UNIT - 11 ELECTROSTATICS

Transcription

1 UNIT - ELETOSTATIS

2 SUMMAY. Electic hage : Just as masses of two paticles ae esponsible fo the gavitational foce, chages ae esponsible fo the electic foce. Electic chage is an intinsic popety of a paticle. hages ae of two types : () Positive chages () Nagative chages. The foce acting between two like chages is epulsive and two unlike chages it is attactive between.. uantization of Electic hage : The magnitude of all chages found in natue ae an integal multiple of a fundamental chage. ne, whee e is the fundamental unit of chage.. onsevation of Electic hage : Iespective of any pocess taking place, the algebaic sum of electic chages in an electically isolated system always emains constant. 4. oulomb's Law : The electic foce between two stationay point chages is diectly popotional to the poduct of thei chages and invesely popotional to the suae of the distance between them. k 4 If then thee is a epulsion between the two chages and fo, thee is a attaction between the chages. 5. Euation fo oce using olumb s Law, when two chages ae placed in a medium having dielectic constant k. () The electic foce expeienced by a test chage ( ) due to a souce chage () when both ae placed in a medium having dielectic constant k and sepaated by a distance, is given by : 4 k ˆ O () Hee ˆ is the unit vecto diected fom to. P ( o ) () The euation of coulomb's foce may be witten as follows : 4 () If the souce chage and test chage ae sepaated by a numbe of medium of thickness d, d, d... having dielectic constants k, k, k... espectively, then the electic foce on chage due to a chage is given by 4 k d k d k d ˆ O k ˆ 4 ki d i In this euation k i is dielectic constant of medium which speads though the distance d i along the line joining and. ˆ

3 o example, see the figue below : Hee, the space between the chages and is filled with medium (,, ). The thickness of medium is d and its dielectic consant is k Similaly the thickness of medium and is d and d of medium and thei dielectic constants ae k and k espectively. 6. onditions fo Euilibium in Vaious ases : Suppose thee chages, and ae situated on a staight line as shown below : If and ae like chages and is of unlike chage then, () oce on 4 () oce on 4 () oce on = 4 Now, fom above euations, it is clea that vaious euilibium conditions can be as follows : (a) ondition fo to be zeo is, (b) ondition fo to be zeo is, (c) ondition fo to be zeo is, If, and ae of same type chages in natue, then, () hage will be in euilibium, if 4 () hages and will not be in euilibium.

4 7. Electic ield Intensity : The electic foce acting on a unit positive chages at a given point in an electic field of a system of chages is called the electic field o the intensity of electic field E at that point. E The SI unit of E is N o Vm. If ae the position vectos of the chages,,... n espectively, then the,,... n esultant electic field at a point of position vecto is, E k n j j j j 8. Electic Dipole : A system of two eual and opposite chage, sepaated by a finite distance is called electic dipole. Electic dipole moment p a The diection of p is fom the negative electic chage to the positive electic chage. 9. Electic field of a dipole on the axis of the dipole at point z = z kp ^ E z p fo z a z Electic field of a dipole on the euato of the dipole at point y = y kp E y p fo y a y ^. The toue acting on the dipole place in an unifom the electic field at an angle, p E, p Esin. Electic lux : Electic flux associated with suface of aea A, placed in the unifom electic field. E A EAcos whee, is the angle between E and A, Its SI unit is Nm o V.m.. Gauss's Law : The total electic flux associated with the closed suface, E d a whee, is the net chage enclosed by the suface. S 4

5 . Electic field due to an infinitely long staight chaged wie, E ˆ, whee, is the pependicula distance fom the chaged wie. 4. Electic field due to bending of chaged od, 5. Electic field due to unifomly chaged thin spheical shell, () Electic field inside the shell E () Electic field at a distance fom the cente outside the shell, E k whee, = adius of spheical shell. 6. Electic field due to a unifomly chaged density sphee of adius, () Electic field inside the egion of the sphee, E 4 5

6 () Electic field outside the sphee, E 4 whee, is the total chage inside the sphee. 7. The infomation about the wok done to take an electic chage fom one point to the othe in a given electic field, obtained fom the uantities called electic potential and electic potential enegy. B 8. E d is the line-integal of electic field between point A and B and it shows the wok done by A the electic field in taking a unit positive chage fom A and B. Moeove, it does not depend on the path and Ed. 9. "The wok euied to be done against the electic field to bing a unit positive chage fom infinite distance to the given point in the electic field, is called the electic potential (V) at that point". Electic potential at point P is V p P Ed Joule It unit is volt. Symbolically coulomb Its dimensional fomula is M L T A V Absolute value of electic potential has no impotance but only the change in it is impotant.. "The wok euied to be done against the electic field to bing a given change () fom infinite distance to the given point in the electic field is called the electic potential enegy of that electic chage at that point." U p P Ed V p 6 J The absolute value of electic potential enegy has no impotance, only the change in it is impotant.. Electic potential at point P, lying at a distance fom a point chage is. The electic potential at a point at distance fom an electic dipole is p, v ( o > > a) 4 p Potential on its axis is V, Potential on its euato is V 4. Electic potential at a point due to a system of point chage,,... n situated at position at position,,... n is V n i k i i V p k

7 The electic potential at point, due to a continuous chage distibution is V 4 volume d' The electic potential due to a spheical shell is i V o 4 and V o 4 4. A suface on which electic potential is eual at all points is called an euipotential suface. The diection of electic field is nomal to the euipotential suface. 5. dv E gives the magnitude of electic field in the diection of dl. dl To find E fom V, in geneal, we can use the euation V ˆ V ˆ V E i j kˆ x y z The diection of electic field is that in which the ate of decease of electic potential with distance dv dl is maximum, and this diection is always nomal to the euipotential suface. 6. The electostatic potential enegy of a system of point chages,,... n situated at positions,,... n U is k n i j i whee ij ij j i i j 7. The electostatic potential enegy of an electic dipole in an extenal electic field E, is U E p Ep cos 8. When a metallic conducto is placed in an extenal electic field, (i) (ii) (ii) A stationay chage distibution is induced on the suface of the conducto. The esultant electic field inside the conducto is zeo. The net electic chage inside the conducto is zeo. (iv) The electic field at evey point on the oute suface of conducto is locally nomal to the suface. (v) (vi) The electic potential inside the egion of conducto is the same evey whee. If thee is a cavity in the conducto then, even when the conducto is placed in an extenal electic field, the esultant electic field inside the conducto and also inside the cavity is always zeo. This fact is called the electostatic shielding. 7

8 When electic chage is placed on the metallic conducto : (i) The electic field inside the conducto is zeo eveywhee. (ii) The chage is distibuted only on the oute suface of the conducto. (iii) The electic field on the suface is locally nomal, and is eual to E nˆ. (iv) If a chage is placed inside the cavity in the conducto, the electic field in the conducto emains zeo. 9. "A device fomed by two conductos sepated fom each othe is called a capacito." Its capacitance is constant. The unit of is V. p 6. The effective capacitance in seies connection is then, coulomb which is also called faad. volt... The effective capacitance in paallel connection is then,.... The capacitance of the paallel plate capacito is A. d. The enegy stoed in the capacito is U V V and the enegy density = enegy stoed pe unit volume E, whee E electic field.. When a dielectic is placed in an extenal electic field E, polaisation of dielectic occus due to electical induction. The electic field poduced by these induced chages is in the opposite diection to the diection of extenal electic field. Hence the esultant electic field E, inside the dielectic is less than the extenal electic field E. The dipole moment poduced pe uint volume is called the intensity of polaisation o in shot polaisation P nˆ b. Since P E, P x E x is called the electic susceptibility of the dielectic medium. e e e x is called the pemittivitty of the dielectic medium. is called the elative pemittivity of that medium and it is also called the dielectic constant K. 8

9 i.e. K E K x, E e. Thus in the dielectic the electic field educes to the th K K pat. D E P is called the electic displacement. Gauss Law in the pesence of dielectic is witten as D ds, whee is only the net fee chage. 4. when thee is ai (o vacuum) between the plates of a paallel plate capacito, the capacitance is A. On placing a medium of dielectic constant K, the capacitance is ' K. Thus the d capacitance becomes K times, due to the pesence of the dielectic. 5. With the help of Van-De-Gaf geneato a potential diffeance of a few nillion volt can be established. 9

10 ONEPT MAP

11 M o the answe of the following uestions choose the coect altenative fom among the given ones.. When a Piece of Polythene is ubbed with wool, a chage of 7 is developed on polythene. The mass tansfeed to polythene is... kg () The potonic chage in gm of wate is... c () A coppe sphee of mass gm contains about atoms. The chage on the nucleus of each atom is 9e. The faction of electons emoved..9 () The ate of alpha patical falls on neutal spheae is pe second. The time in which sphee gets chaged by µc is... sec..5.5 () A chage is divided into two pats and then they ae placed at a fixed distance. The foce between the two chages is always maximum when the chages ae...,, (), 4 4, Two point chages epel each othe with a foce of N. One of the chages is inceased by % and othe is educed by %. The new foce of epulsion at the same distance would be... N. () Given that + = if the between and is maximum, () Two small conducting sphee of eual adius have chages + µc and µc espectively and placed at a distance d fom each othe expeience foce. If they ae bought in contact and sepaated to the same distance, they expeience foce. The atio of to is... 8 : : () : 8 : 9. Thee chages, each of value, ae placed at the vetex of an euilateal tiangle. A fouth chage is placed at the cente of the tiangle. If the chages emains stationey then, =... ()

12 . Tw o small chaged sphees epal each othe w ith a foce N. The chage on one sphee is twice that of the othe. When these two spheaes displaced cm futhe apat the foce 4 is 5 N, then the chages on both the sphees ae ,..4, ()., , 4.4. Thee chages, + and ae placed as shown in figue. The x component of the foce on is popotional to... sinθ b a () + sinθ b a cosθ b a + cosθ b a. Two eual negative chages ae fixed at points (o, a) and (o, a). A positive chage is eleased fom est at the point (a, o) on the X - axis. The chage will... move to the oigin and emain at est thee execute simple hamonic motion about the oigin () move to infinity execute oscillations but not simple hamonic motion. ou chages, each eual to, ae placed at the cones of a suae and a chage + is placed at its cente. If the system is in euilibium, the value of is... 4 () 4 a Y b X + 4. o the system shown in figue, if the esultant foce on is zeo, then (, a) a =... a A A ()

13 5. Two point positive chages each ae placed at ( a, o) and (a, o). A thid positive chage o is placed at (o, y). o which value of y the foce at o is maximum... a a () 6. Two identical chaged sphees suspended fom a common point by two massless stings of length l ae initially a distance d (d << l ) apat because of thei mutual epulsion. The chage begins to leak fom both the sphees at a constant ate. As a esult the sphees appoach each othe with a velocity υ. Then function of distance x between them becomes... v α x a v α x () v α x a v α x 7. Thee identical sphees each having a chage and adius, ae kept in such a way that each touches the othe two spheaes. The magnitude of the electic foce on any sphee due to othe two is π 4 8π () 4π 4 8π 8. Two eual negative chages ae fixed at points (o, a) and (o, a) on the Y axis. A positive chage is eleased fom est at the point x (x < < a) on the X-axis, then the feuency of motion is... π ma 4π ma () 4 π ma π ma 9. Two identical balls having like chages and placed at a cetain distance apat epel each othe with a cetain foce. They ae bought in contact and then moved apat to a distance eual to half thei initial sepaation. The foce of epulsion between them inceases 4.5 times in compaison with the initial value. The atio of the initial chages of the balls is... 4 : 6 : () : :. A point chage is situated at a distance fom one end of a thin conducting od of length L having a chage (unifomly distibuted along its length). The magnitude of electic foce between the two, is... k ( + L) k ( + L) () k ( L) k ( + L). Two point chages of +6µc and 9µc ae placed 8 cm apat in ai.... distance of a point fom 9µc chage at which the esultant electic field is zeo. 4 cm 9 cm () 6 cm 5 cm

14 . Point chages 4µc and µc ae placed at the vetices P and of a ight angle tiangle P espectively. is the ight angle, P = m and = m. The magnitude and diection of the esultant electic field at c is N, 45.8 N, ().7 N, N, 4.7. An inclined plane making an angle of o with the hoizontal is placed in an unifom electic field E = Vm. A 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. the time taken by the paticle to each the bottom is... sec () A small sphee whose mass is. gm caies a chage of and is tieup to one end of a silk fibe 5 cm long. The othe end of the fibe is attached to a lage vetical conducting 6 plate which has a suface chage of 5 m, on each side. When system is feely hanging the angle fibe makes with vetical is () A Semicicula od is chaged unifomly with a total chage coulomb. The electic field intensity at the cente of cuvatue is... K K K 4K () π π π π 6. The electon is pojected fom a distance d and with p v initial velocity υ paallel to a unifomly chaged flat conducting plate as shown in figue. It stikes the plate X afte tavelling a distance l along the diection. The suface chage density of conducting plate is eual to ----l----- Y d mυ el d mυ el () d mυ el d mυ el 7. Two point masses m each caying chage and + ae attached to the ends of a massless igid non-conducting od of length l. The aangement is placed in a unifom electic field E such that the od makes a small angle 5 with the field diection. The minimum time needed by the od to align itself along the field is... t = π ml E π t = ml E () t = ml E t = π ml E 4

15 8. Two unifomaly chaged spheical conductos A and B having adius mm and mm ae sepaated by a distance of 5 cm. If the sphees ae connected by a conducting wie then in euilibium condition, the atio of the magnitude of the electic fields at the sufaces of sphees A and B is... 4 : : () : : 4 9. Let P() = 4 be the chage density distibution fo a solid sphee of adius and total π chage. o a point P inside the sphee at distance fom the cente of the sphee the magnitude of electic field is 4π 4 () 4π π. Two point chages µc and µc ae placed at distance b = cm and a = cm fom the oigin on the y and x axes as shown in figue. The electic field vecto at point P (a, b) will subtend an angle with the X - axis given by, tan θ 4 tan θ () tan θ tan θ 4. A simple pendulum consists of a small sphee of mass m suspended by a thead of length l. The sphee caies a positive chage. The pendulum is placed in a unifom electic field of stength E diected Vetically upwads. If the electostatic foce acting on the sphee is less than gavitational foce the peiod of pendulum is b Y a P(a, b) X l T = π g E m T = π l g () l T = π g +. onside a system of thee chages /, / and a/ placed at points A, B and espectively as shown in the figue. It the adius of the cicle is and AB = 6 then the electic field at cente is... 8π () 6π 54π E m A 6 ml T = E B O 5

16 . In Millikan s oil dop expeiment an oil dop caying a chage is held stationay by a p.d. 4 v between the plates. To keep a dop of half the adius stationay the potential diffeence had to be made 6 v. What is the chage on the second dop? 4 () 4. Eual chages ae placed at the vetices A and B of an euilateal tiangle AB of side a. The magnitude of electic field at the point c is... K a K a K () a t a 5. An electic dipole is placed along the x-axis at the oigin o. A point P is at a distance of cm fom this oigin such that OP makes an angle π with the x-axis. If the electic field at P makes an angle with the x-axis, the value of would be... π tan π () π tan 9 6. A paticle having a chage of.6 entes between the plates of a paallel plate capacite. The initial velocity of the paticle is paallel to the plates. A potential diffeence of v is applied to the capacito plates. If the length of the capacito plates is cm and they ae sepaated by cm, alculate the geatest initial velocity fo which the paticle 4 will not be able to come out of the plates. The mass of the paticle is kg. 4 m s m s () m s m s 7. A chaged paticle of mass kg and chage µc is thown fom a hoizontal gound at an angle o 7 = 45 with speed m/s. In space a hoizontal electic field E = V/m exist. The ange on hoizontal gound of the pojectile thown is.... m 5 m () m m 8. If electon in gound state of H-atom is assumed in est then dipole moment of electon poton system of H-atom is.... Obit adius of H atom in gound state is.56 A cm.848 cm ().5 cm.5 cm 9. At what angle θ a point P must be located fom dipole axis so that the electic field intensity at the point is pependicula to the dipole axis? 5 to 54 5 to 5 () 45 to 46 5 to 5 6

17 4. A n electic dipole is placed at an angle of 6 o with an electic field of intensity 5 N. It expeiences a toue eual to 8 Nm. If the dipole length is cm then the chage on the dipole is... c () An electic dipole coincides on z axis and its mid point is on oigin of the catesian co-odinate system. The electic field at an axial point at a distance z fom oigin is E(z) and electic field at an euatoial point at a distance y fom oigin is E(z) (y = z > > a) =... E(y) () 4 4. An oil dop of excess electons is held stationay unde a constant electic field of 4.55 Vm. If the density of the oil is.6 gm/cm then the adius of the dop is... m () A hage is placed at the cente of the open end of cylindical vessel. The flux of the electic field though the suface of the vessel is... () E(y) Zeo 44. The inwad and outwad electic flux fo a closed suface in units of Nm / ae espectively 8 and 4. Then the total chage inside the suface is... c. 4 4 () A sphee of adius has a unifom distibution of electic chage in its volume. At a distance x fom its cente, (fo x < ), the electic field is diectly popotional to... x x () x x 46. The electic flux fo gaussian suface A that enclose the chged paticles in fee space is... (given = 4nc, = nc, = 56nc) 4 Nm / Nm / () 6. Nm / 4 6. Nm / 7

18 47. A hollow cylinde has a chage coulomb within it. If is the electic flux in units of voltmete associated with the cuved suface B, the flux linked with the plane suface A in units of volt-mete will be... B A () 48. An infinitly long thin staight wie has unifom linea chage density of c/m. Then, the magnitude of the electic intiensity at a point 8 cm away is... N..66. (). 49. Two points ae at distances a and b (a < b) fom a long sting of chage pe unit length. The potential diffeence between the points in popotional to... b ln a b ln a () π b ln a b ln π a 5. A long sting with a chage of pe unit length passes though an imaginay cube of edge l. The maximum possible flux of the electic field though the cube will be... l l () l 5. Two Points P and ae maintained at the Potentials of v and 4 v, espectively. The wok done in moving electons fom P to is J 9.6 J ().4 J 9.6 J 5. The electic Potential V at any Point o (x, y, z all in metes) in space is given by V = 4x volt. The electic field at the point (m,.m) in volt/mete is... 8, along negative x - axis () 6, along negative x - axis 8, along positives x - axis 6, along positives x - axis 9 5. hages of + ae placed at each of the fou cones of a suae of side 8cm. The potential at the intesection of the diagonals is... 5 Volt 9 Volt () 5 Volt 9 Volt 6l 8

19 54. Thee chages,, ae located at the vetices of an euilateal tiangle. At the cente of the tiangle. The ield is Zeo but Potential is non - zeo The ield is non - Zeo but Potential is zeo () Both field and Potential ae Zeo Both field and Potential ae non- Zeo 55. In the electic field of a point chage, a cetain chage is caied fom point A to B,, D and E. Then the wok done... Is least along the Path AB Is least along the Path AD () Is Zeo along all the Path AB, A, and Is least along AE 56. Thee concentic spheical shells have adii a, b and c (a < b < c) and have suface chage densities σ, σ and σ espectively. If V A, V B and V denote the Potentials of the thee shells, then fo c = a + b, we have V = V B = V A V c = VB VA () Vc VB VA Vc VA VB 57. The electic Potential at a point P (x, y, z) is given by V = x y xz + 4 The electic field E at that point is... i ( xy + z ) + j x + k xz i xy + j ( x + y ) + k ( xy y ) B A + D E () i z + j xyz + k z i ( xy z ) + j xy + kz x 58. Thee paticles, each having a chage of µc ae placed at the cones of an euilateal tiangle of side cm. The electostatic potential enegy of the system is Given = 9 N.m /c 9 4π o. J 7 J () Zeo Infinite 59. ou eual chages ae placed at the fou cones of a suae of each side is a. Wok done in emoving a chage - fom its cente to infinity is... π a () 4π a π a 9

20 6. Two chaged sphees of adii and having eual suface chage density. The atio of thei potential is... () 6. Two eual chages ae placed at a distance of a and a thid chage - is placed at the midpoint. The potential enegy of the system is π a 8π a () 7 8π a 6 8π a 6. Two point chages µc and 5µc ae placed at points A and B espectively with AB = 4 cm. The wok done by extenal foce in displacing the chage 5µc fom B to whee B = cm, angle 9 J π AB = and 9 J 5 4π 9 9 Nm /c. () 8 J 9 J 4 6. The electic potential V is given as a function of distance x (mete) by V = (5x + x 9) volt. Value of electic field at x = is... v m v 6 m () v m v m 64. A sphee of adius cm has potential of 8 v, then enegy density nea its suface will be... J 64 5 m J.8 () m J J m 8 m 65. If a chaged spheical conducto of adius cm has potential v at a point distant 5 cm fom its cente, then the potential at a point distant 5cm fom the cente will be... V V () V V 66. Electic chages of +µc, 5µc, µc and 8µc ae placed at the cones of a suae of side m the potential at the cente of the suae is V.8 V ().8 V.8 V 67. Two positive point chages of µc and 8µc ae cm apat each othe. The wok done in binging them 4cm close is J ev () 5.8eV J

21 68. The displacement of a chage in the electic field E = e i + e j + ek is = ai + b j. The wok done is... e + e a + b e + e a + b () ae + be ae + be 69. If an electon moves fom est fom a point at which potential is 5 volt to anothe point at which potential is 7 volt, then its kinetic enegy in the final state will be... 8 N. J (). J dyne 7. Thee chages, + and + ae placed at the veticles of a ight-angled tiangle as shown. The net electostatic enegy of the configuation is zeo if is eual to... + () a + 7. Two electic chages µc and 6µc ae placed cm apat in ai. Thee will be a point P on the line joining these chages and outside the egion between them, at which the electic potential is zeo. The distance of P fom 6µc chage is.... m. m ().5 m.5 m 7. In the ectangle, shown below, the two cones have chages = 5µc and = +.µc. The wok done in moving a chage µc fom B to A is take = Nm /c 4π. 5.5 J.8 J ().5 J 4.5 J 7. 4 Points chages each + is placed on the cicumfeence of a cicle of diamete d in such a way that they fom a suae. The potential at the cente is... 4d () 4 4d d 74. Thee identical chages each of µc ae placed at the vetices of a tiangle AB as shown in the figue. If AB + A = cm and AB. A = cm, the potential enegy of the chage at A is....5 J 5. J ().5 J.5 J B cm A

22 75. A ball of mass gm and chage 8 c moves fom a point A, whee the potential is 6 volt to the point B whee the potential is zeo. Velocity of the ball of the point B is cm/s. The velocity of the ball at the point A will be... m 6.8 s cm.8 s () cm 8 s m 68 s 76. Thee chages, + and + ae placed at the vetices of an euilateal tiangle of side l as shown in the figue. It the net electostatic enegy of the system is zeo, then is eual + l l () Zeo + l Electic potential at any point is V = 5x + y + 5z, then the magnitude of the electic field is... N/. 4 () A small conducting sphee of adius is lying concentically inside a bigge hollow conducting sphee of adius. The bigge and smalle sphee ae chaged with and ( > ) and ae insulated fom each othe. The potential diffeence between the sphes will be... 4π 4π () + 4π 4π 79. If chages ae placed at the vetices of euilateal tiangle of chage each. What is the net potential enegy, if the side of euilateal tiangle is l cm. 4π l 4π l () 4π l 4 4π l 8. If identical chages ( ) ae placed at each cone of a cube of side b, then electic potential enegy of chage (+) which is placed at cente of the cube will be... 4 π b 8 4π b () 8 π b 4 4π b 8. A simple pendulum of peiod T has a metal bob which is negatively chaged. If it is allowed to ascillate above a positively chaged metal plate, its peiod will... emains eual to T () Infinite Less than T Geate than T

23 8. A chaged paticale of mass m and chage is eleased fom est in a unifom electic field E. Neglecting the effect of gavity, the kinetic enegy of the chaged paticale afte t second is... E m t E t m () E t m Em t 8. A thin spheical conducting shell of adius has a chage. Anothe chage is placed at the cente of the shell. The electostatic potential at a point p a distance fom the cente of the shell is... ( + ) 4π 4π 4π () 4π 4π 4π 84. Two point chages and + ae located at points (o, o, a) and (o, o, a) espectively. The potential at a point (o, o, z) whee z > a is... a 4π z a 4π a a () 4π z a 4π z a 85. Point chages = µc and = µc ae kept at points x = and x = 6 espectively. Electical potential will be zeo at points... x =, x = x =, x = 5 () x = 4, x = x =, x = Two thin wie ings each having a adius ae placed at a distance d apat with thei axes coinciding. The chages on the two ings ae + and. The potential diffeence between the centes of the two ings is... π + d () 4π + d 4π d 87. Tw o chages and ae placed cm apat, as shown in the figue. A thid chage is moved along the ac of a cicle of adius 4 cm fom to D. The change in the potential enegy of the system is K, whee k is... 4π 4 cm A cm B D 8 8 () 6 6

24 88. igue shows a tiangula aay of thee point chages. The electic potencial v of these souce chages at the midpoint P of the base of the tiangle is 55 KV () 49 KV 4π 9 9 Nm /c 6 KV 45 KV. 89. hages + and ae placed at point A and B espectively which ae a distance L apat, is the midpoint between A and B. The wok done in moving a chage + along the semicicle D is... π L 6π L () 6π L 4π L 9. N identical dops of mecuy ae chaged simultaneously to volt. when combined to fom one lage dop, the potential is found to be 4 volt, the value of N is () 8 9. Tw o paallel plate ai capacitos have thei plate aeas and 5 cm espectively. If they have the same chage and potential and the distance between the plates of the fist capacito is 9.5 mm, what is the distance between the plates of the second capacito?.5 cm.5 cm ().75 cm cm 9. The effective capacitances of two capacitos ae µ and 6µ, when they ae connected in seies and paallel espectively. The capacitance of each capacito is µ, 4µ 4µ, µ () 6µ, 8µ µ, 6µ 9. An electical technician euies a capacitance of µ in a cicuit acoss a potential diffeence of KV. A lage numbe of µ capacitos ae available to him, each of which can withstand a potential diffeence of not than 4 V. suggest a possible aangement that euies a minimum numbe of capacitos. ows with capacitos () ows with 4 capacitos = -6 4 ows with capacitos 6 ows with capacitos 94. Two spheical conductos of adii and ae at potentials V and V espectively, then what will be the common potential when the conductos ae bought in contant? l =.m l m. m = - A B D v + v + v + v v v + () None of these 4

25 95. A 5µ capacito is chaged by a V supply. It is then disconnected fom the supply and is connected to anothe unchaged.5µ capacito. How much electostatic enegy of the fist capacito is lost in the fom of heat and electomagnetic adiation?. J. J ().4 J.8 J 96. ind the euivalent capacitance of the system acoss the teminals A and B. All the capacitos have eual capacitances. () 4 5 A B 97. apacitance of a paallel plate capacito becomes 4 times its oiginal value if a dielectic slab of thickness t = d/ is inseted between the plates (d is the sepaation between the plates). The dielectic constant of the slab is 8 4 () The plates of a paallel capacito ae chaged up to V. If mm thick plate is inseted between the plates, then to maintain the same potential diffeence, the distance between the capacito plates is inceased by.6mm the dielectic constant of the plate is 5 4 () A paallel plate ai capacito has a capacitance 8µ. If the distance between the plates is tipled and a dielectic medium is intoduced, the capacitance becomes 7µ. The dielectic constant of the medium is 4 () 9. Taking eath to be a metallic sphees, its capacity will appoximately be () 7 7 p. A paallel plate capacito has the space between its plates filled by two slabs of thickness d each and dialectic constant K and K. If d is the plate sepaation of the capacito, then capacity of the capacito is... d K + K A KK A K K d K + K A K + K d KK () 5 A d K + K

26 . o the cicuit shown in figue the chage on 4µ capacito is 4 µc 4µc () µc 54µc. The capacitos of capacitance 4µ, 6µ and µ ae connected fist in seies and then in paallel. What is the atio of euivalent capacitance in the two cases? : : () : : 4. Lage numbe of capactos of ating µ/v V ae available. The minimum numbe of capacitos euied to design a µ/7v capacito is 6 8 () A vaiable condense is pemanently connected to a V battey. If capacito is changed fom µ to µ. then enegy changes is eual to J.5 J () 6.5 J 4 J 6. Two positive point chages of µc and 8µc ae placed cm apat in ai. The wok done to bing them 4 cm close is Zeo 4.8 J ().5 J 5.8 J 7. simila electified ain dops mege togethe into one dop so that thei total chage emains unchanged. How is the electic enegy affected? times times () 4 times times 8. Thee ae condenses each of capacity 5µ. The atio between maximum and minimum capacities obtained fom these condenses will be 4 : 5 : 5 () 6 : : 9. A paallel plate capacito is made by stocking n eually spaced plates connected altenately. If the capacitance between any two plates is x, then the total capacitance is, + V 5 nx nx () n x. o the cicuit shown figue, which of the following statements is tue? With S closed V = 5V, V = V With S closedv = V = V () With S and S closed V = V = With S and S closed V = V, V = V (n )x S S S v =v v =v =pf =pf 6

27 . Two identical metal plates ae given positive chages and (< ) espectively. If they ae now bought close to gethe to fom a paallel plate capacito with capacitance, the potential diffeence between them is ( ) c ( ) c ( ) () c ( ) c. In the cicuit aangement shown in figue, the value of = = = p and 4 = p. If the combination of capacitos is chaged with 4V D supply, the potencial diffeences acoss the fou capacitos will be espectively 8, 4, 4 and V, 4, 4 and 8V () 5, 5, 5 and 5V 8,, and V. In the aangement of capacitos shown in figue, each capacito is of 9µ, then the euivalent capacitance between the points A and B is 8µ 9µ () 5µ 4.5µ A B 4 + 4V 4 4. The electic potencial V at any point x, y, z (all in mete) in space is given by V = 4x volt. The electic field at the point (m,, m) in Vm is + 8i 8i () 6i +6i 5. Two ai capacitos A = µ, B = 4 µ ae connected in seies with 5 V souce. When a medium of dielectic constant K = is intoduced between the plates of A, change on the capacito changes by 6 µc µc () 8 µc 6 µc 6. A paallel plate capacito with ai between the plates has a capacitance of 9 p. The sepaation between its plates is d. The space between the plates is now filled with two dielectics. One of the dielectic constant K = and thickness d/ while the othe one has dielectic constant K = 6 and thickness d/. apacitance of the capacito is now.8 p.5 p () 4.5 p 45 p 7

28 7. A thin spheical shell of adius has change spead unifomly ove its suface. Which of the following gaphs, figue most closely epesents the electic field E () poduced by the shell in the ange <, whee is the distance fom the cente of the shell. E E E E () 8. A paallel plate condense with dielectic of constant K between the plates has a capacity and is chaged to potential V volt. The dielectic slab is slowly emoved fom between the plates and einseted. The netwok done by the system in this pocess is Zeo (K )cv () (K )cv (K ) cv K 9. hages ae placed on the vetices of a suae as shown. Let E be the electic field and V the potential at the cente. If the chages on A and B ae intechanged with those on D and espectively then E hange V emains unchanged E emains unchanged, V changes () Both E and V change E and V emain unchanged. The potencial at a point x (measued in µm ) due to some chages situated on the x-axis is given by V( x) = Volt x 4 5 Vm and in positive x - diection 9 Vm and in positive x - diection () 9 Vm and in positive x - diection 5 Vm and in positive x - diection A D - -. The electic field at x = 5µm is given by B 8

29 . A battey is used to chage a paallel plate capacito till the potential diffeence between the plates becomes eual to the electomotive foce of the battey. The atio of the enegy stoed in the capacito and wok done by the battey will be () 4. Two spheical conductos A and B of adii mm and mm ae sepaated by a distance of 5mm and ae unifomly chaged. If the sphees ae connected by a conducting wie then in euilibium condition, the atio of the magnitude of the electic fields at the sufaces of sphee of A and B is : : () 4 : : 4. The following aangement consists of five identical metal plates maked,,, 4 and 5 paallel to each othe. Aea of each plate is A and sepaation between the successive plates is d. The capacitance between P and is A 5 d 7 A d 9 () 5 A d 4 A d 4. A paallel plate capacito of capacitance 5µ and plate sepaation 6 cm is connected to a V battey and chaged. A dielectic of dielectic constant 4 and thickness 4 cm is intoduced between the plates of the capacito. The additional chage that flows into the capacito fom the battey is µc 5µc () µc µc 5. o cicuit the euivalent capacitance between points P and is 6 4 () 6 6. ou identical capacitos ae connected in seies with a V battey as shown in the figue. Potentials at A and B ae V, V 5 V, 5V () 7.5 V,.5 V 7.5 V,.5 V identical dops of mecuy ae chaged simultaneously to the same potential of volt. Assuming the dops to be spheical, if all the chaged dops ae made to combine to fom one lage dop, then its potential will be V V () 64 V 6 V 8. Two metal plate fom a paallel plate capacito. The distance between the plates is d. A metal sheet of thickness d/ and of the same aea is intoduced between the plates. What is the atio of the capacitance in the two cases? 4 : : () : 5 : P 4 5 p

30 9. The cicula plates A and B of a paallel plate ai capacito have a diamete of. m and ae m apat. The plates and D of a simila capacito have a diamete of. m and ae m apat. Plate A is eathed. Plates B and D ae connected togethe. Plate is connected to the positive pole of a V battey whose negative is eathed, The enegy stoed in the system is.4µj.4µj ().44µJ.4µJ. Two paallel conducting plates of aea A =.5m each ae placed 6 mm apat and ae both eathed. A thid plate, identical with the fist two, is placed at a distance of mm fom one of the eathed plates and is given a chaged of. The potencial of the cental plate is mm 4mm V V () V 4 V. What is the enegy stoed in the capacito between teminals A and B of the netwok shown in the figue? (apacitance of each capacito = ) + v Zeo 5 µj ().5 µj 5 µj. Two identical capacitos and ae connected in seies to a battey as shown in figue. apacito contains a dielectic slab of constant K. and ae the chages stoed in and. Now, the dielectic slab is emoved and the coesponding chages ae and. Then () K + = K K + = K K = K + = K. A paallel plate capacito has plate of aea A and sepaation d. It is chaged to a potential diffeence V o. The chaging battey is disconnected and the plates ae pulled apat to thee times the initial sepaation. The wok euied to sepaate the plates is A V d A V d () A V d A V 4d 4. Two identical capacitos have the same capacitance. one of them is chaged to a potential V and the othe to V. The negative ends of the capacitos ae connected togethe. When the positive ends ae also connected, the decease in enegy of the combined system is c v v 4 c v + v 4 c v v 4 () + E k c v + v 4

31 5. A paallel plate ai capacito has a capacitance. When it is half filled with a dielectic of dielectic constant 5, the pecentage incease in the capacitance will be %. % () 4 % 66.6 % 6. A netwok of six identical capacitos, each of value is made as shown in figue. Euivalent capacitance between points A and B is 4 4 A () 4 B 7. The capacities of thee capacitos ae in the atio : :. Thei euivalent capacity when connected in paallel is 6 µ individual capacitos ae of capacities in moe then that when they ae connected in seies. The 4, 6, 7,, (),, 6,, 4 8. In the given aangement of capacitos euivalent capacitance between points M and N is 5 c 4 () 4 c 4 c 5 c 4 9. An electic cicuit euies a total capacitance of µ acoss a potencial of V. Lage numbe of µ capacitances ae available each of which would beakdown if the potential is moe then 5 V. How many capacitances ae euied to make the cicuit? 4 () 8 4. ead the assetion and eason caefully to mak the coect option out of the options given below : (a) (b) (c) (d) (e) If both assetion and eason ae tue and the eason is the coect explanation of the assetion. If both assetion and eason ae tue but eason is not the coect explanation of the assetion. If assetion is tue but eason is false. If the assetion and eason both ae false. If assetion is false but eason is tue. M N

32 . Assetion : The coulomb foce is the dominating foce in the univese. eason : The coulomb foce is weake than the gavitational foce.. Assetion : If thee capacitos of capacitance < < ae connected in paallel then thei euivalent capacitance p > eason : p. Assetion : A metalic shield in fom of a hollow shell may be built to block an electic field. eason : In a hollow shheical shield, the electic field inside it is zeo at evey point. 4. Assetion : Electons move away fom a low potential to high potential egion. eason : Because electons have negative chage 5. Assetion : If the distance between paallel plates of a capacito is halved and dielectic constant is made thee times, then the capacitance becomes 6 times. eason : apacity of the capacito does not depend upon the natue of the mateial. 6. Assetion : A paallel plate capacito is connected acoss battey though a key. A dielectic slab of constant K is intoduced between the plates. The enegy which is stoed becomes K times. eason : The suface density of chage on the plate emains constant o unchanged. 7. Assetion : Electic lines of foce coss each othe. eason : Electic field at a point supeimpose to give one esultant electic field. 8. Assetion : If a poton and an electon ae placed in the same unifom electic field. They expeience diffeent acceleation. eason : Electic foce on a test chage is independent of its mass. 9. Assetion : Dielectic beakdown occus unde the influence of an intense light beam. eason : Electomagnetic adiations exet pessue.. Assetion : When chages ae shaed between any two bodies, no chage is eally lost, but some loss of enegy does occu. eason : Some enegy disappeaes in the fom of heat, spaking etc.

33 KEY NOTES 7() () 9 8 4() () 99 5() () 4() () 84 6() () 5 7() 5() 69 85() 7 6() () () 7 87 () () 4() () () () 9() 6 8 () () 9 8() () () 77() () 5() () 7 6 () 48() () 4 : () 4 5 6() 7(E) 8 9

34 (9) Distance OA = ca sin 6h =. a. = a The foce on the hage at A due to those at B and have magnitude = The esultant of theses foce is cos = k a K K The foce on the chage at A due to hage at o is, = a Now fo euilibium of the hages, Now alculate. () Let and be the hages and the distance between them. Then So, 5 = b k. g and 5 4 k (. ) =.. m. Now, Substitating the Value of So obtained = 9 9 (.) Now find out and. () oce on - due to is = k along b axis k oce on - due to is at negative diection of yaxis. a x component of foce on -, is x = sin L NM = b i.e = a b O sin P a x sin 5 A a O B k a

35 () oce exeted by chages - at A and B on hage ae amd which ae eual and () K have a magnitude The esultant of these eual foces eually inclined with the X-axis is along the -X diection to wads the oigin. so, ' cos = ( cos ) since, is also Popotional to, Hence chage will move towads the oigin and because of its inetia it will oveshoot the oigin o. Thus chage will oscillate about o but its motion is not simple hamonic. OA O a K a K a K a 4 K The esultant of & is a and act along AP So, find. Now fo euilibium. 4 om this find out. (4) Net attaction foce on due to = epulsion foce due to A fom find out. (5) k k a y a y By symmety, the x components of foce will cancel each othe while along y axis will add up The esultant foce on + is k y cos. a y a y Now, oce on chage will be maximum, when d dy = y (y) k 5 ( a y ) ( a y ) y Now find out y. 5 a y a y c h c h 6

36 (6) T cos = mg and T sin = tan mg = x x x x d dx x dt dt V x d dt onstant (7) AB A AB A (along BA ) (along A ) cos6 Now find out. k k (8) oce on is ( a x) ( a x) ma = k L NM ( a x) ( a x) x a ma Now compae with a = x and find out feuenly. (9) Suppose the balls having chages and espectively. k Initially,... = It is given that 4. 5 Kb g = 4.5 b g = 4.5 Now solve it, and find. O P 7

37 () onside a small element of the od of length dx, at a distance x fom the point chage. The foce between and chage element will be, () d = kd x But d = L dx, d k L dx x L + L k d dx L x Now solve it. k. ( x. 8) k then find x x k () E p N ( P) p k E N ( ) in P, cos P = 6 p p E E E E sin tan E cos p cos 6 and find out E and. () om the fig, Net foce acting along the inclined plance...(i) = mg sin - E cos - f ma = mg sin - E cos So, ma = mg sin E cos - (mg cos + E sin) fom this, find out a. Now, d vot at use this euation and calculate t. (4) om the fig Tsin Tcos mg E E cos Now calculate. T mg o 8

38 (5) Let be the chage Pe unit length so, chage on Potion P... = Intensity at o is de = k d Total intensity E = k d de de and total cahge on the od, = Now find out E. (6) Hee E = and t l / vo Along y axis v o =, a = M ee / M So, d vot at Now Put the values (7) E l sinθ Elθ The moment of inetia of the od is I m H G l I K J mh G l I K J ml Now, = Iα α = I alculate and compale with E then ml So, π T = and od will become Paalled to E in a time T t = 4 Now calculate. K K (8) Afte onnection V V, = The atio of electic fields Now alculate K E E = = K 9

39 (9) onside a Spheical Shell of thickness dx and adius x. The aea of this Spheical shell = 4 x dx L NM O P x 4 x dx x 4 dx 4 4 The hage enclosed in a Sphee of adius is 4 x 4 4 x x = d = = 4 = Now find out E.. 4 L NM K K () E = and E a b Now, E 4 4 a tanθ = E b () Net acceleation g g O P Now, Put the values. ge m So, T l Now Put the Value of g '. g ' () Net electic field due to both chages will get cancelled. So. electic field of O is E = () In balance condition V 4 d E mg g 6 V Now find out (4) Use euation, E E E E cos6 (5),tan tan tan So, tan B A 4

40 (6) d at E x m v Ex V Now find out v md o. (7) Time of flight is t E Whee ax m Now, alculate ange. (8) P e o alculate P. voy Now ange Voxt ax t g tan tan (9) As tan, Shown in fig 9 tan( 9) tan tan cot tan tan tan ( ) 5to 5 (4) PE sin = ()E sin Now find out. ( a) E sin (4) Theoy elated uesition. (4) As the dop is stationay, weight of dop = foce due to electic field. 4 nee g = nee So, now find out. 4 g (4) Theoy elated uestion. (44) = enclosed enclosed f ( 8 4 ) Now find. (45) Applysing gauss law E.4 x v 4 x = E x (46) Applying gauss law (47) total = A B Assume = and = = b A So, = 4

41 (48) E. da E. l (49) E E l dv d vb va dv E dv b a d Now calculate. (5) maximum length of sting = l maximan enclosed chage = Now find out E. 4 dv d l (5) W = e (-4-) = -4ev = -4(-.6-9 ) J =.4-6 J (5) The electic Potential V ( x, y, z) Now Now E iˆ dv ˆj dv kˆ dv dx dy dz dv dx Hence E = E = = 8x, dv dy and dv dz 4x Volt 8xiˆ, So at Point (m,,m) 8iˆ Volt/mete o 8 along negative x-axis l (5) (c) Potential at the cente o, V 4 4 Whee = 9 and a = 8cm = 8 So = = 5 Volt (54) Obviously, fom chage configuation, at the cente electic field is non - zeo. Potential at the cente due to chage Potential due to hage V ( = distance of cente Point) V and Total Potential V V V V = m

42 (55) ABDE is an euipotential suface, on euipotential suface no wok is done in shifing a chge fom one place to anothe. (56) VA a b c ( ) HG a VB b c b e I KJ V = a b c c j on Putting = a +b V A = V V B dv (57) E x xy z dx dv Ey x dy dv Ez xz dz ˆ ˆ ˆ E ˆ ˆ E ˆ xi Ey j zk xy z i x j xz k (58) o Pai of hage U = 4 L NM = = 7J O P L M N + M 6 6 O P + L NM 6 6 O P (59) Potential at ente o of the suae Vo = 4 4 a Wok done in shifting (-) hage fom ente to infinity W V V V 4 a a (6) Let, and ae the hages on Sphee of adii and espectively Suface hage density ch ag e Aea Accoding to given Poblem, = 4 4 =...() 4

43 In ase of a chaged sphee, vs = V = = 4 HG I, V = 4 4 V V KJ...(using (i) ) = ( (6) Usystem = )( ) ( )( ) ( )( ) = 4 4 a 4 a (6) Wok done indisplacing chage of 5lic fom B to is W 5 9 ( V V ) whee c V B v V v b HG I K J 9 So W 5 J dv d E 5x x 9 x dx dx (6) E( x ) v m (64) Enegy density ue = E z 7 8 a v 8 86 H G I. K J =. 8 J / m (65) Potenti al insi de the Sphee w i l l be same as that on its Suf ace i.e. v = V suface o = volt Vout = 5 volt vout Vout V v (66) Length of each side of suae is m so distance of it s cente fom each conet is Potential at the cente V 9 L NM 5 V.8 V O P m 44

44 9 (67) W Uf U 9 L NM = W =. 96 J J E. Δ (68) By using W = O P L NM 4 ˆ ˆ ˆ. ˆ ˆ W e i e j e k ai bj e a e b 6 8 (69) K.E. = ( VA VB ). 6 ( 7 5) J (7) Net electostatic enegy K K K O P U a a a K a Now find out. (7) Point P will lie nea the hage which is smalle in magnitude i.e. -6lic Hence Potential at P. 6 6 ( 6 ) ( ) V x 4 (. x) x. m (7) alculate as M 67. (7) alculate as M 66. (74) () AB+A = cm...(i) = AB. A = cm = AB A ( AB A ) 4 AB. A = AB A 4 om euation (i) and (ii) AB = 8 cm; A = 4cm Potential enegy at Point A V AB A A V. 5 A J (75) Use the euation m V V d i = V 45

45 (76) Potential enegy of the system U k k k l l l K ( ) l dv (77) Ex ( 5) also find E dx y an E z E net = x y z E E E = ( 5) ( ) 5 7 (78) Van - d - glaph geneate pincipal L NM (79) U 4 U net =. 4 πε l O P (8) Length of the diagonal of a cube having each side b is b. So distance of cente b of cube fom each vetex is. Hence Potential enegy of the given system of chage is U = 8 ( )( ) πε b = 4 4 b (8) When a negatively chaged Pendylum oscillates ovea Positively chaged Plate then effective Value of 9 inceases so, accoding to d i T deceases g (8) When chage is eleased in unifom electic field E then its acceleation a = E (is constant) so it s motion will be unifomly acceleated motion and it s m Velocity afte time is given by E V at t = K = m mv (8) Electic Potential at P E t t m m V K K

46 (84) Potential at P due to (+) chage V. ( z a ) Potential at P due to (-) hage = V. 4 ( z a ) Total Potential at P due (AB) electic dipole V V V Now calculate (85) () Potential will be zeo at two Points... At intenal Point(M) = L NM 4 ( 6 l) c l 6 6 ho P P l So distance of M fom oigin; x 6 4 At exteiot Point (N) 6 6 ( ) l ' 4 ( 6 l) l' So distance of N fom oigin x 6 6 (86) Potential at the cente of ings ae V O k. k( ) d, k( ) k VO d V V k O O d d (87) hange in Potential enegy (u) = u f - u d Now calculate (88) Theoy base uestion (89) Potential at, V = Potential diffeence VD -V = W ( V V ) put the values D k L (9) V N V 4 N N 4 N 64 N 8 (9) A A A d d d A d Now find d. 47

47 (9) Hee s l and p 6 l Now in Paallel connection p So, 6 in seies connection 5 So, = b g = b g, o...() o Now, () Adding the uestion () and () we have b g+ b g6+8 a & (9) Suppose that the techinician makes a combination use of N apacitos and connects them in m ows, each ow having m apacitos Then N=mn apacitance of each capacito = l euied apacitance of the combination = l Voltage ating of each < apalito = 4V euied Voltage ating of the combination = V When capacitas ae connected in seies P.d. a coss thei Plates get added. fo n capacitos Voltage...to 4 nv 4n n. 5 o n 4 The total apucitae ' The total apacitanae of m ous mc ' O m ' (94) ommon Potential V V V v v V 4 v 4 v 4 4 (95) U V. J 4 Total chage on the two apacitos V V 6 V 5l. 5l 7. 5l 7. 5 V 9 44 V U V. 8J Now, enegy lost = U U find it. 48