17)

where is the volume charge density and V represents the potential function.

Since, , we can write


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.......................................(3.18)

Using the vector identity,

, we can write

................(3.19)

In the expression , for point charges, since V varies as and D varies as ,

the term V varies as while the area varies as r 2. Hence the integral term varies at least

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

Thus the equation for W reduces to

................(3.20)

, is called the energy density in the electrostatic field.

Boundary conditions for Electrostatic fields

In our discussions so far we have considered the existence of electric field in the homogeneousmedium. Practical electromagnetic problems often involve media with different physical properties.Determination 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, we first consider the field behavior in somecommon material media.

In general, based on the electric properties, materials can be classified into three categories:conductors, 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. Most metals belong to this group. The electrons in the atoms of insulators or dielectrics remain confined to their orbits and under normal circumstances they are notliberated under the influence of an externally applied field. The electrical properties of


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semiconductors fall between those of conductors and insulators since semiconductors havevery few numbers of free charges.

The parameter conductivity is used characterizes the macroscopic electrical property of amaterial medium. The notion of conductivity is more important in dealing with the current flowand hence the same will be considered in detail later on.

If some free charge is introduced inside a conductor, the charges will experience a force dueto mutual repulsion and owing to the fact that they are free to move, the charges will appear on the surface. The charges will redistribute themselves in such a manner that the field

within the conductor is zero. Therefore, under steady condition, inside a conductor .

From Gauss's theorem it follows that

= 0 .......................(3.21)

The surface charge distribution on a conductor depends on the shape of the conductor. The charges on the surface of the conductor will not be in equilibrium if there is a tangential component of the electric field is present, which wouldproduce movement of the charges. Hence under static field conditions, tangential component of the electric field on theconductor surface is zero. The electric field on the surface of the conductor is normal everywhere to the surface

Since the tangential component of electric field is zero, the conductor surface is an equipotential surface . As = 0inside the conductor, the conductor as a whole has the same potential. We may further note that charges require a finite

time to redistribute in a conductor. However, this time is very small sec for good conductor like copper.

Let us now consider an interfacebetween a conductor and freespace as shown in the figure 3.5.

Fig 3.5: Boundary Conditions for at the surface of a Conductor

Let us consider the closed path pqrsp for which we can write,

.................................(3.22)

For and noting that inside the conductor is zero, we can write

=0.......................................(3.23)

E t is the tangential component of the field. Therefore we find that


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E t = 0 ...........................................(3.24)

In order to determine the normal component E n, the normal component of , at the surfaceof the conductor, we consider a small cylindrical Gaussian surface . Let represent thearea of the top and bottom faces and represents the height of the cylinder. Once again,

as , we approach the surface of the conductor. Since = 0 inside the conductor iszero,

.............(3.25)

..................(3.26)

Therefore, we can summarize the boundary conditions at the surface of a conductor as:

E t = 0 ........................(3.27)

.....................(3.28)

Behavior of dielectrics in static electric field: Polarization of dielectric

Here we briefly describe the behavior of dielectrics or insulators when placed in static electricfield. Ideal dielectrics do not contain free charges. As we know, all material media arecomposed of atoms where a positively charged nucleus (diameter ~ 10 -15m) is surrounded bynegatively charged electrons (electron cloud has radius ~ 10 -10m) mo

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