electromagnetic field theory unit 1

8/19/2019 electromagnetic field theory unit 1

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that in turn change the field and hence the separation of cause and effect is not alwaysvisible.

Sources of EMF:

• urrent carrying conductors.

• Mobile phones.

• Microwave oven.• omputer and !elevision screen.

• *igh voltage 'ower lines.

Effects of Electromagnetic fields:

• 'lants and Animals.

• *umans.

• Electrical components.

Fields are classified as

• Scalar field

•

$ector field.

Electric charge is a fundamental property of matter. harge e)ist only in positive or negativeintegral multiple of electronic charge, +e, e -./ 0 -/+-1 coulombs. 2"t may be noted herethat in -13, Murray 4ell+Mann hypothesi(ed Quarks as the basic building bloc5s of matters.#uar5s were predicted to carry a fraction of electronic charge and the e)istence of #uar5shave been e)perimentally verified.6 'rinciple of conservation of charge states that the totalcharge 7algebraic sum of positive and negative charges8 of an isolated system remainsunchanged, though the charges may redistribute under the influence of electric field.9irchhoffs urrent %aw 79%8 is an assertion of the conservative property of charges underthe implicit assumption that there is no accumulation of charge at the ;unction.

Electromagnetic theory deals directly with the electric and magnetic field vectors where ascircuit theory deals with the voltages and currents. $oltages and currents are integratedeffects of electric and magnetic fields respectively. Electromagnetic field problems involvethree space variables along with the time variable and hence the solution tends to becomecorrespondingly comple). $ector analysis is a mathematical tool with which electromagneticconcepts are more conveniently e)pressed and best comprehended. Since use of vectoranalysis in the study of electromagnetic field theory results in real economy of time andthought, we first introduce the concept of vector analysis.

Vector Analysis

!he &uantities that we deal in electromagnetic theory may be either scalar or !ectors 2!hereare other class of physical &uantities called Tensors: where magnitude and direction varywith co ordinate a)es6. Scalars are &uantities characteri(ed by magnitude only and algebraicsign. A &uantity that has direction as well as magnitude is called a vector.


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A vector can be written as, , where, is the magnitude and is the

unit vector which has unit magnitude and same direction as that of .

!wo vector and are added together to give another vector . >e have

................7-.-8

%et us see the animations in the ne)t pages for the addition of two vectors, which has tworules:

1 "arallelogram la# and $ %ea& ' tail rule


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Scaling of a vector is defined as , where is scaled version of vector and is ascalar.Some important laws of vector algebra are:


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ommutative %aw..........................................7-.?8

Associative %aw.............................................7-.@8

=istributive %aw ............................................7-.8

!he position vector of a point P is the directed distance from the origin 7O8 to P , i.e.,

.

(ig 1)* Distance Vector

"f = OP and OQ are the position vectors of the points ' and # then the distancevector

"ro&uct o+ Vectors

>hen two vectors and are multiplied, the result is either a scalar or a vector dependinghow the two vectors were multiplied. !he two types of vector multiplication are:

Scalar product 7or dot product8 gives a scalar.

$ector product 7or cross product8 gives a vector.

!he dot product between two vectors is defined as |A||B|cosθ AB ..................7-.8

$ector product

is unit vector perpendicular to and


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Fig 1.4: Vector dot product

!he dot product is commutative i.e., and distributive i.e.,

. Associative law does not apply to scalar product.

!he vector or cross product of two vectors and is denoted by . is a vector

perpendicular to the plane containing and , the magnitude is given by anddirection is given by right hand rule as e)plained in Figure -..

Be)t

http://nptel.iitm.ac.in/courses/Webcourse-contents/IIT-%20Guwahati/em/modules/chap1/slides/slide11.htmhttp://nptel.iitm.ac.in/courses/Webcourse-contents/IIT-%20Guwahati/em/modules/chap1/slides/slide11.htm


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............................................................................................7-.C8

where is the unit vector given by, .

!he following relations hold for vector product.

= i.e., cross product is non commutative ..........7-.D8

i.e., cross product is distributive.......................7-.18

i.e., cross product is non associative..............7-.-/8

,calar an& !ector tri-le -ro&uct

Scalar triple product .................................7-.--8

$ector triple product ...................................7-.-38

o+ordinate Systems

"n order to describe the spatial variations of the &uantities, we re&uire using appropriate co+

ordinate system. A point or vector can be represented in a curvilinear coordinate systemthat may be orthogonal or non-orthogonal .

An orthogonal system is one in which the co+ordinates are mutually perpendicular. Bon+orthogonal co+ordinate systems are also possible, but their usage is very limited in practice .

%et u constant, v constant and w constant represent surfaces in a coordinate system,

the surfaces may be curved surfaces in general. Furthur, let , and be the unitvectors in the three coordinate directions7base vectors8. "n a general right handed orthogonalcurvilinear systems, the vectors satisfy the following relations :

.....................................7-.-?8

!hese e&uations are not independent and specification of one will automatically imply theother two. Furthermore, the following relations hold


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................7-.-@8

A vector can be represented as sum of its orthogonal components,...................7-.-8

"n general u, v and w may not represent length. >e multiply u, v and w by conversion factorsh-,h3 and h? respectively to convert differential changes du, dv and dw to corresponding

changes in length dl -, dl 3, and dl ?. !herefore

...............7-.-8

"n the same manner, differential volume dv can be written as and

differential area d s- normal to is given by, . "n the same manner,

differential areas normal to unit vectors and can be defined.

In the +ollo#ing sections #e &iscuss three most commonly use& orthogonal co.or&inate systems/ !i0

1) Cartesian or rectangular2 co.or&inate system

$) Cylin&rical co.or&inate system

*) ,-herical -olar co.or&inate system

Cartesian Co.or&inate ,ystem

"n artesian co+ordinate system, we have, (u,v,w8 7 x,y,z 8. A point P 7 x/, y/, z /8 in artesian co+ordinate system is represented as intersection of three planes x x/, y y/ and z z /. !heunit vectors satisfies the following relation:


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"n cartesian co+ordinate system, a vector can be written as . !he

dot and cross product of two vectors and can be written as follows:

.................7-.-18

....................7-.3/8

Since x, y and z all represent lengths, h- h3 h?-. !he differential length, area and volumeare defined respectively as


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................7-.3-8

.................................7-.338

Cylin&rical Co.or&inate ,ystem

For cylindrical coordinate systems we have a point isdetermined as the point of intersection of a cylindrical surface r = r 0, half plane containing the

(+a)is and ma5ing an angle ; with the )( plane and a plane parallel to xy plane locatedat z = z 0 as shown in figure C on ne)t page.

"n cylindrical coordinate system, the unit vectors satisfy the following relations

A vector can be written as , ...........................7-.3@8

!he differential length is defined as,

......................7-.38

.....................7-.3?8


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Trans+ormation 3et#een Cartesian an& Cylin&rical coor&inates

%et us consider is to be e)pressed in artesian co+ordinate as

. "n doing so we note thatand it applies for other components as well.


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!hese relations can be put conveniently in the matri) form as:

.....................7-.?/8

themselves may be functions of as:

............................7-.?-8

!he inverse relationships are: ........................7-.?38


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Fig 1.10: Spherical Polar Coordinate Syste

!hus we see that a vector in one coordinate system is transformed to another coordinatesystem through two+step process: Finding the component vectors and then variabletransformation.

Spherical 'olar oordinates:

For spherical polar coordinate system, we have, . A point isrepresented as the intersection of

7i8 Spherical surface r=r 0

7ii8 onical surface ,and

7iii8 half plane containing (+a)is ma5ing angle with the xz plane as shown in the figure-.-/.

!he unit vectors satisfy the following relationships: .....................................7-.??8

!he orientation of the unit vectors are shown in the figure -.--.


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A vector in spherical polar co+ordinates is written as : and

For spherical polar coordinate system we have h--, h3 r and h? .


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(ig 1)1$32 E4-lo&e& !ie#

>ith reference to the Figure -.-3, the elemental areas are:

.......................7-.?@8

and elementary volume is given by

........................7-.?8

Coor&inate trans+ormation 3et#een rectangular an& s-herical -olar

>ith reference to the figure -.-? ,we can write the following e&uations:


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........................................................7-.?8

4iven a vector in the spherical polar coordinate system, itscomponent in the cartesian coordinate system can be found out as follows:

.................................7-.?C8


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Similarly,

.................................7-.?Da8

.................................7-.?Db8

!he above e&uation can be put in a compact form:

.................................7-.?18

!he components themselves will be functions of . are

related to x, y and z as:

....................7-.@/8

and conversely,

.......................................7-.@-a8

.................................7-.@-b8

.....................................................7-.@-c8

sing the variable transformation listed above, the vector components, which are functionsof variables of one coordinate system, can be transformed to functions of variables of othercoordinate system and a total transformation can be done.

5ine/ sur+ace an& !olume integrals

"n electromagnetic theory, we come across integrals, which contain vector functions. Somerepresentative integrals are listed below:

"n the above integrals, and respectively represent vector and scalar function of spacecoordinates. C ,S and V represent path, surface and volume of integration. All these integrals


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are evaluated using e)tension of the usual one+dimensional integral as the limit of a sum,i.e., if a function f ( x) is defined over arrange a to b of values of x, then the integral is given by

.................................7-.@38

where the interval 7a,b8 is subdivided into n continuous interval of lengths .

5ine Integral %ine integral is the dot product of a vector with a specified C in other

words it is the integral of the tangential component along the curve C .

As shown in the figure -.-@, given a vector around C , we define the integral

as the line integral of E along the curve .

"f the path of integration is a closed path as shown in the figure the line integral becomes a

closed line integral and is called the circulation of around C and denoted as asshown in the figure -.-.


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(ig 1)16 Close& 5ine Integral

Surface !ntegral :

4iven a vector field , continuous in a region containing the smooth surface S , we define

the surface integral or the flu) of through S as assurface integral over surface S.

(ig 1)17 ,ur+ace Integral

"f the surface integral is carried out over a closed surface, then we write

Volue !ntegrals:


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>e define or as the volume integral of the scalar function f 7function of spatial

coordinates8 over the volume V . Evaluation of integral of the form can be carried outas a sum of three scalar volume integrals, where each scalar volume integral is a component

of the vector

The Del O-erator

!he vector differential operator was introduced by Sir >. R. *amilton and later ondeveloped by '. 4. !ait.

Mathematically the vector differential operator can be written in the general form as:

.................................7-.@?8

"n artesian coordinates:

................................................7-.@@8

"n cylindrical coordinates:

...........................................7-.@8

and in spherical polar coordinates:

.................................7-.@8

8ra&ient o+ a ,calar +unction

%et us consider a scalar field V 7u,v,w8 , a function of space coordinates.

4radient of the scalar field V is a vector that represents both the magnitude and direction ofthe ma)imum space rate of increase of this scalar field V .


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(ig 1)19 8ra&ient o+ a scalar +unction

As shown in figure -.-C, let us consider two surfaces S -and S 3 where the function V hasconstant magnitude and the magnitude differs by a small amount dV . Bow as one movesfrom S - to S 3, the magnitude of spatial rate of change of V i.e. d$Gdl depends on the directionof elementary path length dl, the ma)imum occurs when one traverses from S -to S 3along apath normal to the surfaces as in this case the distance is minimum.


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Also we can write,

............................7-.-8


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"t may further be noted that since magnitude of depends on the direction of dl ,

it is called the &irectional &eri!ati!e. "f is called the scalar potential function of

the vector function .

Di!ergence o+ a Vector (iel&

"n study of vector fields, directed line segments, also called flu) lines or streamlines,represent field variations graphically. !he intensity of the field is proportional to the density of lines. For e)ample, the number of flu) lines passing through a unit surface S normal to thevector measures the vector field strength.

(ig 1)1: (lu4 5ines

>e have already defined flu) of a vector field as

....................................................7-.C8

For a volume enclosed by a surface,

.........................................................................................7-.D8

>e define the divergence of a vector field at a point P as the net outward flu) from avolume enclosing P , as the volume shrin5s to (ero.

.................................................................7-.18

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


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(ig 1)1; E!aluation o+ &i!ergence in cur!ilinear coor&inate

%et us consider a differential volume centered on point P(u,v,w) in a vector field . !he flu)through an elementary area normal to u is given by ,

........................................7-./8

Bet outward flu) along u can be calculated considering the two elementary surfaces perpendicular to u .

.......................................7-.onsidering the contribution from all si) surfaces that enclose the volume, we can write

.......................................7-.38

*ence for the artesian, cylindrical and spherical polar coordinate system, the e)pressions for divergencwritten as:

In Cartesian coor&inates

................................7-.?8

"n cylindrical coordinates:


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....................................................................7-.@8

and in spherical polar coordinates:

......................................7-.8

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

• =ivergence of a vector field gives a scalar.

• ..............................................................................7-.8

Di!ergence theorem =ivergence theorem states that the volume integral of the divergence of vector field is e&ualto the net outward flu) of the vector through the closed surface that bounds the volume.

Mathematically,

Proof:

%et us consider a volume V 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 denoted

by S k . "nterior to the volume, cells have common surfaces. Hutward flu) through thesecommon surfaces from one cell becomes the inward flu) for the neighboring cells. !hereforewhen the total flu) from these cells are considered, we actually get the net outward flu)through the surface surrounding the volume. *ence we can write:

......................................7-.C8

"n the limit, that is when and the right hand of the e)pression can be

written as .

*ence we get , which is the divergence theorem.

Curl of a vector field:

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


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Curl of a vector field is a measure of the vector fields tendency to rotate about a

point. url , also written as is defined as a vector whose magnitude isma)imum of the net circulation per unit area when the area tends to (ero and itsdirection is the normal direction to the area when the area is oriented in such away so as to ma5e the circulation ma)imum.

!herefore, we can write:

......................................7-.D8

!o derive the e)pression for curl in generali(ed curvilinear coordinate system, we first

compute and to do so let us consider the figure -.3/ :

(ig 1)$


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Adding the contribution from all components, we can write:

........................................................................7-.C@8

!herefore, ........................7-.C8

"n the same manner if we compute for and we can write,

.......7-.C8

!his can be written as,

......................................................7-.CC8

"n artesian coordinates: .......................................7-.CD8

"n ylindrical coordinates, ....................................7-.C18

"n Spherical polar coordinates, ..............7-.D/8

url operation e)hibits the following properties:


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..............7-.D-8

,toke=s theorem

"t states that the circulation of a vector field around a closed path is e&ual to the integral of

over the surface bounded by this path. "t may be noted that this e&uality holds

provided and are continuous on the surface.

i.e,

..............7-.D38

"roo+%et us consider an area S that is subdivided into large number of cells as shown in thefigure -.3-.

(ig 1)$1 ,tokes theorem

%et k t hcell has surface area and is bounded path "5 while the total area isbounded by path ". As seen from the figure that if we evaluate the sum of the lineintegrals around the elementary areas, there is cancellation along every interiorpath and we are left the line integral along path ". !herefore we can write,


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..............7-.D?8

As /

. .............7-.D@8

which is the sto5es theorem.

A,,I8N>ENT "RO?5E>,

1. "n the artesian coordinate system verify the following relations for a scalar function

and a vector function

a.

b.

c.

2. An electric field e)pressed in spherical polar coordinates is given by .

=etermine and at a point .

3. Evaluate over the surface of a sphere of radius centered at theorigin.

4. Find the divergence of the radial vector field given by .

5. A vector function is defined by . Find around the contour

shown in the figure '-.? . Evaluate over the shaded area and verify that


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(igure "1)*

electromagnetic field theory unit 1

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