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1 Pepaed by: M. S. KumaSwamy, TGT(Maths) Page - -

2 ELECTROSTATICS MARKS WEIGHTAGE 8 maks QUICK REVISION (Impotant Concepts & Fomulas) Chage Quantization: Chage is always in the fom of an integal multiple of electonic chage and neve its faction. q = ±ne whee n is an intege and e =.6 9 coulomb =.6 9 C. Chage on an electon/poton is the minimum chage. Chage on an electon is ve. e =.6 9 C. Chage on a poton is +ve. e = C. Total chage = ± ne. A paticle/body is positively chaged because it loses electons o it has shotage of electons. A paticle is negatively chaged because it gains electons o it has excess of electon. Consevation: The total net chage of an isolated physical system always emains constant. Chage can neithe be ceated no destoyed. It can be tansfeed fom one body to anothe. Coulomb's invese squae law Coulomb s law states that the foce of attaction o epulsion between two point chages is diectly popotional to the poduct of the chages and invesely popotional to the squae of the distance between them. The diection of foces is along the line joining the two point chages. Let q and q be two point chages placed in ai o vacuum at a distance apat (see above Figue). Then, accoding to Coulomb s law, qq F o F k. q q whee k is a constant of popotionality. In ai o vacuum, k 4 qq F. 4 whee F denotes the foce between two chages q and q sepaated by a distance in fee space. is a constant known as pemittivity of fee space. Fee space is vacuum and may be 9 Nm deemed to be ai pactically and 9 4 C One Coulomb is defined as the quantity of chage, which when placed at a distance of mete in ai o vacuum fom an equal and simila chage, expeiences a epulsive foce of 9 9 N. If fee space is eplaced by a medium, then is eplaced by ( K) o ( ) whee K is known as dielectic constant o elative pemittivity o specific inductive capacity (S.I.C.) o dielectic coefficient of the medium/mateial/matte. Thus qq qq qq F K 4 K o K = fo vacuum (o ai), K = fo conducto/metal. = 8.85 C N m. Pepaed by: M. S. KumaSwamy, TGT(Maths) Page - -

3 Vecto fom of the law (q and q ae like chages) qq qq (i) F qq qq (ii) F If is a unit vecto pointing fom q to q, then qq F. 3 = foce on q by q 4 When q q > fo like chages. If is a unit vecto pointing fom q to q, then qq F. 3 = foce on q by q 4 When q q > fo like chages. Intensity/stength of electic field Intensity at a point is numeically equal to the foce acting on a unit positive chage placed at the point. It is a vecto quantity. The units of intensity E ae NC, volt/mete. The dimensions of E ae [MLT 3 A ]. Intensity due to a chage q at distance q (i) E. 4 It acts in the diection in which a +ve chage moves. q (ii) E., if point is in the medium. 4 K Potential (V) and intensity (E) dv (i) E when potential vaies with espect to distance. d potential diffeence V (ii) E, when potential diffeence is constant. distance (iii) Potential at a point distance fom chage q. q V. 4 in fee space q (iv) V. 4 K in medium (v) Potential is a scala quantity E. d dv Fom positively chaged suface, E acts outwads at ight angles i.e. along outwad dawn nomal. Intensity is equal to flux (numbe of electic lines of foce) cossing unit nomal aea. flux( ) E aea( s) Pepaed by: M. S. KumaSwamy, TGT(Maths) Page - 3 -

4 Electic lines of foce Electic lines of foce stat fom positive chage and teminate on negative chage. Fom a positively chaged conducting suface lines of foce ae nomal to suface in outwad diection. Electic lines of foce about a negative point chage ae adial, inwads and about a positive point chage ae adial, outwads. Electic lines of foce ae always pependicula to an equipotential suface. These lines of foce contact along the length but expand at ight angles to thei length. Thee is longitudinal tension and lateal pessue in a line of foce. Contaction shows attaction between opposite chages while expansion indicates that simila chages epel. The numbe of electic lines of foce (flux) passing though unit nomal aea at any point indicates electic intensity at that point. Fo a chaged sphee these lines ae staight and diected along adius. These may be open o closed cuves. They ae not necessaily closed though the magnetic lines of foce ae closed. Two lines of foce neve intesect o cut each othe. Lines of foce ae paallel and equally spaced in a unifom field. Tangent to the cuve at a point shows diection of field. Gauss law Fo a closed suface enclosing a net chage q, the net electic flux emeging out is given by q E. d s S If a dipole is enclosed by a closed suface, flux is equal to zeo. Hee the algebaic sum of chages (+q q = ) is zeo. The flux will come out if +ve chage is enclosed. The flux will ente if negative chage is enclosed. Flux fom a cube (i) If q is at the cente of cube, total flux ( ) = (ii) Fom each face of cube, flux = q 6. q. Electic field due to a chaged shell Q (i) At an extenal point, E. 4 This is the same as the field due to a point chage placed at the cente. (ii) At a point on suface of shell, this is E max. Q E. 4 R Again the shell behaves like a point chage placed at cente. (iii) At an inside point ( < R), E =. Thus a chage q placed inside a chaged shell does not expeience any foce due to the shell. Gaussian suface (i) Fo a sphee o spheical shell a concentic sphee. (ii) Fo a cylinde o an infinite od a coaxial cylinde. (iii) Fo a plate a cube o a cuboid. the Pepaed by: M. S. KumaSwamy, TGT(Maths) Page - 4 -

5 Potential and intensity due to a chaged conducting sphee (o shell) At a point outside the chaged sphee q (i) Intensity, E. 4 ( > adius of sphee R) It is a vecto quantity. q (ii) Potential, V. 4 It is a scala quantity. At a point on the suface of chaged sphee q (i) Intensity, E. 4 R It is a vecto quantity. q (ii) Potential, V. 4 R It is a scala quantity. ( = adius of sphee R) At a point inside the sphee ( < adius of sphee) (i) Intensity E = zeo. q (ii) Potential, V. 4 R Potential is constant inside the sphee. This is same as potential at the suface of sphee. At the cente of sphee (i) Intensity E = zeo. q (ii) Potential, V. 4 R At infinity (i) Intensity E = zeo. (ii) Potential V = zeo. Electic field and potential due to chaged nonconducting sphee Outside the sphee when > adius of sphee R q (i) Electic intensity, E. 4 It is a vecto quantity. q (ii) Electic potential, V. 4 It is a scala quantity. On the suface of the sphee whee = R q (i) Electic intensity, E. 4 R It is a vecto quantity. q (ii) Electic potential, V. 4 R It is a scala quantity. Inside the sphee when < R q (i) Electic intensity, E. 3 4 R Pepaed by: M. S. KumaSwamy, TGT(Maths) Page - 5 -

6 q Vectoially, E. 3 4 R q(3 R ) (ii) Electic potential, V. 3 4 R At the cente of sphee when = (i) Electic intensity E = zeo. 3 q (ii) Electic potential, V.. 4 R Potential at cente = 3 potential at suface At infinity (i) Intensity = zeo. (ii) Potential = zeo. Electic dipole Two equal and opposite chages (q) each, sepaated by a small distance (l) constitute an electic dipole. Many of the atoms/molecules ae dipoles. (i) Dipole moment, p q ( l) (ii) Dipole moment is a vecto quantity. (iii) The diection of p is fom negative chage to positive chage. (iv) Unit of dipole moment = coulombmete = Cm. (v) Dimension of dipole moment = [ATL]. Intensity of electic field due to a dipole (i) Along axis at distance fom cente of dipole p E. 3 4 Diection of E is along the diection of dipole moment. (ii) Along equato of dipole at distance fom cente p E. Diection of E is anti-paallel to diection of p. 3 4 (iii) At any point along diection q p E 3cos. 3 4 The diection of E makes an angle with the line joining the point with cente of dipole whee tan = tan. Toque on a dipole Two foces [qe and ( qe)] equal, opposite and paallel, sepaated by a distance constitute a couple. Pepaed by: M. S. KumaSwamy, TGT(Maths) Page - 6 -

7 toque ( ) = p E pe sin This diection of is pependicula to the plane containing p and E. The toque tends to align the dipole in the diection of field. When dipole is paallel to electic field, it is in stable equilibium. When it is antipaallel to electic field, it is in unstable equilibium. Toque is maximum when = 9. Dipole is pependicula to E. Theefoe maximum toque = pe. Potential enegy of dipole in unifom electic field Wokdone in otating the dipole fom an angle to angle. W d pe d pe pe (i) If = and = 8, W = pe. (ii) If = and = 9, W = pe. sin cos (cos cos ) Potential enegy of dipole, when it is tuned though an angle q fom field diection is U = pecos = - p E (i) If =, U = pe. The dipole oients itself paallel to field. (ii) If = 9, U =. (iii) If = 8, U = pe. Vaiation of potential enegy of dipole with angle, between E and p, is shown in the figue. (i) Potential enegy is negative fom to / and 3 / to. They ae egions of stable equilibium of dipole. (ii) Potential enegy is positive fom / to 3 /. This is the egion of unstable equilibium of the dipole. Dipole in non-unifom electic field In non-unifom electic field, the two ends of dipole ae acted upon by foces qe and qe. They ae not equal as E E in non-unifom field. Hence a foce and a toque both act on the dipole. d E Foce acting on the dipole can be epesented by F p d Boadly speaking, Net foce = (qe qe ) along diection of geate field intensity. On account of net foce upon dipole, it may undego linea motion. In a non-unifom electic field, a dipole may, theefoe, undego otation as well as linea motion. Pepaed by: M. S. KumaSwamy, TGT(Maths) Page - 7 -

8 Potential enegy of chage system Fo two point chages q and q sepaated by a distance, electostatic potential enegy U is given by qq U 4 qq qq3 q3q Fo thee point chages, U 4 3 Fo n chages, conside all pais with due egad of signs of chages, positive o negative. S.I. unit of enegy = joule (J) Anothe popula unit is electon volt (ev). ev =.6 9 joule. Chaged soap bubble Fo equilibium of a chaged soap bubble, pessue due to suface tension = 4T acting inwads. Electic pessue due to chaging = At equilibium, 4T acting outwads. whee s = suface density of chage 4T q q 8 T 4 Hee ai pessues, inside and outside the bubble, ae supposed to be same. Behaviou of a conducto in an electostatic field In the case of a chaged conducto (i) Chage esides only on the oute suface of conducto. (ii) Electic field at any point inside the conducto is zeo. (iii) Electic potential at any point inside the conducto is constant and equal to potential on the suface of the conducto, whateve be the shape and size of the conducto. (iv) Electic field at any point on the suface of chaged conducto is diectly popotional to the suface density of chage at that point, but electic potential does not depend upon the suface density of chage. Capacitance When a conducto is given a chage, its potential gets aised. The quantity of chage given to a conducto is found to be diectly popotional to the potential aised by it. If q is the chage given to conducto and V is potential aised due to it, then q V o q = CV, whee C is a constant, known as capacitance of the conducto. Capacitance = chage potential. Pepaed by: M. S. KumaSwamy, TGT(Maths) Page - 8 -

9 Unit of capacitance is faad. F = coulomb/volt. Faad = 9 x stat faad. Dimensions of capacitance ae [M L T 4 A ]. Capacity of an isolated spheical conducto : Capacitance of an isolated spheical conducto of adius a placed in a medium of dielectic constant K, C 4 Ka faad Fo vacuum o ai, K =, hence C 4 a faad. i.e., capacitance of a spheical conducto µ adius. Capacito is a pai of two conductos of any shape which ae close to each othe and have equal and opposite chages. A capacito is an aangement which can stoe sufficient quantity of chage. The quantity of chage that can be given to a capacito is limited by the fact that evey dielectic medium becomes conducting at a cetain value of electic field. Capacitance of a capacito is (i) Diectly popotional to the aea of the plates (A). (ii) Invesely popotional to distance between plates (d). (iii) Diectly popotional to dielectic constant of the medium filled between its plates (K). Paallel plate capacito : Capacitance of a paallel plate capacito filled completely with some dielectic medium. K A C d Fo ai and vacuum, K =. A C d Capacitance of a paallel plate capacito filled with dielectic slab of thickness t is given by A C d t K Capacitance of a paallel plate capacito filled with a conducting slab of thickness t is given by A C ( d t) The plates of a paallel plate capacito attact each othe with a foce ab Capacitance of a spheical condense/capacito, is C 4 K b a when a and b ae the adii of inne and oute sphees espectively. Q F A Dielectics ae of two types : Nonpola and pola. The nonpola dielectics (like N, O, benzene, methane) etc. ae made up of nonpola atoms / molecules, in which the cente of mass of negative coincides with the cente of mass of negative chage of the atom / molecule. Pepaed by: M. S. KumaSwamy, TGT(Maths) Page - 9 -

10 The pola dielectics (like H O, CO, NH 3, HCl) etc. ae made up of pola atoms / molecules, in which the cente of mass of positive chage does not coincide with the cente of mass of negative chage of the atom / molecule. A non-pola dielectic can be polaized by applying an extenal electic field on the dielectic. The effective electic field E in a polaised dielectic is given by E E E p whee E is stength of extenal field applied and E p is intensity of induced electic field set up due to polaization. It is equal to suface density of induced chage. The atio E / E = K, dielectic constant. When a dielectic slab is placed between the plates of a paallel plate capacito, the chage induced on its sides due to polaization of dielectic is ( K ) qi q K Capacitos in seies: Equivalent capacitance of a seies combination of capacitos is... C C C C3 In seies combination of capacitos, chage is same on each capacito and is equal to chage supplied by souce CV = C V = C V =... Capacitos in paallel : Equivalent capacitance of a paallel combination of capacitos is Cp = C + C + C In paallel combination of capacitos, potential diffeence is same acoss each capacito and is equal to q q q q3 applied potential diffeence... C C C C3 Electic potential enegy stoed in a chaged conducto o capacito is q U CV qv C The electic potential enegy of capacito esides in the dielectic medium between the plates of the condense. When two chaged conductos ae connected togethe, the edistibuted chages on them ae in the atio of thei capacitance. When two chaged conductos having chages q and q and capacitances C and C ae connected q q CV CV togethe, then afte edistibution of chages, the common potential is V whee C C C C V and V ae the initial potentials of the chaged conductos. In case of chaged capacitos, when plates of same polaity ae connected togethe, common potential CV CV V C C CV CV But when plates of opposite polaity ae connected togethe, then common potential is V C C Total enegy stoed in any gouping of capacitos is equal to sum of the enegies stoed in individual capacitos. Pepaed by: M. S. KumaSwamy, TGT(Maths) Page - -

11 If n chaged dops, each of capacity C, chaged to potential V with chage q, suface density and potential enegy U coalesce to fom a single dop, then fo such a dop, total chage = nq total capacity = n /3 C potential = n /3 V Suface density of chage = n /3, and total potential enegy = n /3 U. Shaing of chages (i) Common potential : When two capacitos at diffeent potentials V and V ae connected, chaged q (= C V ) and q (= C V ) ae edistibuted till a common potential V is eached. Then total chage q q CV CV V total capacity C C C C (ii) Loss of enegy : Duing shaing of chages, enegy is lost; mostly as heat, patly as cacking noise and patly as spaking light. CC ( V V ) Loss of enegy = C C Pepaed by: M. S. KumaSwamy, TGT(Maths) Page - -

12 ELECTROSTATICS MARKS WEIGHTAGE 5 maks Impotant Questions & Answes VERY SHORT ANSWER TYPE QUESTIONS ( MARK). Why must electostatic field at the suface of a chaged conducto be nomal to the suface at evey point? Give eason. Ans. The wok done in moving a chage fom one point to anothe on an equipotential suface is zeo. If electic field is not nomal, it will have a non-zeo component along the suface which would cause wok to be done in moving a chage on an equipotential suface.. Figue shows the field lines due to a positive point chage. Give the sign of potential enegy diffeence of a small negative chage between the points Q and P. qq Ans. U. 4 ' Hee, UQ U P Theefoe, UQ U P is negative 3. Why do the electostatic field lines not fom closed loops? Ans. The electostatic field lines stat fom positive chage and end on negative chage. 4. Why do the electic field lines neve coss each othe? Ans. If the field lines coss each othe, then at the point of intesection, thee will be two diections fo the same electic field which is not possible. 5. Figue shows the field lines on a positive chage. Is the wok done by the field in moving a small positive chage fom Q to P positive o negative? Give eason. Pepaed by: M. S. KumaSwamy, TGT(Maths) Page - -

13 Ans. The wok done by the field is negative. This is because the chage is moved against the foce exeted by the field. 6. At what position is the electic dipole in unifom electic field in its most stable equilibium position? [AI 8] Ans. When = between P and E 7. If the adius of the Gaussian suface enclosing a chage is halved, how does the electic flux though the Gaussian suface change? [AI 8] Ans. The electic flux emains the same, as the chage enclosed emains the same. 8. Define the tem electic dipole moment of a dipole. State its S.I. unit. [AI 8] Ans. Stength of an electic dipole is measued by its electic dipole moment, whose magnitude is equal to poduct of magnitude of eithe chage and sepaation between the two chages i.e., p q.a and is diected fom negative to positive chage, along the line joining the two chages. Its SI unit is Cm. 9. A chage q is placed at the cente of a cube of side l. What is the electic flux passing though each face of the cube? [AI ] q Ans. Flux though whole of the cube, q Flux though each face of the cube, ' 6 6. Two chages of magnitudes Q and +Q ae located at points (a, ) and (4a, ) espectively. What is the electic flux due to these chages though a sphee of adius 3a with its cente at the oigin? [AI 3] qinside Q Ans. Electic flux. What is the electostatic potential due to an electic dipole at an equatoial point? [AI 9] Ans. Zeo. Name the physical quantity whose S.I. unit is J C. Is it a scala o a vecto quantity? [AI ] Ans. J C is the S.I. unit of electostatic potential. It is a scala quantity. 3. What is the value of the angle and between the vectos p and E fo which the potential enegy of an electic dipole of dipole moment p, kept in an extenal electic field E, has the maximum value. Pepaed by: M. S. KumaSwamy, TGT(Maths) Page - 3 -

14 Ans. Potential enegy = p. E pe cos Theefoe, Potential enegy is the maximum when cos i. e. o 8 4. A point chage Q is placed at point O as shown in the figue. Is the potential diffeence V A V B positive, negative o zeo, if Q is (i) positive (ii) negative? [AI ] Ans. (i) If Q is positive, V A V B is positive. (ii) If Q is negative, V A V B is negative. SHORT ANSWER TYPE QUESTIONS ( MARKS/3 MARKS) 5. Define electic flux. Wite its S.I. unit. A chage q is enclosed by a spheical suface of adius R. If the adius is educed to half, how would the electic flux though the suface change? [AI 9] Ans. Electic flux linked with a suface is the numbe of electic lines of foce cutting though the suface nomally. It s SI unit is Nm C o Vm on deceasing the adius of spheical suface to half thee will be no effect on the electic flux. 6. A positive point chage (+q) is kept in the vicinity of an unchaged conducting plate. Sketch electic field lines oiginating fom the point on to the suface of the plate. Deive the expession fo the electic field at the suface of a chaged conducto. [AI 9] Ans. Let us conside an infinite plane sheet of chage of unifom chage density whee q is chage in aea A on sheet of chage. q A Let P be any point on the one side of sheet and P on the othe side of sheet, at same distance fom it. We daw a Gaussian cylindical suface S of coss section aea A cutting though the plane sheet of chage, such that points P and P lie on its plane faces. Then electic flux linked with cylindical suface S is Pepaed by: M. S. KumaSwamy, TGT(Maths) Page - 4 -

15 This gives the electic field intensity at any point nea o on the suface of the infinite thin plane sheet of chage. 7. A paallel plate capacito is chaged by a battey. Afte some time the battey is disconnected and a dielectic slab of dielectic constant K is inseted between the plates. How would (i) the capacitance, (ii) the electic field between the plates and (iii) the enegy stoed in the capacito, be affected? Justify you answe. [AI 9] Ans. (i) On filling the dielectic constant of K in the space between the plates, capacitance of paallel plate capacito becomes K times i.e. C = KC (ii) As the battey was disconnected, so the chage on the capacito emains the same i.e. Q = Q. So, E Q E the electic field in the space between the plates becomes E o E KA KA K i.e. electic field becomes K times. Q Q (iii) Enegy stoed in capacito becomes U o U U C KC i.e. becomes K times. 8. A spheical conducting shell of inne adius and oute adius has a chage Q. A chage q is placed at the cente of the shell. (a) What is the suface chage density on the (i) inne suface, (ii) oute suface of the shell? (b) Wite the expession fo the electic field at a point x > fom the cente of the shell. [AI ] q Ans. (a) (i) Suface chage density on the inne suface of shell is in 4 Q q 4 (ii) Suface chage density on the oute suface of shell is out Pepaed by: M. S. KumaSwamy, TGT(Maths) Page - 5 -

16 Q q (b) Using, Gauss s law, E( x). 4 x 9. Explain the meaning of the statement electic chage of a body is quantized. Ans. The electic chage of a body is quantized means that the chage on a body can occu in some paticula values only. Chage on any body is the integal multiple of chage on an electon because the chage of an electon is the elementay chage in natue. The chage on any body can be expessed by the fomula q ne, whee, n = numbe of electons tansfeed and e = chage on one electon. The cause of quantization is that only integal numbe of electons can be tansfeed fom one body to othe.. Why can one ignoe quantization of electic chage when dealing with macoscopic, i.e., lage scale chages? Ans. We can ignoe the quantization of electic chage when dealing with macoscopic chages because the chage on one electon is.6 x 9 C in magnitude, which is vey small as compaed to the lage scale change.. An electostatic field line is a continuous cuve. That is, a field line cannot have sudden beaks. Why not? Ans. An electostatic field line epesents the actual path tavelled by a unit positive chage in an electic field. If the line have sudden beaks it means the unit positive test chage jumps fom one place to anothe which is not possible. It also means that electic field becomes zeo suddenly at the beaks which is not possible. So, the field line cannot have any sudden beaks.. Explain why two field lines neve coss each othe at any point? Ans. If two field lines coss each othe, then we can daw two tangents at the point of intesection which indicates that (as tangent dawn at any point on electic line of foce gives the diection of electic field at that point) thee ae two diections of electic field at a paticula point, which is not possible at the same instant. Thus, two field lines neve coss each othe at any point. 3. Show that the electic field at the suface of a chaged conducto is given by E n whee is the suface chage density and is a unit vecto nomal to the suface in the outwad diection. [AI ] Ans. Conside an elementay aea S on the suface of the chaged conducto. Enclose this aea element with a cylindical gaussian suface as shown in figue. Now electic field inside a chaged conducto is zeo. Theefoe, diection of field, just out side S will be nomally outwad i.e. in diection of n. Accoding to Gauss s theoem, total electic flux coming out is S E. S [ E is electic field at the suface] Pepaed by: M. S. KumaSwamy, TGT(Maths) Page - 6 -

17 E S cos E S 4. Using Gauss s law obtain the expession fo the electic field due to a unifomly chaged thin spheical shell of adius R at a point outside the shell. Daw a gaph showing the vaiation of electic field with, fo > R and < R. [AI ] Ans. Conside a thin spheical shell of adius R caying chage Q. To find the electic field outside the shell, we conside a spheical Gaussian suface of adius (> R), concentic with given shell. The electic field E is same at evey point of Gaussian suface and diected adially outwads (as is unit vecto n so that q = ) Accoding to Gauss s theoem, Q o E ds s Q E(4 ) E 4 Q s.. Q E d s E nd s s Hence, electic field outside a chaged thin spheical shell is the same as if the whole chage Q is concentated at the cente. The vaiation of electic field E with distance fom cente of a unifomly chaged spheical shell is shown in figue. 5. Daw 3 equipotential sufaces coesponding to a field that unifomly inceases in magnitude but emains constant along Z diection. How ae these sufaces diffeent fom that of a constant electic field along Z diection? [AI 9] Ans. Fo constant electic field, equipotential sufaces ae equidistant fo same potential diffeence between these sufaces. Fo inceasing electic field, sepaation between equipotential sufaces deceases, in the diection of inceasing field, fo the same potential diffeence between them. Pepaed by: M. S. KumaSwamy, TGT(Maths) Page - 7 -

18 6. A netwok of fou capacitos each of F capacitance is connected to a 5 V supply as shown in the figue. Detemine (a) equivalent capacitance of the netwok and (b) chage on each capacito. Ans. Hee C, C and C 3 ae in seies, hence thei equivalent capacitance is C given by C ' C C C3 C ' F C ' 4F 3 The cicuit can be edawn as shown, above. Since C and C 4 ae in paallel C C ' C 4F F 6F net 4 Pepaed by: M. S. KumaSwamy, TGT(Maths) Page - 8 -

19 (b) Since C and C 4 ae in paallel, potential diffeence acoss both of them is 5 V. Chage acoss C 4 is Q 4 = C 4 5 C = 6 5 C = 6 mc and Chage acoss C, Q = C 5 C = C = mc C, C, C 3 ae in seies, chage acoss them is same, which is Q = mc 7. Figue shows two identical capacitos C and C, each of F capacitance connected to a battey of 6 V. Initially switch S is closed. Afte sometime S is left open and dielectic slabs of dielectic constant K = 3 ae inseted to fill completely the space between the plates of the two capacitos. How will the (i) chage and (ii) potential diffeence between the plates of the capacitos be affected afte the slabs ae inseted? Ans. When the switch S is closed, the two capacitos in paallel will be chaged by the same potential diffeence V. So, chage on capacito C q = C V = 6 = 6 mc and chage on capacito C q = C V = 6 = 6 mc q = q + q = = mc. When switch S is opened and dielectic is intoduced. Then Capacity of both the capacitos becomes K times i.e., C = C = KC = 3 = 3 mf Capacito A emains connected to battey V = V = 6 V q = Kq = 3 x 6 mc = 8 m C Capacito B becomes isolated q = q o C V = C V o (KC)V = CV V 6 V ' V K 3 8. A test chage q is moved without acceleation fom A to C along the path fom A to B and then fom B to C in electic field E as shown in the figue. (i) Calculate the potential diffeence between A and C. (ii) At which point (of the two) is the electic potential moe and why? [AI ] Pepaed by: M. S. KumaSwamy, TGT(Maths) Page - 9 -

20 Ans. In the elation dv VC VA E E d ( 6) (i)v C V A = 4E (ii) Hence V C > V A Also electic field is diected fom points of high potential to low potential. 9. Deduce the expession fo the electostatic enegy stoed in a capacito of capacitance C and having chage Q. How will the (i) enegy stoed and (ii) the electic field inside the capacito be affected when it is completely filled with a dielectic mateial of dielectic constant K? [AI ] q Ans. Potential diffeence between the plates of capacito V C Wok done to add additional chage dq on the capacito dw = V dq = q C dq Total enegy stoed in the capacito Q q Q U dw dq C When battey is disconnected (i) Enegy stoed will be deceased o enegy stoed = times the initial enegy. K (ii) Electic field would decease E o E ' K Altenatively, if a student attempts to answe by keeping the battey connected, then (i) enegy stoed will incease o become K times the initial enegy. (ii) electic field will not change. 3. Daw the equipotential sufaces due to an electic dipole. Locate the points whee the potential due to the dipole is zeo. Ans. Pepaed by: M. S. KumaSwamy, TGT(Maths) Page - -

21 Altenatively Any point on the equatoial plane (AB) of the dipole. 3. A slab of mateial of dielectic constant K has the same aea as that of the plates of a paallel plate capacito but has the thickness d/, whee d is the sepaation between the plates. Find out the expession fo its capacitance when the slab is inseted between the plates of the capacito. [AI 3] Ans. Capacitance of a capacito patially filled with a dielectic A C t d t K A AK C d d d d( K ) K 3. A capacito, made of two paallel plates each of plate aea A and sepaation d, is being chaged by an extenal ac souce. Show that the displacement cuent inside the capacito is the same as the cuent chaging the capacito. [AI 3] Ans. The displacement cuent within capacito plates de Id dt q q whee E EA A A dq Id Id I dt 33. A point chage (+Q) is kept in the vicinity of unchaged conducting plate. Sketch electic field lines between the chage and the plate. Ans. Pepaed by: M. S. KumaSwamy, TGT(Maths) Page - -

22 The lines of foce stat fom + Q and teminate at metal place inducing negative chage on it. The lines of foce will be pependicula to the metal suface. 34. Deive the expession fo the electic field of a dipole at a point on the equatoial plane of the dipole. Ans. It is the poduct of magnitude of eithe chage and the distance between the two equal and opposite chages. Altenatively, p qaa It is a vecto quantity. E E cos E cos E cos E q a. 4 ( a ) ( a ) / qa qa E 4 ( a ) 4 ( a ) p E whee p qa 3/ 4 ( a ) 3/ 3/ 35. Using Gauss law deduce the expession fo the electic field due to a unifomly chaged spheical conducting shell of adius R at a point (i) outside and (ii) inside the shell. Plot a gaph showing vaiation of electic field as a function of > R and < R. ( being the distance fom the cente of the shell) Ans. By Gauss Law, > R (outside) q E. d S q E d S q E4 E Similaly, < R (inside) E q d S As inside the shell q = E4 R As R, E 4 q Pepaed by: M. S. KumaSwamy, TGT(Maths) Page - -

23 36. Two infinitely lage plane thin paallel sheets having suface chage densities and ( > ) ae shown in the figue. Wite the magnitudes and diections of the net fields in the egions maked II and III. Ans. (i) Net electic field in egion II = ( ) Diection of electic field in fom sheet A to sheet B. (ii) Net electic field in egion III = ( ) Diection is away fom the two sheets i.e. towads ight side. 37. In a paallel plate capacito with ai between the plates, each plate has an aea of 6 3 m and the sepaation between the plates is 3 mm. (i) Calculate the capacitance of the capacito. (ii) If this capacito is connected to V supply, what would be the chage on each plate? (iii) How would chage on the plates be affected, if a 3 mm thick mica sheet of K = 6 is inseted between the plates while the voltage supply emains connected? Ans. Hee, A = 6 3 m, d = 3 mm = 3 3 m 3 A (i) Capacitance, C 7.7 F 3 d 3 (ii) Chage, Q = CV 7.7 C 7.7 C (iii) New chage, Q = KQ = 67.7 C 6. C 38. In a paallel plate capacito with ai between the plates, each plate has an aea of 5 3 m and the sepaation between the plates is.5 mm. (i) Calculate the capacitance of the capacito. (ii) If this capacito is connected to V supply, what would be the chage on each plate? (iii) How would chage on the plates be affected, if a.5 mm thick mica sheet of K = 8 is inseted between the plates while the voltage supply emains connected? Ans. Hee, A = 5 3 m, d =.5 mm =.5 3 m 3 A (i) Capacitance, C 7.7 F 3 d.5 (ii) Chage, Q = CV 7.7 C 7.7 (iii) New chage, Q = KQ = 87.7 C 4.6 Pepaed by: M. S. KumaSwamy, TGT(Maths) Page C C

24 39. Two equal balls having equal positive chage q coulombs ae suspended by two insulating stings of equal length. What would be the effect on the foce when a plastic sheet is inseted between the two? Ans. Foce will decease. Reason: Foce between two chages each q in vacuum is q F. 4 On inseting a plastic sheet (a dielectic K > ) q Then F. 4 K i.e. Foce F F K The foce between chaged balls will decease. 4. A paallel plate capacito of capacitance C is chaged to a potential V. It is then connected to anothe unchaged capacito having the same capacitance. Find out the atio of the enegy stoed in the combined system to that stoed initially in the single capacito. Ans. The chage on the capacito q = CV and initial enegy stoed in the capacito q U CV (i) C (a) If anothe unchaged capacito is connected in seies then the same amount of the chage will tansfe as shown in figue. Keeping chage constant, and final voltage v v q q q U f C C C q q U f : U i : : C C 4. Deduce the expession fo the toque acting on a dipole of dipole moment p in the pesence of a unifom electic field E. Ans. Expession fo toque An electic dipole having chages ± q, and of size a is placed in unifom electic field E as shown in figue. The foces, acting on the chages ae +q E and -q E. The net foce on the dipole is F = +q E + (-q E ) = Both foces povides an equivalent toque with magnitude t = qe x Pependicula distance (AC) Pepaed by: M. S. KumaSwamy, TGT(Maths) Page - 4 -

25 = q E.a sin = P E sin The diection of the toque can be given by p E 4. Conside two hollow concentic sphees, S and S, enclosing chages Q and 4Q espectively as shown in the figue. (i) Find out the atio of the electic flux though them. (ii) How will the electic flux though the sphee S change if a medium of dielectic constant is intoduced in the space inside S in place of ai? Deduce the necessay expession. q( T) Ans. Using Gauss s Theoem E. ds ( q) Electic flux though sphee S = (Q 4 Q) 6Q Electic flux though sphee S = Q Ratio = 6Q 3 If a medium of dielectic constant K(= ) is filled in the sphee S, electic flux though Q Q sphee, ' K 43. Fo any chage configuation, equipotential suface though a point is nomal to the electic field. Justify. Ans. The wok done in moving a chage fom one point to anothe on an equipotential suface is zeo. If electic field is not nomal to the equipotential suface, it would have non-zeo component along the suface. In that case wok would be done in moving a chage on an equipotential suface. 44. An electic dipole of length 4 cm, when placed with its axis making an angle of 6 with a unifom electic field, expeiences a toque of 4 3 Nm. Calculate the potential enegy of the dipole, if it has chage ±8 nc. Ans. Toque, t = pe sin q 4 3 = pe sin pe pe 8 Pepaed by: M. S. KumaSwamy, TGT(Maths) Page - 5 -

26 Now, potential enegy, U = -pe cos = 8 cos 6 = -8 = -4 J 45. Given a unifom electic field E = 5 3 i N/C, find the flux of this field though a squae of cm on a side whose plane is paallel to the y-z plane. What would be the flux though the same squae if the plane makes a 3 angle with the x-axis? Ans. Hee, E = 5 3 i N/C, i.e. field is along positive diection of x-axis. Suface aea, A = cm cm =. m. m = m- (i) When plane paallel to y - z plane, the nomal to plane is along x axis. Hence = = EA cos = 5 x 3 x - cos = 5NC m (ii) When the plane makes a 3 angle with the x-axis, the nomal to its plane makes 6 angle with x-axis. Hence = 6 = EA cos = 5 x 3 x - cos 6 = 5 NC m 46. The electic field inside a paallel plate capacito is E. Find the amount of wok done in moving a chage q ove a closed ectangula loop abcda. Ans. Wok done in moving a chage q fom a to b = Wok done in moving a chage q fom c to d = This is because the electic field is pependicula to the displacement. Now, wok done fom b to c = wok done fom d to a Theefoe, total wok done in moving a chage q ove a closed loop =. 47. Obtain the expession fo the enegy stoed pe unit volume in a chaged paallel plate capacito. Ans. When a capacito is chaged by a battey, wok is done by the chaging battey at the expense of its chemical enegy. This wok is stoed in the capacito in the fom of electostatic potential enegy. Conside a capacito of capacitance C. Initial chage on capacito is zeo. Initial potential diffeence between capacito plates = zeo. Let a chage Q be given to it in small steps. When chage is given to capacito, the potential diffeence between its plates inceases. Let at any instant when chage on q capacito be q, the potential diffeence between its plates V C Now wok done in giving an additional infinitesimal chage dq to capacito q dw Vdq V dq C The total wok done in giving chage fom to Q will be equal to the sum of all such infinitesimal woks, which may be obtained by integation. Theefoe total wok Pepaed by: M. S. KumaSwamy, TGT(Maths) Page - 6 -

27 q q Q Q W Vdq dq C C C C If V is the final potential diffeence between capacito plates, then Q =CV ( CV ) W CV QV C This wok is stoed as electostatic potential enegy of capacito i.e., Q Electostatic potential enegy, U CV QV C Enegy density: Conside a paallel plate capacito consisting of plates, each of aea A, sepaated by a distance d. If space between the plates is filled with a medium of dielectic constant K, then Q Q Q K A Capacitance of capacito, C d If is the suface chage density of plates, then electic field stength between the plates E K E K Chage on each plate of capacito Q = A = K EA Q ( K EA) Enegy stoed by capacito, U K E Ad C K A d But Ad = volume of space between capacito plates Enegy stoed, U K E Ad U Electostatic Enegy stoed pe unit volume, ue K E Ad This is expession fo electostatic enegy density in medium of dielectic constant K. In ai o fee space (K = ), theefoe enegy density, ue E 48. Two chaged spheical conductos of adii R and R when connected by a conducting wie acquie chages q and q espectively. Find the atio of thei suface chage densities in tems of thei adii. Ans. When two chaged spheical conductos ae connected by a conducting wie, they acquie the same potential. kq kq R R q q q R R R q R Hence, the atio of suface chage densities q 4 R qr q qr 4 R 49. Deive the expession fo the capacitance of a paallel plate capacito having plate aea A and plate sepaation d. Ans. Pepaed by: M. S. KumaSwamy, TGT(Maths) Page - 7 -

28 In the egion between the plates the net electic field is equal to the sum of the electic fields due to the two chaged plates. Thus, the net electic field is given by E The electic field is constant in the egion between the plates. Theefoe, the potential diffeence between the plates will be d V Ed Q Now, capacitance C Q V d Suface chage density Q A A C Qd d Q A, whee A is the aea of coss-section of the plates. R R R R R R 5. Deive an expession fo the enegy stoed in a paallel plate capacito. On chaging a paallel plate capacito to a potential V, the spacing between the plates is halved, and a dielectic medium of E = is intoduced between the plates, without disconnecting the d.c. souce. Explain, using suitable expessions, how the (i) capacitance, (ii) electic field and (iii) enegy density of the capacito change. Ans. k A k A (i) C kc C d d C C (ii) As battey emains connected so, potential diffeence V emains same acoss the capacito. Pepaed by: M. S. KumaSwamy, TGT(Maths) Page - 8 -

29 V V E d E E d (iii) Initial enegy density = E Final enegy density = E ( E) 4 E o Final enegy density = 4 Initial enegy density. 5. Deive the expession fo the electic potential at any point along the axial line of an electic dipole? Ans. Electic Potential due to an electic dipole at axial point. Conside an electic dipole AB, having chages q and + q at points A and B espectively. The sepaation between the chages is l. Electic dipole moment, p q.l, diected fom q to + q. Conside a point P on the axis of dipole at a distance fom mid-point O of dipole. The distance of point P fom chage + q is BP = l The distance of point P fom chage - q is AP = + l Let V and V be the potentials at P due to chages + q and - q espectively. Then q q V and V 4 ( l) 4 ( l) Resultant potential at P due to dipole q q V V V 4 ( l) 4 ( l) q q ( l) ( l) V 4 ( l) ( l) 4 ( l)( l) q.l V 4 ( l ) As q. l = p (dipole moment) p V 4 ( l ) If point P is fa away fom the dipole, then > >l p V 4 Pepaed by: M. S. KumaSwamy, TGT(Maths) Page - 9 -

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