Objective: Introduction:

Transcription

1 Electostatics 39

2 Objective: This chapte intoduces the fundamental equation of electostatics - Coulomb's law, which descibes the foce between two point chages. The electic field intensity fo vaious chage distibutions is explained and its expession is deived using coulomb s law. The eade is intoduced to the elation between the distibutions of electic chage and the esulting electic field and electic potential, though Gauss s law. Late this chapte discusses the electic field in fee space, conductos and dielectics and also about the behavio of the electic fields at thei boundaies. It will be inteesting to see how Poisson s equation can be used to find the electic potential fo a given chage distibution and how Laplace s equation can be used to accuately descibe the behavio of electic potentials. Capacitance and enegy stoed in a capacito is also discussed biefly towads the end of the chapte. Intoduction: The ancient Geeks obseved that when the fossil esin ambe was ubbed, small light-weight objects wee attacted. Yet, upon contact with the ambe, they wee then epelled. No futhe significant advances in the undestanding of this mysteious phenomenon wee made until the eighteenth centuy when moe quantitative electification expeiments showed that these effects wee due to electic chages, the souce of all effects we will study in this text. 4

3 . Electostatics: Electostatics is the banch of science that deals with the phenomena and popeties of stationay o slow-moving (without acceleation) electic chages. Electostatic phenomena aise fom the foces that electic chages exet on each othe. Such foces ae descibed by Coulomb's law. Electostatics involves the buildup of chage on the suface of objects due to contact with othe sufaces. Although chage exchange happens wheneve any two sufaces contact and sepaate, the effects of chage exchange ae usually only noticed when at least one of the sufaces has a high esistance to electical flow. Example: A od of plastic ubbed with fu o a od of glass ubbed with silk will attact small pieces of pape and is said to be electically chaged. The chage on plastic ubbed with fu is defined as negative, and the chage on glass ubbed with silk is defined as positive... Coulomb s law Coulomb's law o Coulomb's invese-squae law states that: "The foce between any two point chages is. Diectly popotional to the magnitude of each chage. Invesely popotional to the squae of the distance between them 3. Diected along the staight line joining the two chages Mathematically the magnitude of the foce is given by, F QQ QQ R Newtons In vecto fom, foce between two point chages can be witten as, 4

4 F QQ = k QQ R a R = 4πε QQ R a R = 4πε QQ 3 R RNewtons Whee, Q and Q ae the magnitude of point chages (Coulombs), R is the distance between the point chages (metes), ε o is the pemittivity of the medium in fee space (F/m) [ε =8.854x - ], F QQ is the foce exeted by one chage ove the othe (Newton) and a R is the unit vecto in the diection of foce As shown in figue below, if the two chages ae of same polaity then the foce is epulsive and if the chages ae of opposite polaities then the foce is attactive as shown in figue below. Conside point chages q and q ae at a distance of and espectively fom a point of efeence (oigin) as shown in figue below The coulomb s law can be modified to F qq = F qq ' qq = Newtons 3 ' 4πε 4

5 43.. Foce in a System of Discete Point Chages: Fo the sytem of point chages shown below, the foce expeienced by chage q due to all othe foces is given by supeposition pinciple. Pinciple of supeposition: fo all linea systems, the net esponse at a given place and time caused by two o moe stimuli is the sum of the esponses which would have been caused by each stimulus individually. n F F F F F = ( ) ( ) ( ) ( ) ( ) = = = n j j j j n n n q q q q q q q q q q F πε πε πε πε πε ( ) ( ) = + = n i j j j q q q q F πε πε and so on. In geneal foce on the i th chage is given by ( ) = = n i j j j i j i j i i q q F 4πε

6 . Electic field intensity: Electic field intensity is the foce expeienced by a unit positive test chage that is bought into the electic field ceated by a chage of +q coulombs. The foce on the unit test chage acts along the adial (flux) line emanating fom the chage and it acts outwads fom chage +q (epulsive foce). The electic field intensity has the same diection as that of foqce and in measued in V/m. It is expessed as F q = = ar V/m; 4πε E qtest R.. Electic flux lines: Electic flux line o line of foce is the path followed by an electic chage (fee to move in an electic field) when placed inside an electic field. It is an imaginay continuous line o cuve dawn in an electic field such that tangent to it at any point gives the diection of the electic foce at that point. Popeties of Electic Flux Lines: A tangent to the electic flux line at any point gives the electic field diection at that point. The density of the electic flux lines gives the magnitude of the field. Electic field lines ae consideed to oiginate on positive electic chages and teminate on negative chages as shown in the figue below. 44

7 Field lines diected into a closed suface ae consideed negative; those diected out of a closed suface ae positive. If thee is no net chage within a closed suface, evey field line diected into the suface continues though the inteio and is diected outwad elsewhee on the suface. The negative flux just equals in magnitude the positive flux, so that the net, o total, electic flux is zeo. If a net chage is contained inside a closed suface, the total flux though the suface is popotional to the enclosed chage, positive if it is positive, negative if it is negative. 45

8 Two electic lines of foce cannot intesect each othe. Two electic lines of foce poceeding in the same diection epel each othe. Two electic lines of foce poceeding in the opposite diection attact each othe... Electic field intensity at a point due to point chages: Conside the figue shown below. q and q ae two positive chages at distances R and R fom the oigin. Electic field intensity due to these two chages at a point P, at a distance R fom the oigin can be found by the pinsiple of supeposition. 46

9 q R R E = 4πε R R ( ) The electic field intensity at a point P due to the chage q is ( ) The electic field intensity at a point P due to the chage q is E q R R = 4πε R R The electic field intensity at a point P due to the chage q and q is E = E + E ( R R ) q ( R R ) q = + 4πε R R 4πε R R..3 Electic field intensity at a point due to an aay point chages: q 3 q q 4 q 5 q P n q n Electic field intensity at point P due to chage q is E = q 4πε Electic field intensity at point P due to chage q is E q = and so on. By 4πε supeposition pinciple, Eat P due to the aay of chages q, q, q n is E = 4πε n k = q k k k 47

10 ..4 Electic field intensity at a point due to continuous distibution of chages: Conside a line chage as shown above with line o linea chage density ρ L C/m. the foce exeted by such a line chage on a test chage q at a point P, placed at a pependicula distance m fom the line chage is given by F = q πε 4 ( L ρ L dl) newtons. Continuously distibuted chages Line chage Suface chage Volume chage So the electic field intensity at that point is the foce pe unit test chage placed at that point o ( dl F ρ L ) L E = = V/m. q 4πε 48

11 Similaly, we can wite the electic field fo a suface chage distibution as ρ ( S ds) F S E = = and fo a volume chage distibution as q 4πε ρ ( V dv) F V E = = and the q 4πε coesponding figues ae shown below. In the figue below, linea chage density is denoted as λ, suface chage density as σ and volume chage density as ρ. 49

12 ..5 Electic field intensity due to infinitely chaged conducto: 5

13 5

14 ..6 Electic field due to chaged cicula ing: 5

15 ..7 Electic field due to infinite sheet of chage: 53

16 54

17 .3 Electic flux density: The net flux passing nomal though a unit suface aea is called electic flux density D. It has a specific diection which is nomal to the suface aea unde consideation; hence it is a vecto field. If Ψ is the total flux and S is the total suface aea, then D = Ψ. S If ds is the diffeential suface aea, dψ is the total flux lines cossing nomal though the diffeential suface aea ds and an is the unit vecto in the diection nomal to the diffeential suface aea, then dψ D = anc/m. ds Relation between electic field intensity and flux density is D = E = ε ε E. Note that F, Eand Dact in the same diection. ε 55

18 Ddue to a point chage = Q 4π a C/m Ddue to a line chage = L ρ dl L 4π a C/m Ddue to infinite line chage = ρ L a C/m π Ddue to a suface chage = ρ S S 4π ds a C/m Ddue to infinite sheet of chage = ρ S an C/m Ddue to a volume chage = ρ V V 4π dv a C/m poblem: 56

19 57

20 .4 Gauss s Law: Gauss's law, also known as Gauss's flux theoem, is a law elating the distibution of electic chage to the esulting electic field. It poves that the electic flux though any closed suface is popotional to the enclosed electic chage. Statement: Suface integal of the nomal component of the electic flux density vecto Dove any closed suface is equal to the total chage enclosed by the suface ds = Ψ = S D Q (O) an altenate definition can be given as, Suface integal of the electic field vecto Eove any closed suface in fee space is given by Q total chage enclosed by the suface = Ψ = Poof: S E ds Q ε ε, whee Q is the The field lines emanating fom a point chage in all diections is shown below. We ll be consideing spheical co-odinate system to pove gauss law.. 58

21 The numbe of flux lines emanating fom a point chage is equal to its magnitude of chage, that is, Ψ=Q. the following can be witten about spheical co-odinate system. θ φ a n = a ; ds dsan sinθd d a Q Q = ; D = a 4πε 4π E a = = Taking LHS of gauss law, S S D ds = D ds = D ds = sinθdθ S S S Q 4π Q 4π Q 4π a sinθdθdφa sinθdθdφ π dφ ds Q E θ a n ds E 59

22 S S S S Q D ds = *π * 4π Q D ds = Q D ds = * [ cosθ ] D ds = Q = Ψ π sinθdθ Hence, gauss law is veified. π Gaussian suface: A Gaussian suface is a closed suface in thee dimensional spaces though which the flux of an electomagnetic field is calculated. It is an abitay closed suface, used in conjunction with Gauss's law in ode to calculate the total enclosed electic chage. Gaussian pillbox, sphee and cylinde ae examples of Gaussian suface..4. Electic field intensity of infinite line chage using gauss law: 6

23 6

24 .4. Electic field intensity of infinite sheet of chage using gauss law: 6

25 .4.3 Electic field intensity due to co-axial cable using gauss law: 63

26 64

27 .4.4 Electic field intensity due to spheical shell of chage using gauss law: 65

28 66

29 67

30 .4.5 Electic field intensity due to unifomly chaged sphee using gauss law: 68

31 69

32 7

33 Wok done: If we have two chages of opposite sign, wok must be done to sepaate them in opposition to the attactive Coulomb foce. This wok can be egained if the chages ae allowed to come togethe. Similaly, if the chages have the same sign, wok must be done to push them togethe; this wok can be egained if the chages ae allowed to sepaate. A chage gains enegy when moved in a diection opposite to a foce. This is called potential enegy because the amount of enegy depends on the position of the chage in a foce field. The wok W, equied to move a test chage q t, along any path fom the adial distance a, to the distance b with a foce that just ovecomes the coulombic foce fom a point chage q, as shown in Figue above is W a = F dljoules. The minus sign in font of b the integal is necessay because the quantity W epesents the wok we must exet on the test chage in opposition to the coulombic foce between chages qqt = 4πε a W b a dl, Whee, dl = da qq = a Newtons. 4πε t F 7

34 W a a qqt a da qqt d qqt = = 4πε 4πε 4 b πε b = a b W = qq t πε a 4 b Joules Electic potential: Definition: It is the amount of wok needed to move a unit test chage fom a efeence point to a specific point against and inside an electic field. Typically, the efeence point is any point beyond the influence of the electic field. It is a scala. E Equipotential lines +q C a Moving q t fom B to A A B +q t C b Wok done in moving the test chage q t fom a distance b to a isw = qq t πε a 4 b 7

35 But electic potential is the wok done pe unit test chage. So q t =. Also, the unit test chage is moved into the electic field fom infinity. So b =. V = W q qq t 4πε a = q b q = 4πε a q = 4πε t t a volts Potential diffeence: Definition: It is the amount of wok needed to move a unit test chage fom one point inside electic field to anothe point inside electic field and at the same time moving against the electic field. It is a scala. E Equipotential lines +q C a A Moving q t fom B to A B +q t C b 73

36 Wok done in moving the test chage q t fom B to A (both A and B ae inside the electic field) isw = qq t πε a 4 b Theefoe, the potential diffeence between points A and B can be witten as, qq t W 4πε q q q V a b AB = = = = [ ] = [ VA VB ]volts qt qt 4πε a b 4πε a 4πε a Equi-potential sufaces: V>V>..>V8 +qc V V V3 V4 V5 V6 V7 V8 E An equi-potential line is a closed contou aound a chage, along which at all points the potential is the same. The potential diffeence between one point on an equi-potential line and any point on a second equi-potential line is same. The dop in potential when a chage moves fom ed to oange line is V- V volts. 74

37 Similaly an equi-potential suface is a closed suface aound a chage and at all points on the suface the potential is the same. LAPLACE'S AND POISSON'S EQUATIONS A useful appoach to the calculation of electic potentials is to elate that potential to the chage density which gives ise to it. The electic field is elated to the chage density by the divegence elationship and the electic field is elated to the electic potential by a gadient elationship Theefoe the potential is elated to the chage density by Poisson's equation In a chage-fee egion of space, this becomes LaPlace's equation This mathematical opeation, the divegence of the gadient of a function, is called the LaPlacian. Expessing the LaPlacian in diffeent coodinate systems to take advantage of the symmety of a chage distibution helps in the solution fo the electic potential V. Fo example, if the chage distibution has spheical symmety, you use the LaPlacian in spheical pola coodinates. 75

38 Since the potential is a scala function, this appoach has advantages ove tying to calculate the electic field diectly. Once the potential has been calculated, the electic field can be computed by taking the gadient of the potential. DIELECTRICS A dielectic is a nonconducting substance, i.e. an insulato. The tem was coined by William Whewell in esponse to a equest fom Michael Faaday. Whewell consideed "diaelectic", fom the Geek "dia" meaning "though", since an electic field passes though the mateial but felt that "dielectic" was easie to ponounce. Although "dielectic" and "insulato" ae geneally consideed synonymous, the tem "dielectic" is moe often used when consideing the effect of altenating electic fields on the substance while "insulato" is moe often used when the mateial is being used to withstand a high electic field. "Dielectics... ae not a naow class of so-called insulatos, but the boad expanse of nonmetals consideed fom the standpoint of thei inteaction with electic, magnetic, of electomagnetic fields. Thus we ae concened with gases as well as with liquids and solids, and with the stoage of electic and magnetic enegy as well as its dissipation." Dielectics is the study of dielectic mateials and involves physical models to descibe how an electic field behaves inside a mateial. It is chaacteised by how an electic field inteacts with an atom and is theefoe possible to appoach fom eithe a classical intepetation o a quantum one. In the classical appoach to the dielectic model, a mateial is made up of atoms. Each atom consists of a cloud of negative chage bound to and suounding a positive point chage at its cente. Because of the compaatively huge distance between them, none of the atoms in the dielectic mateial inteact with one anothe. In the pesence of an electic field the chage cloud is distoted, as shown in the figue. This can be educed to a simple dipole using the supeposition pinciple. A dipole is chaacteized by its dipole moment, a vecto quantity shown in the figue as the blue aow 76

39 labeled M. It is the elationship between the electic field and the dipole moment that gives ise to the behavio of the dielectic. When the electic field is emoved the atom etuns to its oiginal state. Electic field inteaction with an atom unde the classical dielectic model. Dielectic model applied to vacuum The popety which defines how a dieletic behaves is the elationship between the applied electic field and the induced dipole moment. Fo a vacuum the elationship is a eal constant numbe. This constant is called the pemittivity of fee space, ε. CAPACITORS Commecially manufactued capacitos typically use a solid dielectic mateial with high pemittivity as the intevening medium between the stoed positive and negative chages. This mateial is often efeed to in technical contexts as the "capacito dielectic. The most obvious advantage to using such a dielectic mateial is that it pevents the conducting plates on which the chages ae stoed fom coming into diect electical contact. Moe significantly howeve, a high pemittivity allows a geate chage to be stoed at a given voltage. This can be seen by teating the case of a linea dielectic with 77

40 pemittivity ε and thickness d between two conducting plates with unifom chage density σ ε. In this case, the chage density is given by and the capacitance pe unit aea by Fom this, it can easily be seen that a lage ε leads to geate chage stoed and thus geate capacitance. Dielectic mateials used fo capacitos ae also chosen such that they ae esistant to ionization. This allows the capacito to opeate at highe voltages befoe the insulating dielectic ionizes and begins to allow undesiable cuent flow. Cable insulation The tem "dielectic" may also efe to the insulation used in powe and RF cables. Some pactical dielectics Dielectic mateials can be solids, liquids, o gases. In addition, a high vacuum can also be a useful, lossless dielectic even though its elative dielectic constant is only unity. Solid dielectics ae pehaps the most commonly used dielectics in electical engineeing, and many solids ae vey good insulatos. Some examples include pocelain, glass, and most plastics. Ai, nitogen and sulfu hexafluoide ae the thee most commonly used gaseous dielectics. Industial coatings such as paylene povide a dielectic baie between the substate and its envionment. Mineal oil is used extensively inside electical tansfomes as a fluid dielectic and to assist in cooling. Dielectic fluids with highe dielectic constants, such as electical gade casto oil, 78

41 ae often used in high voltage capacitos to help pevent coona dischage and incease capacitance. Because dielectics esist the flow of electicity, the suface of a dielectic may etain standed excess electical chages. This may occu accidentally when the dielectic is ubbed (the tiboelectic effect). This can be useful, as in a Van de Gaaff geneato o electophous, o it can be potentially destuctive as in the case of electostatic dischage. Specially pocessed dielectics, called electets (also known as feoelectics), may etain excess intenal chage o "fozen in" polaization. Electets have a semipemanent extenal electic field, and ae the electostatic equivalent to magnets. Electets have numeous pactical applications in the home and industy. Some dielectics can geneate a potential diffeence when subjected to mechanical stess, o change physical shape if an extenal voltage is applied acoss the mateial. This popety is called piezoelecticity. Piezoelectic mateials ae anothe class of vey useful dielectics. Some ionic cystals and polyme dielectics exhibit a spontaneous dipole moment which can be evesed by an extenally applied electic field. This behavio is called the feoelectic effect. These mateials ae analogous to the way feomagnetic mateials behave within an extenally applied magnetic field. Feoelectic mateials often have vey high dielectic constants, making them quite useful fo capacitos. CAPACITANCE Capacitance is a measue of the amount of electic chage stoed (o sepaated) fo a given electic potential. The most common fom of chage stoage device is a two-plate capacito. If the chages on the plates ae +Q and Q, and V gives the voltage diffeence between the plates, then the capacitance is given by The SI unit of capacitance is the faad; faad = coulomb pe volt. 79

42 Capacitos Capacitance is typified by a paallel plate aangement and is defined in tems of chage stoage: Whee,Q = magnitude of chage stoed on each plate. V = voltage applied to the plates. The capacitance can be calculated if the geomety of the conductos and the dielectic popeties of the insulato between the conductos ae known. Fo example, the capacitance of a paallel-plate capacito constucted of two paallel plates of aea A sepaated by a distance d is appoximately equal to the following: whee (in SI units) C is the capacitance in faads, F A is the aea of each plate, measued in squae metes ε is the elative static pemittivity (sometimes called the dielectic constant) of the mateial between the plates, (vacuum =) ε is the pemittivity of fee space whee ε = 8.854x - F/m 8

43 d is the sepaation between the plates, measued in metes The equation is a good appoximation if d is small compaed to the othe dimensions of the plates. In CGS units the equation has the fom: whee C in this case has the units of length. The dielectic constant fo a numbe of vey useful dielectics changes as a function of the applied electical field, e.g. feoelectic mateials, so the capacitance fo these devices is no longe puely a function of device geomety. If a capacito is diven with a sinusoidal voltage, the dielectic constant, o moe accuately efeed to as the elative static pemittivity, is a function of fequency. A changing dielectic constant with fequency is efeed to as a dielectic dispesion, and is govened by dielectic elaxation pocesses, such as Debye elaxation capacitance. Capacito Geometies The capacitance of a system depends only on its shape and on the insulatos it contains. In geneal, the capacitance is quite difficult to calculate, but if the geomety is symmetic, Gauss's law makes it possible to find fomulas fo C. Paallel Plates: The simplest geomety is a pai of paallel plates, each with aea A and sepaated fom each othe by a distance d which is small compaed to the width of the plates. The capacitance of this system is Cylindical Capacito Anothe simple geomety is the coaxial cylinde in which an inne cylindical conducto of length L and adius a is suounded by an oute cylindical conducto of length L and adius b. We assume that the length of the cylinde is much geate than its 8

44 adius. (The ound cable you use to connect a VCR to a TV set is an example of such a capacito.) This system has capacitance Spheical Capacito Finally, a spheical capacito fomed of two concentic spheical conducting shells, one with lage adius b and the othe with small adius a, has capacitance If the oute conducto is at infinity, we take the limit isolated sphee of adius a: to get the capacitance of an Electic Cuent Electic cuent is the flow (movement) of electic chage. A solid conductive metal contains a lage population of mobile, o fee, electons. These electons ae bound to the metal lattice but not to any individual atom. Even with no extenal electic field applied, these electons move about andomly due to themal enegy but, on aveage, thee is zeo net cuent within the metal. Given an imaginay plane though which the wie passes, the numbe of electons moving fom one side to the othe in any peiod of time is on aveage equal to the numbe passing in the opposite diection. 8

45 A typical metal wie fo electical conduction is the standed coppe wie. When a metal wie is connected acoss the two teminals of a DC voltage souce such as a battey, the souce places an electic field acoss the conducto. The moment contact is made, the fee electons of the conducto ae foced to dift towad the positive teminal unde the influence of this field. The fee electons ae theefoe the cuent caie in a typical solid conducto. Fo an electic cuent of ampee, coulomb of electic chage (which consists of about electons) difts evey second though any imaginay plane though which the conducto passes. The cuent I in ampees can be calculated with the following equation: whee Q is the electic chage in coulombs (ampee seconds) It follows that: t is the time in seconds Q = I t and t = Q/I Moe geneally, electic cuent can be epesented as the time ate of change of chage, o Cuent Density. Cuent density is a measue of the density of flow of a conseved chage. Usually the chage is the electic chage, in which case the associated cuent density is the electic 83

46 cuent pe unit aea of coss section, but the tem cuent density can also be applied to othe conseved quantities. It is defined as a vecto whose magnitude is the cuent pe coss-sectional aea. In SI units, the electic cuent density is measued in ampees pe squae mete. Definition Electic cuent is a coase, aveage quantity that tells what is happening in an entie wie. If we want to descibe the distibution of the chage flow, we use the concept of the cuent density: whee is the cuent density vecto (SI unit ampees pe squae mete) is the paticle density in count pe volume (SI unit m -3 ) is the individual paticles' chage (SI unit coulombs) is the chage density (SI unit coulombs pe cubic mete) is the paticles' aveage dift velocity (SI unit metes pe second) The cuent density can also be defined as: whee is the electic field stength and σ is the electical conductivity. The cuent though a suface S can be calculated by the following elation: 84

47 whee the cuent is in fact the integal of the dot poduct of the cuent density vecto and the diffeential suface element flowing though the suface S., i.e. the net flux of the cuent density vecto field The cuent density is an impotant paamete in Ampèe's cicuital law (one of Maxwell's equations), which show the diect link between cuent density and magnetic field. Cuent density is an impotant consideation in the design of electical and electonic systems. Most electical conductos have a finite, positive esistance, making them dissipate powe in the fom of heat. The cuent density must be kept sufficiently low to pevent the conducto fom melting o buning up, o the insulating mateial failing. In supeconductos excessive cuent density may geneate a stong enough magnetic field to cause spontaneous loss of the supeconductive popety. Continuity Equation In electomagnetic theoy, the continuity equation can eithe be egaded as an empiical law expessing (local) chage consevation, o can be deived as a consequence of two of Maxwell's equations. It states that the divegence of the cuent density is equal to the negative ate of change of the chage density, Deivation fom Maxwell's equations One of Maxwell's equations, Ampèe's law, states that 85

48 Taking the divegence of both sides esults in but the divegence of a cul is zeo, so that Anothe one of Maxwell's equations, Gauss's law, states that Substitute this into equation () to obtain which is the continuity equation. 86

49 Summay: Coulomb's law o Coulomb's invese-squae law states that: "The foce between any two point chages is diectly popotional to the magnitude of each chage, invesely popotional to the squae of the distance between them and diected along the staight line joining the two chages QQ = arnewtons 4πε R F Pinciple of supeposition: fo all linea systems, the net esponse at a given place and time caused by two o moe stimuli is the sum of the esponses which would have been caused by each stimulus individually. Electic field intensity is the foce expeienced by a unit positive test chage that is bought into the electic field ceated by a chage of +q coulombs. The foce on the unit test chage acts along the adial (flux) line emanating fom the chage and it acts outwads fom chage +q (epulsive foce). Electic flux line o line of foce is the path followed by an electic chage (fee to move in an electic field) when placed inside an electic field. It is an imaginay continuous line o cuve dawn in an electic field such that tangent to it at any point gives the diection of the electic foce at that point. E = E + E The electic field intensity at a point P due to the chage q and q is ( R R ) q ( R R ) q = + 4πε R R 4πε R R Eat P due to the aay of chages q, q, q n is E = 4πε n k= q k k k V/m 87

50 ( dl F ρ L ) L Electic field fo a line chage distibution is E = = V/m, fo a suface q 4πε chage ρ ( S ds) F S distibution E = = V/m and fo a volume chage q 4πε ρ ( V dv) F V distibution E = = V/m. q 4πε The net flux passing nomal though a unit suface aea is called electic flux density D. It has a specific diection which is nomal to the suface aea unde consideation; hence it is a vecto field. If Ψ is the total flux and S is the total suface aea, then D = Ψ S Gauss law states that Suface integal of the nomal component of the electic flux density vecto Dove any closed suface is equal to the total chage enclosed by the suface ds = Ψ = S D Q (O) an altenate definition can be given as, Suface integal of the electic field vecto Eove any closed suface in fee space is given by Q the total chage enclosed by the suface E ds = Ψ = S Q ε ε, whee Q is 88

51 Review questions: Two mak questions:. Define coulomb s law. Define gauss law 3. Define electic field intensity 4. What is pinciple of supeposition? 5. Define electic flux density 6. Find the foce of inteaction between chages spaced cm apat in a vacuum. The chages ae 4x -8 and 6x -5 coulombs. If the same chages ae sepaated by the same distance in keosene (ε =), what is the coesponding foce of inteaction? 7. 3 equal positive chages, each of 4x -9 coulombs ae located at thee cones of a squae, side cm. detemine the magnitude and diection of the electic field at the vacant cone of the squae. 8. Calculate the foce on a unit positive chage at P on x-axis whose co-odinates ae (x=m,y=) due to the following two chages. A positive chage of -9 C is situated in ai at the oigin and the negative chage of -x -9 C is situated on the x-axis (x=m,y=). 9. An infinitely long line oiented paallel to z-axis caies a unifomly distibuted chage of. mico-coulomb pe mete length. It is situated at x=m, y=5m. Detemine the electic field intensity Eat (-,-,5m).. Two infinite plane sheets ae a sepaated by a distance d m. the fist has a chage +σ pe unit aea and the second -σ pe unit aea. Find the electic field intensity at any point between them. 89

52 6 mak questions:. State and pove gauss law. A cicula disc of adius a m chaged unifomly with a chage density of σ C/m. Find the electic field intensity at a point h m fom the disc along its axis. 3. Find the electic field intensity E at any point in the space above an infinite sheet of chage with unifom density σ C/m, by consideing the plane sheet as composed of infinitely long lines with unifom line chage density. 4. Detemine the foce on a point chage of mico-coulomb at a point (,,4m) due to the total chage of 6 mico-coulomb unifomly distibuted ove the suface of a cicula disc with oveall adius of 4 m centeed at the oigin and situated in the X-Y plane. 5. A unifom sheet of chage with suface chage density σ = -.nc/m is located at z=4m and a unifomly distibuted line chage with linea chage density λ=3nc/m is situated at z=-4m, y=3m. Detemine the esultant electic field E at (x,-,m). 9