Charge is that property of material due to which it shows electric and magnetic phenomena.

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1 Electostatics Electostatics deals with the study of foces, fields and potentials aise fom static chages. In othe wods it is study of chages at est. Chage Chage is that cetain something possessed by mateial objects that make it possible fo them to exet electical foce and to espond to electical foce. William Gilbet Chage is that popety of mateial due to which it shows electic and magnetic phenomena. Kinds of electic chages: Electic chages exists just in two vaieties, (+) and ( ). When two ae ubbed against each othe, the chages on them appea due to tansfe of electons fom one object to othe and they ae said to acquie fictional electicity. Fictional electicity leads to chages on two bodies ubbing each othe. Chage (+) Chage ( ) Glass Rod Silk Fu Ebonite Wool Plastic Wool Rubbe Dy hai Comb The popety which diffeentiates the two kinds of chages is called polaity of chage. Like chages epel and unlike chages attact each othe. Oigin of Chage: By tansfe of electons fom/to a body, a body gets chaged. s potons ae also chaged paticle but thei tansfe is not possible. Electons can tavel fom one atom to anothe o fom one body to anothe. If a body loses one electon, it becomes positively chaged with +e chage & if it gains one electon it becomes negatively chaged with -e. body cannot lose o gain any poton by odinay methods. Basic unit of chage: Basic unit of chage is e, whose magnitude is equal to the magnitude of chage on an electon o poton. e =.6 9 C. SI unit of chage is coulomb: coulomb is defined as the amount of chage which is tansfeed by a cuent of one ampee in one second. (Because q = I t). Smalle units C(C = -6 C) is often used. Othe units ae nc = -9 C; pc = - C; C = - C; In CGS system of units, unit of chage is esu (electo static unit). Coulomb = 3 9 esu. Electon Poton Neuton Chage e + e Mass 9. 3 kg kg kg Detection of electic chage: simple appaatus to detect chage on a body is the goldleaf electoscope. It consists of a vetical metal od housed in a box, with two thin gold leaves attached to its bottom end. When a chaged object touches the metal knob at the top of the od, chage flows on to the leaves and they ELECTROSTTICS divege. The degee of divegence is an indicato of the amount of chage. When an electified body is touched to the metal knob of the metal od, chage is tansfeed to the metal od and to the attached gold foil. Both the halves of the foil get simila chage and theefoe epel each othe. The divegence in the leaves depends on the amount of chage on them i.e. highe the chage moe is the angle of divegence and vice-vesa. Popeties of Chages. Thee ae two types of chages positive and negative. Like chages epel each othe and unlike chages attact each othe.. Chage is additive: Chage is a scala quantity. It is added like numbes. Total electic chages of a system is equal to algebaic sum of all the positive and negative chages contained in that system. 3. Chage is quantized : The basic unit of chage is the chage that an electon o poton caies. When chage is tansfeed, only integal numbe of electons can be tansfeed fom one body to anothe. The fact that the electic chage is always an integal multiple of e is temed as quantization of chage. Symbolically, q = ne Whee q = total chage on a body e = chage on an electon Quantization of chage is meaningful only at micoscopic level. 4. Total chage of an isolated system is conseved. Consevation of chage means that total chage of an electically isolated system always emains constant. The following example explains the law of consevation of chage : RIF CDEMY (D 64/94 3 Siga (Nea Shubham Hospital, Vaanasi)) (94544) Page

2 [ELECTROSTTICS] () When glass od is ubbed with silk, glass od becomes positively chaged and silk becomes negatively chaged. Thus the system of glass od and silk, which had zeo net chage befoe ubbing, still posses zeo net chage afte ubbing. () In all nuclea tansfomation, poton numbe is found to emain unchanged. (Poton is only chaged paticle in nucleus) Fo example, in nuclea fission of uanium ( 9U 35 ) by a neuton ( n ) is as 9U 35 + n 56 Ba K n + enegy Poton numbe befoe fission = + 9 = 9 Poton numbe afte fission = () = 9 (3) nnihilation (to vanish o cease to exist) of matte (a positon and an electon combine to poduce a photon enegy) is an impotant phenomenon in which chage is conseved. 5. Chage is always associated with mass. It is possible that an object having mass has no chage but a chaged object cannot be massless. 6. Chage is invaiant with speed i.e. the magnitude of chage on a body does not depend upon the speed of the body which cay the chage. NOTES: (a) The basic cause of quantization of change lies in the fact that chaging of a body is always due to the tansfe of integal numbe of electon. (b) The quantization effect of chage can be obseved only at micoscopic level. It means fo bigge chage we can teat it as continues instead of discete (i.e. all possible value of chage is possible). Compaison of chaged and mass. Chage Electic foce oiginating fom chages may be attactive o epulsive Thee ae two kinds of chages positive and negative Chage of a body does not depend upon its speed. Mass Gavitational foce is always attactive. Thee is only one kind of Mass The mass changes accoding to the fomula m m / (v / c ) Whee m is est mass, m is at speed v, c is speed of light in vacuum. Methods of chaging body can be chaged by (i) fiction, (ii) induction, (iii) conduction (iv) photoelectic emission, (v) themionic emission o (vi) field emission. Polaization: If the cente of the negative chages i.e., electons does not coincide with the cente of the positive chages i.e., potons in an atom/molecule of an insulato/dielectic, the atom/molecule is called a pola atom/ molecule. Pola atoms/molecules align them-selves accoding to the diection of the extenal electic field. This pocess of alignment is called polaization. Coulomb s law of electic foce: ccoding to Coulomb s law, The electic foce between two point chages is diectly popotional to the poduct of the magnitude of the two chages and vaies invesely as the squae of distance between the chages. It two point chages q, q ae sepaated by a distance, the magnitude of this foce (F) between them is accoding to Coulomb s law F q q and F combining we get, q q F qq o, F = k Whee k is the constant of popotionality called electostatic constant which depends upon the medium between the chages. Coulomb s law is applied to static point chages only (limitation of Coulomb s law). Value of k The value of the constant k depends upon the medium between the chages. If the medium is vacuum then k = 9 9 Nm C in SI system. In CGS system value of k is one. Pemittivity (absolute pemittivity): In SI system k is witten as k = / 4πε (fo vacuum) whee is called the pemittivity of fee space o vacuum. Value of = C N m & k = 9 9 Nm C qq Coulomb s law is witten as F 4πε If the medium between the two chages is othe than vacuum. The qq fomula becomes F 4πε The foce between the chages gets educed. is called the pemittivity (also known as absolute pemittivity) of the medium. Relative pemittivity/dielectic constant: Dielectic is an insulating mateial, which tansmits electic effects without actually conducting electicity. If chages ae situated in a medium othe than vacuum, is eplaced by, whee is called the pemittivity of the medium and the atio ε ε/ε, is called the elative pemittivity o dielectic constant (K) of the medium. s it is a atio, it does not have any unit. o K = fo vacuum ; fo ai; 8 fo wate RIF CDEMY (D 64/94 3 Siga (Nea Shubham Hospital, Vaanasi)) (94544) Page

3 [ELECTROSTTICS] We have impotant elation fo dielectic constant of a medium K Fo vacuum,k is one and fo conducto it is infinity. Thus if F is the electostatic foce between two points chages when they ae placed in vacuum and F m when they ae placed in some medium having elative pemittivity o dielectic constant K, then Fm F/ε o F/K. The value of K fo conducto tends to infinity. Fvacuum q q F = = m K 4πε K Coulomb s law in vecto fom: Coulomb s law is stated in vecto fom as follows: 3. Two fixed point chages 4Q and Q ae sepaated by a distance x. Whee should the thid point chage q be placed fo it to be in equilibium? [ns:.59 x] 4. Two chages + 6 C and 6 C attact each othe with a foce of 4 N. Calculate the distance between them. [ns:.5 mm] 5. Conside thee chages q, q, q 3 ; each equal to q at the vetices of an equilateal tiangle of side l. What is the foce acting on a chage Q (with the same sign as q) placed at the centoid of the tiangle? [ns: zeo] 6. Two similaly equally chaged identical metal sphees and B epel each othe with a foce of. 5 N. thid identical unchaged sphee C is touched to, then placed at the midpoint between and B. Calculate the net electostatic 5 foce on C. [ns:. N, along BC ] kqq F ˆ (Put chages with thei sign) Whee F = Foce on q due to q = Vecto distance between q and q, diection being fom q to q. ˆ = Unit vecto between q and q, diection being fom q to q. If souce chage q is at the oigin and the test chage q is at a point whose position vecto is then foce kqq F 3 If q and q ae placed at positions and, kqq then F ( 3 ) Hee, q and q ae values of the two chages along with thei algebaic signs, i.e., eithe with a (+) sing o with a ( ) sing. You have to put these sings along with thei values while using in the fomula. PROBLEMS FOR PRCTICE. body has a net chage of. C. How many excess electons ae contained in the body?. What is the total chage on 5. kg of electons? [ns: 4.4 C] 7. chage q is placed at the cente of the line joining two equal chages, each equal to Q. Show that the system of thee chages will be in equilibium if q = Q/4. 8. chage Q is to be divided on two objects. What should be value of the chages on the objects so that the foce between the objects can be maximum? [ns: Q/, Q/] 9. Two positive chages distant. m apat, epel each othe with a foce of 8 N. If the sum of the chages be 9 micocoulomb (C), then calculate thei sepaate values. [ns: 5 C, 4 C]. Two point-chages of + C and + 6 C epel each othe with a foce of N. If each is given an additional chage of 4 C, then what will be the new foce? [ns: 4 N (attaction)] Chage Densities Chage can be distibuted unifomly o non unifomly ove a line o on a suface o within a volume. Censequently, change densites is of thee types. Linea change density (). ea o suface change density () 3. Volume change density () Linea change density () Change on unit length is called linea change density () dq dl Total chage dq dl [ns: ] Q dq o L dl Note: if = constant, then chage distibution is unifom. Ex. RIF CDEMY (D 64/94 3 Siga (Nea Shubham Hospital, Vaanasi)) (94544) Page 3

4 [ELECTROSTTICS] Q Suface chage density () It is Chage on unit suface aea Q C / m o dq d Ex. If Q chage is distibuted unifomly ove the conducting sphee of adius R. Find Q 4 R Volume chage Density () It is chage pe unit volume Q C / m V Pinciple of Supeposition: Foce on any chage due to a numbe of othe chages is the vecto sum of all the foces on that chage due to the othe chages, taken one at a time. Note that the individual foces ae unaffected due to the pesence of othe chages. Suppose we have to calculate the esultant net foce on chage q because of othe point chage, q, q 3, q 4, q 5, & q 6. If F = Foce on q due to q, F 3 = Foce on q due to q 3, etc. Then net foce on q F F 3 F 4 F 5 F 6 Question: Thee chages (each q) ae placed at the cones of an equilateal tiangle. fouth chage Q is placed at centoid of the tiangle. Fo what value of Q will all the fou chages emain stationay? Solution: Since all the thee chages at the vetices ae equal, so the net foce on the chage at the cente will be always zeo. So let us discuss the equilibium of any one of the chages at the vetices. Foce due to chages at and B will have equal magnitude q equal to F 4 o. So esultant of these foces will be o 3 q FR F F FFcos 6 3F 4 o Foce between the chage at cente and the chage at C should be equal and opposite the above foce, so the chage at cente should have negative sign. So, 3 q qq q Q (neagtive) 4 o 4 o / 3 3 Electic field chaged paticle cannot diectly inteact with anothe paticle kept at a distance. chage poduces something called an electic field in the space aound it and this electic field exets a foce an any othe chage (except the souce chage itself) placed in it. Electic field due to a given chage is the space aound a chage in which an electic foce of attaction o epulsion due to that chage can be expeienced by any othe chage. Electic field stength is the intensity of the electic field at a given point which is defined as the electic foce pe unit chage as expeienced by a small electic positive test chage if placed thee. F E = lim q q Fom equation F q E, keep in mind that the diection of foce on chage is same as that of electic field if chage is positive and foce is in opposite diection to field if chage is negative. Electic field intensity due to a point chage: Electic field intensity at any point P due to a point chage q at O, whee OP = is E q = ˆ 4πε If q is positive E is diected away fom q. on the othe hand if q in negative, then E is diected towads q. To find the E fo a goup of point chages : (a) Calculate En due to each chage at the given point as if it wee the only chage pesent. (b) dd these sepaately calculated fields vectoially to find the esultant field E at the point. In equation fom, Enet E E E 3... En RIF CDEMY (D 64/94 3 Siga (Nea Shubham Hospital, Vaanasi)) (94544) Page 4

5 Electic field intensity due to a goup of chage: It is equal to the vecto sum of the electic field intensities due to the individual chages at the same point in qi E = ˆ 4πε i i [ELECTROSTTICS] Note that if R x then E o Electic field due to a line of chage: If the chage distibution is a continuous one, the field it sets up at any point P an be computed by dividing the chage into infinitesimal element dq. The field de due to each elements at the point in question is then calculated, teating the elements as point chages. The magnitude of de is given by dq de 4 whee is the distance fom the chage element dq to the point P. The esultant field at P is then fount by adding (that is, integating) the field contibution due to all the chage elements, o, Enet de The integation hee is a vecto opeation. Some Impotant esults: E x q 4o x x a Note that fo infinitely along line of chage q q λ a x E k 4 o xa o x a x Fo semi-infinite long line of chage Electic field due to a point chage q E. 4 o If q is positive E is diected away fom q. On the othe hand if q in negative, then E is diected towads q. E y & Ex 4 y 4 y o o Electic field due to an ac of adius R and with linea chage density is E k sin along the angle bisecto R of the ac. Electic field due to a Ring of chage E qx 4 x R x 3/ o Electic field due to a disc of chage Electic field due to a finite unifomly chaged line as shown in the figue is given by x Ex o x R of disc) ;( is suface chage density RIF CDEMY (D 64/94 3 Siga (Nea Shubham Hospital, Vaanasi)) (94544) Page 5

6 [ELECTROSTTICS] Question: Find the electic field at the point O, the cente of the semi-ing, due the shown infinite aangement of chage. The linea chage density is. Solution: 6. Find the time taken by a paticle of mass 8 kg and caying a chage 3. 9 C to fall though a distance of 8 m in a unifom electic field of intensity 8 N C. [ns..5 s] 7. n electon is eleased with a velocity of 5 6 m s in an electic field of 3 N C which has been applied so as to oppose its motion. What distance would the electon tavel and how much time could it take befoe it is bought to est? [ns. 7. m, s] 8. n electon falls though a distance of.5 cm in a unifom electic field of magnitude. 4 N C. Field due to the two infinite chage distibutions E and the semi-ing E` ae shown in the diagam. Hee E and E`. So it is obvious that all 4 R o or the fields get cancelled out. So, net field at the point O will be zeo. PROBLEMS FOR PRCTICE. In the electic field shown in Figue the electic field lines on the left have twice the sepaation as that between those on the ight. If the magnitude of the field at point is 4 N C, calculate the foce expeienced by a poton placed at point. lso find the magnitude of electic field at the point B. [ns N, N C ]. Calculate the magnitude and diection of the electic field, which keeps a poton just floating. Given that mass of poton.67 7 kg, chage on poton =.6 9 C and g = 9.8 m s. [ns..3 7 N C (vetically upwad)] 3. wate paticle of mass mg and having a chage of.5 6 C stays suspended in a oom. What is the magnitude and diection of the electic field in the oom? [ns N C (vetically upwad)] 4. n electon above the eath is balanced by the gavitational foce and the electic field of the eath. Find the electic field of the eath. [ns N C ] 5. How many electons should be emoved fom a coin of mass.6 g, so that it may float in an electic field of intensity 9 N C diected upwad. [ns ] Figue (a) (Page No. 6) The diection of the field is evesed keeping its magnitude unchanged and a poton falls though the same distance Figue (b). Compute the time of fall in each case. Given that mass of electon = 9. 7 kg. [ns. (a).9 9 s, (b).5 7 s] 9. chaged paticle of mass g is suspended though a silk thead of length 4 cm in a hoizontal electic field of 4. 4 N C. If the paticle stays at a distance of 4 cm fom the wall in equilibium, find the chage on the paticle. [ns C]. Two point chages of + C and + 8 C ae placed 8 cm apat. Find the position of the point, whee electic field is zeo. [ns. at a distance.6 m fom 9 C chage (between the two chages)]. Two point chages of 6 C and + 8 C ae placed 8 cm apat. Find the position of the point, whee electic field is zeo. [ns. at a distance of 6.47 cm fom 6 C chage and 4.47 cm fom + 8 C]. Figue shows fou point chages at the cones of a squae of side cm. Find the magnitude and diection of the electic field at the cente O of the squae, if Q =. C. Use [ns. 4 9 Nm C NC (paallel to side B)] 3. Thee chages, each equal to q ae placed at the thee cones of a squae of side a. Find the electic field at the fouth cone of the squae. RIF CDEMY (D 64/94 3 Siga (Nea Shubham Hospital, Vaanasi)) (94544) Page 6

7 q [ns.. 4 a 4. Fou chages + q, + q, q, q ae placed espectively at the fou cones of a squae of side a. Find the magnitude and diection of the electic field at the cente of the squae. ] 4 q [ns.., paallel to the side having 4 a chages + q and q at its two ends.] 5. BC is an equilateal tiangle of side 5 cm. Chages of + 6 statc and 3 statc ae placed at points and B espectively. Calculate completely the electic field at point C. Given, C = 3 9 statc. [ns N C making 9 o with CB and 3 o with C (when poduced)] 6. BC is a ight-angled tiangle, the ight angle being at point B. Chages of.4,.5, C ae placed at points, B and C espectively. If B = 4 cm and BC = 3 cm, calculate the magnitude and diection of the esultant electic field at the foot D of the pependicula dawn fom point B on the side C. [ns. 8 N C, inclined equally with both D and BD (when poduced)] 7. coppe ball of density 8.6 g cm 3, cm in diamete is immesed in oil of density.8 g cm 3. If the ball emains suspended in oil in a unifom electic field of intensity 36, N C acting in upwad diection, what is the chage on the ball? [ns.. 6 C] 8. chage of 4 9 C is distibuted unifomly ove the cicumfeence of a conducting ing of adius.3 m. Calculate the field intensity at a point on the axis of the ing at.4 m fom its cente. lso calculate the electic field at the cente of the ing. [ns. 5. N C, Zeo] 9. dipole consists of two chages + C and C sepaated by a cetain distance. Let they be located at x = 6. cm, y = espectively. Calculate the field stength at a point x =, y = 8 cm. [ns..8 7 N C paallel to the line joining the two chages] 3. n inclined plane making an angle of 3 o with the hoizontal is placed in a unifom hoizontal electic field of V m as shown in figue paticle of mass kg and chage. C is allowed to slide down fom est fom a height of m. If the coefficient of fiction is. find the time it will take the paticle to each the bottom. [ELECTROSTTICS] [ns..3 s] Physical significance of electic field. The tue physical significance of the concept of electic field, howeve, emeges only when we go beyond electostatics and deal with time-dependent electomagnetic phenomena. Suppose we conside the foce between two distant chage q, q in acceleated motion. Now the geatest speed with which a signal o infomation can go fom one point to anothe is c, the speed of light. Thus, the effect of any motion of q on q cannot aise instantaneously. Thee will be some time delay between the effect (foce on q ) and the cause (motion of q ). The field pictue is this: the acceleated motion of chage q poduces electomagnetic waves, which then popagate with the speed c, each q and cause a foce on q. The notion of field elegantly accounts fo the time delay. Thus, even though electic and magnetic fields can be detected only by thei effects (foce) on chages, they ae egaded as physical entities, not meely mathematical constucts. They have an independent dynamics of thei own, i.e., they evolve accoding to laws of thei own. They can also tanspot enegy. Thus, a souce of time-dependent electomagnetic field, tuned on biefly and switched off, leaves behind popagating electomagnetic fields tanspoting enegy. The concept of field was fist intoduced by Faaday and is now among the cental concepts in physics. SI unit of electic field: Since E = foce/chage S.I. unit of electic field E is Newton pe coulomb (NC ) Shell Theoem. shell of unifom chage attacts o epels a chaged paticle that is outside the shell as if all the shell s chage wee concentated at its cente.. If a chaged paticle is located inside a shell of unifom chage, thee is no net electostatic foce on the paticle fom the shell. Vecto & Scala Fields Vecto field: Those fields which have both magnitude as well as diection ae vecto field. e.g. electostatic, magnetic, gavitational fields. Scala field. Those fields which has magnitude only and no diection, e.g., the fields of fluid tempeatue o pessue. Consevative & non-consevative fields If the wok done by a field depends only upon the final and initial positions of the object (and not upon the path taken to each the final position), the field is called consevative field, e.g., electostatic, magnetic, gavitational fields. If the wok done depends upon the length of the path taken, it is called non-consevative field, e.g., fictional field. Electic field lines (o electic lines of foce): Electic field lines ae gaphical epesentation of electic field. Field lines ae imaginay lines used to get the idea about electic field. n electic field line is a cuve dawn in such a way that the tangent at each point on the cuve gives the diection of electic field at that point. RIF CDEMY (D 64/94 3 Siga (Nea Shubham Hospital, Vaanasi)) (94544) Page 7

8 [ELECTROSTTICS] Popeties of electic fields lines (i) Electic field lines ae hypothetical lines o cuves; tangent to which give diection of electic field at that point. (ii) The elative closeness of field lines indicates the elative stength of electic field at diffeent points. (iii) ll electic field lines oiginate fom a positive chage and teminate on a negative chage. They ae open cuves. (iv) The numbe of electic field lines oiginating fom o ending on a chage is popotional to the magnitude of the chage. (v) No two electic field lines eve coss each othe because field cannot have two diections at the same point. (vi) They ae continuous cuves. (vii) They emege fom o ente into a conducto pependiculaly. Theefoe, thee is no component of electic field intensity paallel to the suface of the conducto. Electic field lines of isolated chages It is adially away to infinite distance if chage is positive and adially towad the chage if chage is negative though field weaken away fom the chage. Electic field lines of multiple chages Electic field lines of unifom electic field Equidistant paallel staight lines with pope diection epesent a unifom electic field. Electic field lines of non-unifom electic field.. ea Vecto plane aea, such as in Fig., can be egaded as possessing both magnitude and diection. Its magnitude is the amount of the aea and its diection is the diection of the outwad dawn nomal to the plane of the aea. Hence, the aea can be epesented by a vecto along the outwad dawn nomal to the aea, the length of the vecto epesenting the magnitude. Thus, nˆ, is the definition of an aea vecto, whee ˆn is a unit vecto along the outwad dawn nomal.. Solid ngle The ac of a cicle subtends an angle at the cente of the cicle. This angle is called a plane angle. Its unit is adian (ad). adian is the angle which an ac of length equal to the adius of a cicle subtends at the cente of the cicle. Similaly, the aea of a spheical suface subtends an angle at the cente of the sphee. This angle is called the solid angle and is epesented by ω. Let O (Fig. ) be the cente and the adius of a sphee. Let d be a small aea element of the suface of the sphee. If the points situated on the bounday of this aea be joined to O, then the lines so dawn will subtend a solid angle dω at O. Since the spheical aea d is diectly popotional to the adius ( ), the atio d/ is a constant. This atio is called the solid angle dω subtended by the aea d at the cente O of the sphee. Thus dω = d/..(i) The unit of solid angle is steadian (s). In eq. (i), if d =, then dω =. Theefoe, steadian is the solid angle subtended by a pat of the suface of a sphee at the cente of the sphee, when the aea of the pat is equal to the squae of the adius of the sphee. The entie suface aea of a sphee is = 4. Hence the solid angle subtended by the entie suface of the sphee at its cente is 4 4 steadian. In fact, the solid angle subtended by a closed suface of any shape at a point inside it is 4 steadian. Now, let us conside an aea element d situated at a distance fom a point O (Fig. 3). Let ˆ be a unit vecto along the line joining the point O to the aea element d. Let be the angle between ˆ and the vecto d epesenting the aea element d. If we daw a sphee with O as cente and as adius, then the pojection of aea element d on the suface of the sphee will be d cos (shown shaded). Theefoe, the solid angle subtended by aea element d at the cente O of the sphee is d cos d d ˆ o d. RIF CDEMY (D 64/94 3 Siga (Nea Shubham Hospital, Vaanasi)) (94544) Page 8

9 [ELECTROSTTICS] 3. The Flux of a Vecto Field The concept of Flux is vey impotant in undestanding Gauss theoem and its applications. The flux (symbol Φ) is a popety of any vecto field and is a measue of the flow (o penetation) of the field vectos though an imaginay suface placed in the field. s an example, let us conside the velocity (vecto) field of a unifomly flowing liquid (Fig. 4) and imagine a wie bent into a squae loop of aea placed in the steam. The flux Φ of the velocity field though the loop of aea can be consideed as the volume ate of liquid flow. If the plane of the loop is pependicula to the diection of flow, as in Fig. 4(a), then Φ = v, whee v is the magnitude of the velocity. In tems of field concept, the flux Φ though the loop can be consideed as a measue of the numbe of vecto field lines passing though the loop. In Fig. 4 (b), the loop of aea is inclined to the diection of flow. In this case, the numbe of field lines passing though the loop is smalle. The same (smalle) numbe of lines, howeve, pass though the loop aea cos pojected pependicula to liquid flow. The flux in this case is Φ = v cos. If the loop is placed paallel to the diection of flow ( = 9 ), no field lines pass though the loop (Fig. 4 c). Hence, in this case we have Φ =. Now, the ight side of eq. (ii) can be witten as a dot poduct of velocity vecto v and aea vecto (Fig. 4 d). That is Φ = v. Since the diection of is the diection of the outwad nomal to the plane of the loop, the flux (field lines) leaving a suface is consideed positive, while that enteing a suface is consideed negative. If, instead of a loop, we immese a closed suface consisting of seveal individual plane sufaces in the liquid steam, then v. Fo non-unifom fields and sufaces of abitay shape and oientation, the last expession fo flux Φ is genealized as v d The flux Φ is a scala quantity, because it is defined in tems of the dot poduct of two vectos. 4. Element Flux The electic flux is a popety of electic field. We known that an electic field can be visualized by lines of foce ae close, and vice-vesa. The electic flux is a measue of the numbe of lines of foce passing though some suface held in the electic field. It is denoted by Φ E. Let thee be an abitay suface immesed in an electic field (not necessaily unifom), as shown by lines of foce (Fig. 5). Let us conside an imaginay small suface element d, the suface ove the element being pactically plane and the electic field unifom. The suface element may be epesented by a vecto d of magnitude d, diected along the outwad dawn nomal to the element. Let E be electic field at the location of the element d. Then, the scala poduct E d is defined as electic flux though the element. The electic flux though the entie suface is theefoe E E d, whee is the (suface) integal ove the entie suface. Φ E is positive fo the lines of foce pointing outwads (leaving the suface); and negative fo those pointing inwads (enteing the suface). Thus, the electic flux linked with a suface in an electic field may be defined as the suface integal of the electic field ove that suface. In case of a closed suface (Fig. 6), that is, a suface that completely encloses a volume (like the suface of a balloon), the net flux though the suface is given by E E d. The electic flux Φ E, being the scala poduct of two vectos, is a scala. Let us now conside a plane suface of aea placed in a unifom electic field E. The nomal to the suface makes an angle with the field. By definition, the electic flux though the suface is given by E E d, whee d is the aea vecto of a small element d on the suface. Since d is nomal to the suface, the angle between vectos E and d is and But E d Ed cos. E d cos E cos d. E d (given). RIF CDEMY (D 64/94 3 Siga (Nea Shubham Hospital, Vaanasi)) (94544) Page 9

10 [ELECTROSTTICS] Gauss law, is one of the fou fundamental equations of electomagnetic theoy (Maxwell s equations). Poof. Let us conside a point-chage +q situated at O inside a closed suface (Fig. 8). Φ E = E cos. If the plane suface is nomal to the electic field E (Fig. 7), that is, =, then Φ E = E cos = E. If the plane suface is paallel to the electic field ( = 9 ), then Φ E = E cos 8 = E. Electic Flux Density : In an electic field, the atio of electic flux Φ E though a suface to the aea of the suface is called the electic flux density at the location of the suface. That is E electic flux density =. Fo a plane suface nomal to the electic field, Φ E = E. Thus electic flux density = E E The unit of electic flux density is same as that of electic field. Since electic field is a vecto quantity, the electic flux density is also a vecto quantity (although electic flux is a scala quantity). Unit and Dimensions of Electic Flux : By definition Φ E = E cos. Theefoe, the SI unit of Φ E is N m N m C. C E is also expessed in V/m. V Theefoe, anothe SI unit of Φ E is m V m. m The dimensions of Φ E ae dimensions of E dimensions of = [M L T 3 ] [L ] = [M L 3 T 3 ]. 5. Gauss Theoem The Gauss theoem is one among, the fou fundamental laws of electomagnetism say; (i) Gauss theoem in electostatics (ii) Gauss s theoem in magnetism (iii) mpee s cicuital law (iv) Faaday s law of electomagnetic induction. The Gauss s theoem in electostatics gives a elation between the electic flux though any closed hypothetical suface (called a Gaussian suface) and the chage enclosed by the suface. It states that the electical flux Φ E though any closed suface is equal to /ε times the net chage q enclosed by the suface. That is, q E E d, whee ε is pemittivity of fee space. This is the integal fom of Gauss theoem. The Gauss theoem of the Let d be a small aea element suounding a point P on the suface. Let O P =. The aea element may be epesented by a vecto d dawn outwad along the nomal to the element. Let E be the electic field intensity at P due to chage +q at O. Its diection is along OP. The electic flux though the aea element d is de E d = E d cos, whee is the angle between the vectos E and d. Now, E = q. 4 q d cos d. E 4 d cos But is the solid angle dω subtended by the aea d at the point O. Theefoe, q de d. 4 The total flux E Now, d 4, suface at the point O. though the entie suface is q E d. 4 the solid angle subtended by the entie closed q. E This is what we had to pove. If thee ae seveal chages +q, + q, +q 3, q 4, q 5.etc., the same agument can be applied to each in tun. Hence the total flux though the suface due to all of them is q q q q q... q, whee q is the algebaic sum of all the chages enclosed by the closed suface. If the chage q be outside the closed suface (Fig. 9), the total flux though the suface is zeo because the cone with vetex at q cuts off aea d whee it entes, and aea d whee it q leaves the suface. The flux though d is d (inwad) 4 RIF CDEMY (D 64/94 3 Siga (Nea Shubham Hospital, Vaanasi)) (94544) Page

11 [ELECTROSTTICS] q and that though d is + d (outwad). Hence, the 4 total flux though the two aeas is zeo. This is tue fo two aeas cut off by any such cone. Hence fo the whole closed suface the total flux is zeo. Gauss theoem is vey useful in computing the electic fields due to a system of chages o symmetical continuous distibution of chage. 6. Electic Field due to a Point-Chage (Deduction of Coulomb s Law fom Gauss Law) Suppose, an isolated point-chage + q is placed at a point O (Fig. ) in vacuum (o ai). In ode to find the electic field due to this chage at a distance fom it, we conside an imaginay spheical suface of adius with O as cente. This is known as Gaussian suface. By symmety, the electic field E at any point on the Gaussian spheical suface is along the outwad nomal at that point and has the same magnitude at evey point on the suface. Thus, fo any suface element d, both the electic field vecto E and the aea vecto d ae along the same diection (adially outwad), that is, the angle between them is zeo. Theefoe, E d Ed cos Ed. Hence the total electic flux leaving the Gaussian suface is given by E E d E d E d, because E is constant thoughout the suface. But d 4 (aea of the sphee). E 4 E By Gauss s theoem, E q /, whee is pemittivity of fee space. o E 4 q / q E. 4 This is the magnitude of electic intensity E at a distance fom a point-chage +q, which is diected away fom the chage. If we put a test-chage q at that point, then the magnitude of foce expeienced by q will be q q F qe. 4 This is Coulomb s law, as deived fom Gauss law. Thus, Coulomb s law and Gauss law in electostatics ae mutually equivalent. They ae not two independent physical laws but the same law expessed in diffeent ways. Gauss law holds only because the exponent in Coulomb s law is exactly : The Gauss law would not hold if the exponent in the Coulomb s law of foce wee diffeent fom. Let us see what happens if the exponent q wee 3, that is, if E. Then, the electic flux 3 4 fom a point-chage q though a sphee of adius with q as cente would be E E d E d (because E and d ae along the same diection) = E d = 3 3 q q q d This means that the flux would depend on the size of the sphee which is against the Gauss law. Hence, fo the Gauss law to hold, the Coulomb s law must be an invese-squae law. 7. Electic Field due to an Infinite Line of Chage Let us conside a unifomly-chaged (say, positive) wie of infinite length having a constant linea chage density (that is, chage pe unit length) λ coulomb/mete (C m ). Let P (Fig. ) be a point, distant mete fom the wie at which the electic intensity E is equied. Let us daw a coaxial Gaussian cylindical suface of length l though the point P. By symmety, the magnitude E of the electic field intensity will be the same at all points on this suface and diected adially outwads. Thus, fo any aea element d taken on the suface, both the electic field vecto E and the aea vecto d ae along the same diection (adially outwads). Theefoe, E d E d cos = E d. Hence the electic flux though the Gaussian suface is E d E d E = E d ( E is same eveywhee) = E( l). The flux though the plane ends of the suface is zeo because E and d ae at ight angles eveywhee on these faces... E d. Hence the total flux though the Gaussian suface is E E l. But, by Gauss law, this must q /, whee q is the net chage enclosed by the Gaussian suface. Hee, q = λ l, so that l E. Compaing the last two expessions, we get l E( l) = o E. In vecto notation, E, ˆ whee ˆ is unit vecto in the diection of. The diection of E is adially outwads (fo positive chage). RIF CDEMY (D 64/94 3 Siga (Nea Shubham Hospital, Vaanasi)) (94544) Page

12 [ELECTROSTTICS] 8. Electic Field due to an Infinite Plane Sheet of Chage Let us conside a thin, non-conducting sheet of chage, infinite in extent and having a suface chage density (chage pe unit aea) σ. Let P be a point distant fom the sheet at which the electic intensity is equied (Fig. ). Let us choose a point P, symmetical with P on the othe side of the sheet. Let us now daw a Gaussian cylinde cutting though the sheet, with its plane ends paallel to the sheet and passing though the points P and P. Let be the aea of each plane end. Since the sheet is infinite in extension, the electic intensity E at all points on eithe side nea the sheet will be pependicula to the sheet, diected outwads (if the sheet is positively-chaged). Thus, E is pependicula to the plane ends of the Gaussian cylinde and paallel to the cuved suface. lso its magnitude will be same at P and P. Theefoe, the flux though the two plane ends is E d + E d, E whee d is the aea vecto at the plane ends. Since E and d ae paallel, E d Ed. Theefoe E d E d E E E E. The flux though the cuved suface of the Gaussian cylinde is zeo because E and d ae at ight angles eveywhee on the cuved suface. Hence, the total flux though the Gaussian cylinde is E E. But, by Gauss theoem, E q /, whee q is the net chage enclosed by the Gaussian cylinde. Hee q = σ, and so E /. Compaing the last two equations, we get E / σ o E =. ε This is independent of. It means that the electic field intensity is same fo all points close to the plane sheet of chage. This esult is not supising. s we move away fom the sheet, moe and moe chage comes in out field of view and this offsets the decease in the field due to inceasing distance. E at any point (as P o P ) is diected away fom the sheet of positive chage. If the sheet be of negative chage, then E would be diected towads the sheet. Electic Chage Resides on the Oute Suface of the Conducto conducto has fee electons. When an excess chage is given to an insulated conducto, it sets up electic fields inside the conducto. These fields set the fee electons in motion which constitute intenal cuents. These moving electons immediately aange themselves in such a way that the electic fields inside the conducto become zeo eveywhee, theeafte the electons stop moving. Let us now conside a Gaussian suface inside the conducto as close as possible to the suface of the conducto as shown in Fig 3 (a). Since the electic intensity E is zeo eveywhee inside the conducto, it must be zeo fo evey point on the Gaussian suface. Obviously, the flux though this suface, E d will be q zeo. Theefoe, accoding to Gauss s law E d, the net chage inside the Gaussian suface, and hence inside the conducto, must be zeo. Since thee can be no chage in the inteio of the conducto, any chage supplied to the conducto will eside on the suface of the conducto. Fig. 3 (b) shows the same isolated conducto but now with a cavity totally within the conducto. Let us now daw a Gaussian suface suounded the cavity, close to its suface but inside the conducto. Because E is zeo inside the conducto, thee can be no flux though this new Gaussian suface. Theefoe, Gauss law demands that thee can be no net chage inside the Gaussian suface and hence no chage on the cavity walls. Clealy, any chage given to the conducto will entiely move out to the suface of the conducto. Electic Field due to two Infinite Paallel Sheets of Chage Suppose two infinite, plane, non-conducting sheets of positive chage, and, ae placed paallel to each othe in vacuum, o ai (Fig. 4). Let σ and σ be the suface densities of chage on sheets and espectively. We know that the magnitude of electic intensity E on eithe side close to a plane sheet of chage of density σ is E. E acts pependicula to the sheet, diected away fom the sheet (if chage is positive) o towads the sheet (if chage is negative). Let E and E be the electic intensities at any point due to sheets and espectively. Then, fo points outside the sheets, like P, we have E (away fom sheet ) RIF CDEMY (D 64/94 3 Siga (Nea Shubham Hospital, Vaanasi)) (94544) Page

13 and E (away fom sheet ). Since E and E ae in the same diection, the magnitude of the esultant intensity at points P is given by E E E, away fom both and. t a point in between the sheets, like P, we have E (away fom sheet ) and E (away fom sheet ). Now E and E ae oppositely-diected, and so E E E. In the special case of, the intensity at P is zeo. Let us now conside two sheets, and (Fig. 5) of positive and negative chages densities E. t points like P, we have E (away fom sheet ) (Fig. 5) and E (towads sheet ). Since E and E at P ae oppositely diected, the esultant intensity due to both sheets and is E E E. In the special case of, we have E =, that is, the field outside the sheets is zeo eveywhee. t a point P in between the sheets, we have E (away fom sheet ) and E (towads sheet ). Since E and E ae in the same diection, the esultant intensity at P due to both the sheets is given by E E E. gain, in the special of, we have E The magnitude of E is fee fom the position of the point P between the sheets. Hence, the electic field between the sheets (except at nea the edges) is unifom eveywhee, independent of the sepaation between the sheets. It points fom the positive towads the negative sheet. Thus, in this case, a unifom electic field exists only in the egion between the sheets and is zeo elsewhee. In fact, this is the way of poducing a unifom field in a limited egion of space; the only othe equiement is that [ELECTROSTTICS] the sheets should be much lage in linea dimension than the sepaation between them. Electic Field due to a Unifomly Chaged Thin Spheical Shell Let O (Fig. 6) be the cente and R the adius of a thin, isolated spheical shell caying a chage +q which is unifomly distibuted on its suface. We have to detemine electic field intensity due to this shell at points outside the shell, on the suface of the shell and inside the shell. t an Extenal Point : Let P be a point, distant (> R) fom the cente O, at which electic intensity is equied. Let us daw a concentic spheical Gaussian suface (of adius ) though the point P. ll points on this suface ae equidistant fom the suface of the chaged shell. Because of the spheical symmety, the magnitude E of the electic field intensity will be the same at all points on the Gaussian suface, and diected adially outwad. Let us conside an aea element d aound the point P. Both the electic intensity adially outwads, that is, the angle between them is zeo. Theefoe, the electic flux though the aea element d is d E E d Ed cos d. The flux though the entie Gaussian suface is. E E d E d E 4 But, by Gauss theoem, E q /, whee q is the total chage enclosed by the Gaussian suface. RIF CDEMY (D 64/94 3 Siga (Nea Shubham Hospital, Vaanasi)) (94544) Page 3 E 4 q / q o E. (fo > R) 4 This is also the fomula fo the electic intensity at a point distant fom a point-chage q. It theefoe, follows that fo points outside the chaged spheical shell, the shell behaves as if all the chage on its suface wee concentated at its cente. If σ be the unifom chage density at the suface of the shell, then q = 4R σ. Substituting this value of q in eq. (i), we get R E Eq. (i) and (ii) ae the equied elations. t the Suface : If the point P lies just on the suface of the shell, then = R. Eq. (i) and (ii) now give q E. 4 R t an Intenal Point : Fo points inside the shell, such as P (Fig. 6), the Gaussian suface though P does not enclose any chage. Hence, accoding to Gauss theoem, we have o E = E E 4

14 [ELECTROSTTICS] The electic field intensity is zeo eveywhee inside the chage shell. The vaiation of electic field E of a unifomly chaged spheical shell with distance fom the cente of the shell is shown in Fig. 7. The electic field is zeo eveywhee inside the shell (fom = to = R), is maximum at the suface of the shell and deceases apidly outside the shell (E / ). BORD LEVEL QUESTIONS:. Define electic flux. O What do you undestand by electic flux?. Is electic flux a scala o a vecto? Give the SI unit of electic flux. 3. State Gauss theoem in electostatics. 4. n electic dipole of dipole moment 6 C m is enclosed by a closed suface. What is the net flux coming out of the suface? [ns:.5 N m C, 4.45 C]. lage plane sheet of chage having suface chage density 5. 6 C m lies in the XY-plane. Find the electic flux though a cicula aea of adius. m, if the nomal to the cicula aea makes an angle of 6 with the Z-axis. Given that = 8.85 C N - m. [ns: N m C ] 3. Two lage thin metal plates ae paallel and close to each othe as shown in the Fig. On thei inne faces, the plates have suface chage densities of opposite signs and of magnitude 7. C m. What is E (a) to the left of the plates, (b) to the ight of the plates and (c) between the plates? [ns:.9 N C ] 4. The electic field in a egion is adially outwad and vaies with distance as E = 5 V m Calculate the chage contained in a sphee of adius. m cented at the oigin. [ns:. C] 5. The flux of the electostatic field, though the closed suface S, is found to be fou times that though the closed spheical suface S[Fig.]. Find the magnitude of the chage Q. Given q = C, q = C and q 3 = C. 5. Define electic flux. Give its SI unit and dimensional fomula. 6. What is a Gaussian suface? Explain in bief. 7. Explain the tem electostatic shielding. O What is the pinciple of electostatic shielding? [ns: 6.56 C] 6. S and S ae two paallel concentic sphees enclosing chages Q and Q espectively as shown in Fig. 8. sensitive instument is to be shielded fom the stong electostatic field in its envionment. Suggest a possible way. 9. Is it possible to tansfe all the chage fom a conducto to anothe insulated conducto?. Calculate the numbe of electic lines of foce oiginating fom a chage of C. Given, = C N m. [ns:.9 ]. The electic field components due to a chage inside the cube of side. m ae as shown in Fig. : (a) What is the atio of the electic flux though S and S? (b) How will the electic flux though the sphee S change, if a medium of dielectic constant 5 is intoduced in the space inside S in place of ai? Q [ns: ' ] 5 Electic o Electostatic Potential (V): The electic potential at a point in an electic field is the wok done by an extenal foce in bining a unit positive chage fom infinity to that point without impating any acceleation to it. s wok o enegy is a scala quantity, so is the potential. Unit of electic potential is volt. volt = NmC = JC E x = x, whee = 5 N C m, E y = and E z =. Calculate (a) the electic flux though the cube and (b) the chage inside the cube. RIF CDEMY (D 64/94 3 Siga (Nea Shubham Hospital, Vaanasi)) (94544) Page 4

15 [ELECTROSTTICS] Electic potential at a point due to isolated point chage Suppose we have to find electic potential due to a point chage q at a distance. Let a small positive test chage q is bought to fom with a negligibly small velocity. q V = 4πε and is same fom the chage fo all the points on the suface of the same sphee. Let, Electic field at any point P be E PQ = an infinitesimally small path element dl Electic foce due to the field E on q at P = q E Foce to be applied on q to move it fom P to Q without impating any acceleation to it q coesponding wok, dw qo E. dl Potential at, ( q E. dl) V [s V ] q V 4 o q q V 4o NOTE: While calculating the potential at a point, put the value of chage with its sign. It means positive chage poduces positive potential and negative chage poduces negative potential. E Potential diffeence between points & B will be, B V V = E.dl B VB V = kq kq B B Electic potential at a point due to many point chages q q q qn n 3 V P =... 4πε 3 Popeties of equipotential suface. Electic potential is same at evey point on equipotential suface.. Electic field is always nomal to a equipotential suface 3. No wok is equied to be done in moving a chage fom one point to anothe on an equipotential suface. 4. The line integal of E along an equipotential suface is zeo, because V V B = 5. No two equipotential sufaces can intesect each othe. 6. In a family of equipotential sufaces, the sufaces ae close togethe whee the electic field is stonge and fathe apat whee the field is weake. Examples of some othe equipotential suface (i) Equipotential suface fo a negative chage i.e., fo q < will be as follows: (ii) Equipotential sufaces in unifom electic field will be as follows: (iii) Equipotential suface between two point chage +q and q is as follows: Equipotential suface: n equipotential suface is a suface with a constant value of potential at all points on the suface. Fo example, the suface of each sphee whose cente lies on an isolated chage is an equipotential suface, since (iv) Equipotential sufaces fo an electic dipole ae shown in figue given below: RIF CDEMY (D 64/94 3 Siga (Nea Shubham Hospital, Vaanasi)) (94544) Page 5

16 [ELECTROSTTICS] Electic potential enegy (U): Potential enegy of chage q at a point (in the pesence of field due to any chage configuation) is the wok done by the extenal foce (equal and opposite to the electic foce) in binging slowly the chage q fom infinity to that point. Electic potential enegy of a point chage in extenal electic field: Potential enegy of a single chage q at in an extenal electic field is U q V(). Electic potential enegy of two point chages: When q is bought fom to P, no wok is needed to be done on it because thee is no foce against which some wok would be equied to be done. PE of q = (a) Electic potential enegy of a single chage in a nonunifom extenal electic field: Potential enegy of a single chage q at in an extenal electic field q.v( ). (b) Electic potential enegy of a system of two chages in an extenal field: Potential enegy of a system of two chages q and q located at and espectively qq qv( ) qv( ) 4πε whee is the distance between q and q. Electic potential enegy of system of n point chages n n qq U = 4πε i j j ij j > i so that we don t count the same expession (o tem) twice. Change in P.E. of a point chage in moving it fom one point to anothe B in a fixed electic field P.E. = chage potential existing at its position Change in P.E. = chage p.d. between its two positions U = q(vb V ) Question: What should be the value of Q, so that potential enegy of the system is zeo. Howeve, when q is bought fom to P, the field of q aleady exists at P, against which wok has to be done to bing q to P. If q is a unit chage, this wok would be = potential at P. P.E. of q q V P q q. 4πε o q q U = + 4πε q q 4πε o o qq U = 4πεo Note:. In above fomula both the chages q & q ae to be substituted with sign.. Fo a Electic potential enegy of a system of chages is given by q iq i U 4 o i j ij Fo example U of fou point chages q, q, q3 and q 4 would be given by qq qq 4 q q3 q q 4 q 3q 4 U 4 o If thee be n chages in space, then the total numbes of pai of possible combination n C = n(n-)/. Solution: The electic potential enegy of the system will be q qq qq U 4 a 4 a 4 a o o o Q Fo U =, q Q q Q Question: Thee point chages q, q and 8q ae to be placed on a 9cm long staight line. Find the positions whee the chages should be placed such that the potential enegy of this system is minimum. In this situation, what is the electic field at the chage q due to the othe two chages? Solution: The maximum contibution will come fom the chage 8q foming pais with othes. To educe its effect, the two lage chages should be placed at the exteme ends and the smallest chage q in between them. The aangement shown in figue ensues that the chages in the stongest pai q, 8q ae at the lagest sepaation. The potential enegy is q 6 8 U 4 x 9cm 9cm x. o RIF CDEMY (D 64/94 3 Siga (Nea Shubham Hospital, Vaanasi)) (94544) Page 6