1... Multiple Choice uestions with One Correct Choice A hollow metal sphere of raius 5 cm is charge such that the potential on its surface to 1 V. The potential at the centre of the sphere is (a) zero 1 V the same as that at a point 5 cm away from the surface () the same as that at a point 5 cm away from the surface Two equal negative charges - q are fixe at points (, a) an (, - a) on the y-axis, A positive charge is release from rest at a point (a, ) on the x-axis. The charge will (a) execute simple harmonic motion about the origin move to the origin an remain at rest there move to infinity () execute oscillatory but not simple harmonic motion. Four capacitors, each of capacitance 5 F are connecte as shown in Fig. If the voltmeter reas 1 V, the charge on each capacitor is (a) 1 - C 5 1 - C. C ().5 C 4. 5. 6. 7. Three point charges 4q, an q are place in a straight line of length l at points istant,l/ an l respectively. The net force on charge q is zero. The value of is (a) -q -q 1 q () 4q Two positive point charges of 1 an 8 microcoulornbs respectively are place 1 cm apart in air. The work one to bring them 4 cm closer is (a) zero.8 J 4.8 J () 5.8 J The work one is carrying a charge q once roun a circle of raius r with a charge at the centre is (a) q 4 r q 1 4 r q 1 4 r () zero A capacitor of capacitance C = F is connecte as shown in Fig. If the internal resistance of the ceil is.5 O, the charge on the capacitor plates is (a) zero C 4 C () 6 C
8. A charge q is place at the centre of the line joining two equal charges. The system of the three charges will be in equilibrium if q is equal to (a) 4 () 9. The electric potential V (in volt) varies with x (in metre) accoring to the relation F = 5 + 4x 4 The force experience by a negative charge of 1-6 C locate at x =.5 m is (a) 1-6 N 4 1-6 N 6 1-6 N () 8 1-6 N 1. Two parallel plate capacitors of capacitances C an C are connecte in parallel an charge to a potential ifference V by a battery. The battery is then isconnecte an the space between the plates of capacitor C is completely fille with a material of ielectric constant K. The potential ifference across the capacitors now becomes (a) V K1 V K V K () V K 11. The force of attraction between the plates of air fille parallel plate capacitor having charge an area of each plate A is given by (a) A A A () 4 A 1. In the network shown in Fig., C 1 = 6 F an C = 9 F. The equivalent capacitance between points P an is (a) F 6 F 9 F () 1 F 1. Three capacitors, each of capacitance C = F, are connecte as shown in Fig. The equivalent capacitance between points P an S is (a) 1 F F 6 F () 9 F
14. A parallel plate capacitor of capacitance 1 pf is to be constructe by using paper sheets of 1. mm thickness as ielectric. If the ielectric constant of paper is 4., the number of circular metal foils of iameter. cm each require for this purpose is (a) 1 () 4 15. One thousan spherical water roplets, each of raius r an each carrying a charge q. coalesce to form a single spherical rop. If v is the electrical potential of each roplet an V that of the bigger rop, then (a) V 1 V 1 V 1 v 1 v 1 v () V 1 v 16. A parallel plate air fille capacitor shown in Fig. (a) has a capacitance of F. When it is half fille with a ielectric of ielectric constant k = as shown in Fig., its capacitance becomes (a) 1 F 1F F () 9 F 17. A parallel plate air fille capacitor shown in Fig. (a) has a capacitance of F. When it is half fille with a ielectric of ielectric constant k = as shown in Fig., its capacitance becomes (a) 4F 4F 1.5F ().5F 18. Three point charges + q, - q an + q are place at the vertices P, an R of an equilateral triangle 1 q as shown in Fig. If F =, where r is the sie of the triangle, the force on charge at P ue to 4 r charges at an R is (a) F along positive x-irection F along negative x-irection F along positive x-irection () F along negative x-irection.
19. In. 18, the force on charge at ue to charges at P an R is (a) F along positive x-irection. F along negative x-irection. F along with the x-irection. () F along with the x-irection.. Two small ientical balls P an, each of mass /1 gram, carry ientical charges an are suspene by threas of equal lengths. At equilibrium, they position themselves as shown in Fig. What is the charge on each ball. Given 1 4 = 9 1 9 Nm C - an take g = 1 ms -. (a) 1 - C 1-5 C 1-7 C () 1-9 C 1. Two point charges q 1 = C an q = 1 C are place at istances b = 1 cm an a = cm from the origin on the y an x axes as shown in Fig. The electric fiel vector at point P (a, b) will subten an angle with the x-axis given by (a) tan = 1 tan = tan = () tan = 4. An electric ipole place with its axis in the irection of a uniform electric fiel experiences (a) a force but no torque a force as well as a torque a torque but no force () neither a force nor a torque
. An electric ipole place with its axis incline at an angle to the irection of a uniform electric fiel experiences (a) a force but no torque a torque but no force a force as well as a torque () neither a force nor a torque 4. An electric ipole place in a non-uniform electric fiel experiences (a) a force but no torque a torque but no force a force as well as a torque () neither a force nor a torque. 5. Four point charges + q, + q,- q an - q are place respectively at the comers A,B,C an D of a square of sie a. The electric potential at the centre O of the square is (a) 1 q. 4 a 1 q. 4 a 1 4q. 4 a () zero 6. A cube of sie b has a charge q at each of its vertices. What is the electric potential at the centre of the cube? (a) 4q b q b q b 7. In. 6, the Electric fiel at the centre of the cube is 4q (a) b q b q b () zero () zero 8. Two point charges - q an + q are locate at points (,, - a) an (,, a) respectively. What is the electric potential at point (,, z)? qa (a) 4 z q 4 a qa 4 (z a ) qa () 4 (z a ) 9. In. 8, how much work is one in moving a small test charge q from point (5,, ) to a point (- 7,, ) along the x-axis? (a) 5 qq 7 4 a 7 qq 5 4 a 1 qq 1 4 a () zero. A neutral hyrogen molecule has two protons an two electrons. If one of the electrons is remove we get a hyrogen molecular ion (H + ). In the groun state of H + the two protons are separate by roughly 1.5 Å an the electron is roughly 1 Å from each proton. What is the potential energy of the system? (a) -8.4 ev - 19. ev -9.6 ev () zero 1. In a hyrogen atom, the electron an the proton are boun together at a separation of about.5 Å. If the zero of potential energy is taken in an infinite separation of the electron from the proton, the potential energy of the electron-proton system is (a) - 54.4 ev - 7. ev -1.6 ev () zero. In. 1, what is the minimum work require to free the electron from the proton if the kinetic energy of the electron in its orbit is half the potential energy of the electron-proton system?
(a). 1-1 J. 1-14 J. 1-16 J (). 1-18 J. In. 1, what will be the potential energy of the electron-proton system if the zero of potential energy is taken at a separation of 1.6 Å? (a) zero -1.6 ev -1. ev () -54.4eV 4. What is the answer to. if the zero of potential energy is taken at a separation of 1.6 Å? (a) zero 1.1 1-14 J 1.1 1-16 J () 1.1 1-18 J 5. What is the equivalent capacitance between A an D of the network shown in Fig.? (a) pf 1 pf pf () 5 pf 6. Figure shows a network of capacitors where the numbers inicate capacitances in microfara. What must be the value of capacitance C if the equivalent capacitance between points A an B is to be 1 F? (a) 1 F F F () 4 F 7. A F capacitor C 1 is charge to a voltage 1 V an a 4 F capacitor C is charge to a voltage 5 V. The capacitors are then connecte in parallel. What is the loss of energy ue to parallel connection? (a) 1.7 J 1.7 1-1 J 1.7 1 - J () 1.7 1 - J 8. A positive charge (+ q) is locate at the centre of a circle as shown in Fig. W 1 is the work one in taking a unit positive charge from A to B an W is the work one in taking the same charge from A to C. Then (a) W 1 > W W 1 < W W 1 = W () W 1 = W =
9. Two concentric spheres of raii r 1 an r carry charges q 1 an q respectively. If the surface charge ensity () is the same for both spheres, the electric potential at the common centre will be r1 (a). r r. r 1 (r1 r ) () (r1 r ) 4. The magnitue of the electric fiel on the surface of a sphere of raius r having a uniform surface charge ensity is (a) 41. The electric fiel ue to an extremely short ipole at a istance r from it is proportional to (a) 1 r 1 r 1 r r () 1 () 4 r 4. The electric potential ue to an extremely short i pole at a istance r from it is proportional to (a) 1 r 1 r 1 r 1 () 4 r 4. A soap bubble of raius r is charge to a potential V. If the raius is increase to n r, the potential on the bubble will become (a) nv n V V n r V () n 44. If n rops, each of capacitance C, coalesce to form a single big rop, the capacitance of the big rop will be (a) n C nc n 1/ C () n 1/ C 45. If n rops, each charge to a potential V, coalesce to form a single rop, the potential of the big rop will be V (a) / n V 1/ n Vn 1/ () Vn / 46. If n rops, each of capacitance C an charge to a potential V, coalesce to form a big rop, the ratio of the energy store in the big rop to that in each small rop will be (a) n : 1 n 4/ : 1 n 5/ : 1 () n : 1 47. A parallel plate capacitor is mae by stacking 1 ientical metallic plates equally space from one another an having the same ielectric between plates. The alternate plates are then connecte. If the capacitor forme by two neighbouring plates has a capacitance C, the total capacitance of the combination will be (a) C 1 C 9 9C () 1 C 48. Figure shows four capacitors connecte to an 8 V power supply. What is the potential ifference across each 1F capacitor? (a) 1 V V V () 4 V
49. Figure shows three capacitors connecte to a 6 V power supply. What is the charge on the F capacitor? (a) 1C C C () 4C 5. Figure shows five capacitors connecte across a 1 V power supply. What is the charge on the F capacitor? (a) 6C 8C 1 C () 1 C 51. Six charges, each equal to + q, are place at the corners of a regular hexagon of sie a. The electric potential at the point where the iagonals of the hexagon intersect will be given by (a) zero 1 q. 4 a 1 6q. 4 a 5. In. 51, the electric fiel at the point of intersection of iagonals is 1 q (a) zero. 4 a 1 6q. 4 a () 1 q. 4 a 1 q (). 4 a 5. A parallel plate capacitor with air as ielectric is charge to a potential V. It is then connecte to an uncharge parallel plate capacitor fille with wax of ielectric constant k. The common potential of both capacitors is (a) V kv (1 + K)V () V (1 k)
54. A capacitor of capacitance C is fully charge by a V supply. It is then ischarge through a small coil of resistance wire embee in a thermally insulate block of specific heat.5 1 Jkg -1 K -1 an of mass.1 kg. If the temperature of the block rises by.4 K, what is the value of C? (a) 5 F 4 F F () F 55. A charge having magnitue is ivie into two parts q an ( - q) which are hel a certain istance r apart. The force of repulsion between the two parts will be maximum if the ratio q/ is (a) 1 1 1 4 () 1 5 56. A charge is given to a hollow metallic sphere of raius R. The electric potential at the surface of the sphere is (a) zero 1. 4 R 1. 4 R () 4 /R 57. In. 56, the potential at a istance r from the centre of the sphere where r < R is (a) zero 1. 4 (R r) 1. 4 R r 4 () (R r) 58. The electric potential V at any point (x, y, z) in space is given by V = 4x volt where x, y an z are all in metre. The electric fiel at the point (1 m,, m) in Vm -1 is (a) 8 along negative x-axis 8 along positive x-axis 16 along negative x-axis () 16 along positive x-axis 59. A charge is situate at the centre of a cube. The electric flux through one of the faces of the cube is (a) 6. Eight ipoles of charges of magnitue q are place insie a cube. The total electric flux through the cube will be (a) 8q 16q 4 q () 6 () zero 61. The magnitue of the electric fiel in the annular region of a charge cylinrical capacitor (a) is the same throughout is higher near the outer cyliner than near the inner cyliner varies as 1lr where r is the istance from the axis () varies as 1/r where r is the istance from the axis. 6. An ientical capacitors are joine in parallel an the combination is charge to voltage V. The total energy store is U. The capacitors are now isconnecte an joine in series. The total energy store in the series combination will be (a) U n U nu () n U
6. Two spheres of raii r an R carry charges q an respectively. When they are connecte by a wire, there will be no loss of energy of the system if (a) qr = R qr = r qr = R () qr = r 64. Two equal point charges of 1 C each are locate at points (i + j + k) m an (i + j - k) m. What is the magnitue of electrostatic force between them? (a) 1 - N 1-6 N 1-9 N () 1-1 N 65. Two point charges q an 4q are hel at a separation r. The electric fiel ue to them is zero at a istance (a) r form charge 4q r from charge 4q r from charge 4q r () from charge 4q 66. The introuction of a metal plate between the plates of a parallel plate capacitor increases its capacitance by 4.5 times. If is the separation of the two plates of the capacitor, the thickness of the metal plate introuce is (a) 5 9 7 9 67. If the potential ifference between the plates of a capacitor is increase by %, the energy store in the capacitor increases by exactly () (a) % % 4% () 44% 68. If the potential ifference between the plates of a capacitor is increase by.1%, the energy store in the capacitor increases by very nearly (a).1%.11%.144% ().% 69. Three capacitors connecte in series have an effective capacitance of F. If one of the capacitors is remove, the effective capacitance becomes F. The capacitance of the capacitor that is remove is (a) 1F F F () 6 F 7. The effective capacitance of two capacitors of capacitances C 1 an C (with C > C 1 ) connecte in parallel is 5 6 times the effective capacitance when they are connecte in series. The ratio C /C 1 is (a) 4 5 () 5 6 71. Three equal point charges q are place at the corners of an equilateral triangle. Another charge is place at the centroi of the triangle. The system of charges will be in equilibrium if equals (a) q - q q () - q
7. A metallic sphere A of raius a carries a charge. It is brought in contact with an uncharge sphere B of raius b. The charge on sphere A now will be (a) a b b a b a b () a a b 7. A soli conucting sphere having a charge is surroune by an uncharge concentric conucting hollow spherical shell. The potential ifference between the surface of the soli sphere an the outer surface of the hollow shall is V. If the shell is now given a charge of-, the new potential ifference between the same two surfaces is (a) V V 4V () -V 74. Two ientical thin rings, each of raius R are coaxially place at a istance R apart. If 1 an are the charges uniformly sprea on the two rings, the work one in moving a charge q from the centre of one ring to the centre of the other is q (a) zero (1 )( 1) 4 R q ( 1 ) 4 R ( 1)q( 1 ) () 4 R 75. An electron of mass m e, initially at rest, moves through a certain istance in a uniform electric fiel in time t 1. A proton of mass m p,, also initially at rest, takes time t to move through an equal istance in this uniform electric fiel. Neglecting the effect of gravity, the ratio t /t 1 is nearly equal to (a) 1 m m p e 1/ m m e p 1/ () 186 76. A metallic soli sphere is place in a uniform electric fiel. In Fig., which path will the lines of force follow? (a) 1 () 4 77. A charge + q is fixe at each of the points x = x, x = x, x = 5x... upto infinity an a charge - q is fixe at each of the points x = x, x = 4x, x = 6x... upto infinity. Here x is a positive constant. The potential at the origin of this system of charges is (a) zero q 4 x In() infinity () q ln() 4 x 78. Three charges, + q an + q are place at the vertices of a right-angle isosceles triangle as shown in Fig. The net electrostatic energy of the configuration is zero if is equal to
q (a) 1 q -q () +q 79. A parallel plate capacitor of capacitance C is connecte to a battery an is charge to a potential ifference V. Another capacitor of capacitance C is similarly charge to a potential ifference V. The charging battery is then isconnecte an the capacitors are connecte in parallel to each other in such a way that the positive terminal of one is connecte to the negative terminal of the other. The final energy of the configuration is (a) zero CV 5 CV 6 () 9 CV 8. A ielectric slab of thickness is inserte in a parallel plate capacitor whose negative plate is at x = an positive plate is at x =. The slab is equiistant from the plates. The capacitor is given some charge. As x goes from to, (a) the magnitue of the electric fiel remains the same the irection of the electric fiel changes continuously the electric potential increases continuously () the electric potential increases at first, then ecreases an again increases. 81. Two ientical metal plates are given positive charges 1 an (< 1 ) respectively. If they are brought close together to form a parallel plate capacitor with capacitance C. the potential ifference between them is 1 (a) C 1 C 1 C 8. For the circuit shown in Fig. which of the following statements is true? 1 () C (a) With S 1 close, V 1 = 15 V, V = V With S close, V 1 = V = 5 V With S 1 an S close V 1 = V = () With S 1 an S close V 1 = V an V = V 8. A parallel plate capacitor of area A, plate separation an capacitance C is fille with three ifferent ielectric materials having ielectric constant K 1, K an K as shown in Fig. If a single ielectric material is to be use to have the same capacitance C in this capacitor, then its ielectric constant K is given by
1 1 1 1 (a) K K K K KK 1 K K K 1 1 K 1 1 1 K K K K 1 () K K1 K K L 84. A quantity X is given by L where is the permittivity of free space, L is a length, V is a t potential ifference an t is a time interval. The imensional formula for X is the same as that of (a) resistance charge voltage () current 85. Consier the situation shown in Fig. The capacitor A has a charge q on it whereas B is uncharge. The charge appearing on the capacitor B a long time after the switch is close is (a) zero q/ q () q 86. A uniform electric fiel pointing in positive x-irection exists in a region. Let A be the origin, B be the point on the x-axis at x = + 1 cm an C be the point on the y-axis at y = + 1 cm. Then the potentials at the points A, B an C satisfy: (a) V A < V B V A > V B V A < V C () V A > V C 87. Two equal point charges are fixe at x = - a an x = + a on the x-axis. Another point charge is place at the origin. The change in the electrical potential energy of, when it is isplace by a small istance x along the x-axis, is approximately proportional to (a) x x x () 1/x 88. There is a uniform electric fiel of strength 1 Vm -1 along the y-axis. A boy of mass 1 g an charge 1-6 C is projecte into the fiel from the origin along the positive x-axis with a velocity of 1 ms -1. Its spee (in ms -1 ) after 1 secon will be (neglect gravitation) (a) 1 5 1 () 89. Two ientical charges are place at the two corners of an equilateral triangle. The potential energy of the system is U. The work one in bringing an ientical charge from infinity to the thir vertex is (a) U U U () 4 U
9. A parallel plate capacitor of capacitance 5 F an plate separation 6 cm is connecte to a 1 V battery an charge. A ielectric of ielectric constant 4 an thickness 4 cm is introuce between the plates of the capacitor. The aitional charge that flows into the capacitor from the battery is (a) C C 5 C () 1 C 91. A capacitor of capacitance 4 F is charge to 8 V an another capacitor of capacitance 6 F is charge to V. When they are connecte together, the energy lost by the 4 F capacitor is (a) 7.8 mj 4.6 mj. mj ().5 mj 9. The magnitue of electric fiel at a istance x from a charge q is E. An ientical charge is place at a istance x from it. Then the magnitue of the force it experiences is (a) qe qe qe () qe 4 9. The flux of electric fiel E = i NC -1 through a cube of sie 1 cm, oriente so that its faces are parallel to the co-orinate axes is (a) zero NC -1 m 6NC -1 m () 1NC -1 m 94. Figure shows a spherical Gaussian surface an a charge istribution. When calculating the flux of electric fiel through the Gaussian surface, the electric fiel will be ue to (a) +q alone + q 1 an + q + q 1 + q an -q () +q 1 an -q 95. Three infinite long plane sheets carrying uniform charge ensities 1 = -, = + an = + are place parallel to the x-z plane at y = a, y = a an y = 4a as shown in Fig. The electric fiel at point P is (a) zero j j () j
96. A metallic spherical shell of raius R has a charge - istribute uniformly on it. A point charge + is place at the center of the shell. Which graph shown in Fig. represents the variation of electric fiel E with istance r from the centre of the shell? (a) () 97. A metallic sphere of raius R is charge to a potential V, The magnitue of the electric fiel at a istance r (> R) from the center of the sphere is (a) V r Vr R VR () zero r 98. Two point charges q 1 = 1 C an q = C are place at points A an B 6 cm apart as shown in Fig. A thir charge = 5 C is move from C to D along the arc of a circle of raius 8 cm as shown. The change in the potential energy of the system is (a). J.6 J 5. J () 7. J 99. A partical of mass m an charge + q is miway between two fixe charge particles, each having a charge + q an at a istance L apart. The mile charge is isplace slightly along the line joining the fixe charges an release. The time perio of oscillation is proportional to. (a) L 1/ L L / () L 1. The potential ifference between points A an B in the circuit shown in Fig. is (a) 6 V V 1 V () 14 V
11. An electric fiel of Vm -1 exists in the region between the plates of a parallel plate capacitor of plate separation 5 cm. The potential ifference between the plates when a slab of ielectric constant 4 an thickness 1 cm is inserte between the plates is (a) 7.5 V 8.5 V 9. V () 1 V 1. A parallel plate capacitor is maintaine at a certain potential ifference. When a ielectric slab of thickness mm is introuce between the plates, the plate separation ha to be increase by mm in orer to maintain the same potential ifference between the plates. The ielectric constant of the slab is (a) 4 () 5 1. A capacitor of capacitance C 1 is charge by connecting it to a battery. The battery is now remove an this capacitor is connecte to a secon uncharge capacitor of capacitance C. If the charge istributes equally on the two capacitors, the ratio of the total energy store in the capacitors after connection to the total energy store in them before connection is (a) 1 1 14. Four metal plates numbere 1,, an 4 are arrange as shown is Fig. The area of each plate is A 1 () 1 an the separation between ajacent plates is. The capacitance of the arrangement is (a) A A A () 4A 15. Four metal plates numbere 1,, an 4 are arrange as shown in Fig. The area of each plate is A an the separation between the plates is. The capacitance of the arrangement is (a) A A A () 4A
16. The equivalent capacitance between points A an B in the network shown in Fig. is (C 1 = F an C = F) (a) 1 F F F () 4 F 17. A capacitor of capacitance C 1 = C is charge to a voltage V. It is then connecte in parallel with a series combination of two uncharge capacitors of capacitances C = C an C = C. The charge that will flow through the connecting wires is (a) CV CV CV () zero 18. The capacitance of a sphere of raius R 1 is increase times when it enclose by an earthe sphere of raius R - The ratio R /R 1 is (a) 4 () 19. A parallel plate capacitor of plate area A an plate separation is charge by a battery of voltage V. The battery is then isconnecte. The work neee to pull the plates to a separation is AV (a) AV AV AV () 11. One plate of a parallel plate capacitor of plate area A an plate separation is connecte to the positive terminal to a batter}' of the voltage V. The negative terminal of the battery an the other plate of the capacitor are earthe as shown in Fig. The charge that flows from the battery to the capacitor plates is (a) zero AV V A () AV Answers 1.. (). 4. (a) 5. () 6. () 7. 8. 9. () 1. 11. 1. (a) 1. () 14. (a) 15. 16. 17. (a) 18. 19. (). 1.. (). 4. 5. () 6. (a) 7. () 8. 9. ().
1.. (). 4. (a) 5. 6. 7. () 8. () 9. () 4. (a) 41. 4. 4. 44. () 45. () 46. 47. 48. 49. 5. 51. 5. (a) 5. () 54. (a) 55. (a) 56. 57. (a) 58. (a) 59. () 6. () 61. 6. 6. 64. (a) 65. () 66. 67. () 68. () 69. () 7. (a) 71. 7. () 7. (a) 74. 75. 76. () 77. () 78. 79. 8. 81. () 8. 8. 84. () 85. (a) 86. 87. 88. 89. 9. 91. (a) 9. () 9. (a) 94. 95. 96. (a) 97. 98. 99. 1. (a) 11. 1. 1. 14. 15. 16. (a) 17. (a) 18. 19. 11. SOLUTIONS 1. The potential insie a spherical conuctor is constant an is the same as that on the surface. Hence the correct choice is.. Let the charge be at P, with OP = x. The resultant force F is along the x-axis irecte towars the origin. The charge moves to O, an acquires kinetic energy. It will cross O an move to -ve x-axis until it comes to rest. It is again attracte towars O an cross it an this process continues. Therefore charge executes perioic motion (see Fig.). Let AP = BP = r. Then q F F 1 4 r The resultant force on is q F F cos F cos cos 1 4r F qx q x 4 r 4 (a x ) / Thus F is not of the form F = kx (where k = constant) an hence the motion is not simple harmonic. Hence the correct choice is ().. Each parallel combination of capacitors is equivalent to a capacitance of 1 F connecte in series. Potential rop across each of them will be 5 V. Charge = CV= 1 1-6 5 = 5 1 - C Hence the correct choice is. 4. Refer to Fig.
The net force on q will be zero if q.4q q 4 l 4 (l / ) 4q + 4q = or 4 q (q + ) = = -q Hence the correct choice is (a). 5. Electrostatic potential energy when the charges are 1 cm =.1 m apart is qq 11 81 6 6 1 W1 4 r 4.1 961 4 11 Potential energy when the charges are brought 4 cm closer, i.e., when they are 6 cm =.6 m apart is 11 81 161 W 4.6 4 Work one = 6 6 1 1 1 W W (16-9.6) 4 1 = 9 1 9 1-1 6.4 = 5.76 J = 5.8 J Hence the correct choice is (). 6. The work one in carrying a charge roun a close path is zero. Hence the correct choice is (). 7. When the capacitor is fully charge, no current flows in the 1 resistor. The current in the circuit is.5 I 1A.5 Potential rop across resistor = 1A = V. This is also the potential rop across the capacitor plates. Therefore, the charge on capacitor plates is = CV= 1-6 = 4 1-6 C = 4 C Hence the correct choice is. 8. Refer to Fig.
The three charges will be in equilibrium if no net force acts on each charge. The charge q is in equilibrium because the forces exerte on q by charge at A an charge at B are equal an opposite. The charge at A will be in equilibrium if the forces exerte on it by charge q an charge at B are equal an opposite, i.e. if Or q 4 r 4 (r) q 4 Similarly, charge at B will be in equilibrium if q = 9. Electric fiel E = V x x (5 + 4x ) = - 8x Force on charge (-q)= -q E = + 8qx At x =.5m,force = 8 1-6.5 = 8 1-6 N Hence the correct choice is ().. Hence the correct choice is. 4 1. Original capacitance of the parallel combination of C an C = C + C = C. Total charge = CV. When the capacitor C is fille with ielectric, its capacitance becomes KC. Therefore, the capacitance of the combination after the capacitor C is fille with ielectric, C' = KC + C = (K+ )C. Since the charge remains the same, = CV, the potential ifference across the capacitors will be CV V C' (K )C K Hence the correct choice is. 11. The capacitance of the capacitor is C = A/x where x is the istance between the plates. The energy store in the capacitor is 1 AV U CV x Differentiating w.r.t x we get U AV 1 AV x x x x The force of attraction between the plates is F U x AV x (i)
Now AV CV or x Using (ii) in (i) we get F A Hence the correct choice is. x V A 1. The last three capacitors on the right, each of capacitance C = 9 F are in series an are equivalent to a capacitance C" given by 1 1 1 1 1 or C' F. C' 9 9 9 Since C is in parallel with C 1, the equivalent capacitance of the last part of the network is C" = C' + C 1 = + 6 = 9 F. Continuing this process of calculation towars the left, we notice that we are finally left with the combination whose equivalent capacitance is F. Hence the correct choice is (a). 1. The three capacitors can be rearrange as shown in Fig. The capacitance between points P an S or between points an R = sum of the three capacitances = C = 9 F. Hence the correct choice is (). (ii) 14. C = 1 pf = 1 1-1 F = 1-1 F Let the number of sheets of foils require be n. They will form (n - 1) capacitors. If K is the ielectric constant of the ielectric, the capacitance is given K (n 1)A C C C 4 Or n 1. K A K.4 r 4C K.4 r 41 11 91 4 (1. 1 ) 1 9 9 or n = 1 Hence the correct choice is (a). 15. If R is the raius of the big rop, we have
4R 4r 1 which gives R = 1 r. The electrical potential of each roplet is q v 4 r an that of the big rop is V 1q 4 R V 1r 1 (R = 1r) v R Hence the correct choice is. 16. If A is the area of each plate, the capacitance of the air capacitor shown in Fig.(a) is C A, where C = F (given). The capacitance of air capacitor in Fig. is A A C C / 1 The capacitance of the ielectric fille capacitor in Fig. is C k A k A kc / where k is the ielectric constant. Now capacitors C 1 an C are in series. Therefore, the capacitance C of the capacitor shown in Fig. is given by 1 1 1 1 1 (k 1) C C C C kc C k 1 Ck F Or C F (k 1) (1) Hence the correct choice is. 17. If A is the area of each plate, the capacitance of the air-fille capacitor shown in Fig. (a) is A C, where C = F (given). The capacitance of air capacitor in Fig. is A / A C C1 The capacitance of ielectric fille capacitor in Fig. is C k A / k A kc Since C 1 an C are in parallel, the capacitance C of the capacitor shown in Fig. is
C kc C C1 C C F (1 k) (1 ) 4 F Hence the correct choice is (a). 18. Refer to Fig. The charge at exerts an attractive force F on charge at P along P. The charge at R exerts a repulsive on charge at P along PS of magnitue F. The angle between these two forces is 1. From parallelogram law, the magnitue of the resultant force is F 1 = F + F + F cos 1 = F - F = F or F r = F. As shown in the figure, the irection of the resultant force is along the negative x- irection. Hence the correct choice is. 19. Refer to Fig. Charges at P an R both exert an attractive force on charge at. The angle between these forces is 6. The resultant force has a magnitue F = F + F + F cos 6 = F + F = F F r = F The angle is given by tan =. o Fsin 6 1 o F Fcos 6 Hence the correct choice is ().. Refer to Fig. Let us consier forces on a ball, say,. Three forces act on it: (i) tension T in the threa, (ii) force mg ue to gravity an (iii) force F ue to Coulomb repulsion along +ve x- irection. For equilibrium, the sum of the x an y components of these forces must be zero, i.e. T cos 6 - F=
an T sin 6 - mg= These equations give F = mg cot 6 = 1 1-1 1 = 1- N. Now 1 q F. 4 r Putting F = 1 - N, r =. m an 1 4 = 9 19, we get q = 1-7 coulomb. 1. Refer to Fig. The electric fiel E 1 at (a, b) ue to q 1 has a magnitue 1 q E. 1 1 4 a an is irecte along + x-axis. The electric fiel E at (a, b) ue to q has a magnitue 1 q E. 4 b an is irecte along +y-axis. The angel subtene by the resultant fiel E with the x-axis is given by E q a 1 E1 q1 b 1 tan. Hence the correct choice is.. The correct choice is (). The electric fiel E exerts a force qe on charge + q an a force - qe on charge - q of the ipole. Since these forces are equal an opposite, they a upto zero.. The correct choice is. A torque acts on the ipole which tens to align it along the fiel. 4. The correct choice is. In a non-uniform electric fiel, a ipole experiences a force which gives it a translational motion an a torque which gives it a rotational motion. 5. Refer to Fig. Potential at O is
1 q q q q V 4 r r r r Hence the correct choice is (). 6. The istance of a vertex from the centre of the cube of sie b is r = b/. Now the potential ue to charge q at the centre is q/4 r. Hence the potential ue to the arrangement of eight charges (each of magnitue q) at the centre is 8q 4q V 4 r b 7. We know that electric fiels are to be ae vectorially. From the symmetry of the eight charges with respect to the centre of the cube, it is evient that the electric fiels at the centre ue to two opposite charges cancel in pairs (being equal an opposite). Hence the net electric fiel at the centre of the cube will be zero. 8. Refer to Fig. The istance of point P 1 from charge + q is r 1 = z - a an from charge - q is r = z + a. 1 q q Potential at P l = 4 r1 r q r r. 4 r r 1 1 qa 4 (z a ), which is choice. 9. Refer to Fig. again. Any point on the perpenicular bisector passing through the centre of the ipole is at the same istance from the two charges. Hence the potentials at point P (5,, ) an that at point P (- 7,, ) are zero. Since P an P are at the same potential (zero), the potential
ifference between them is zero. Hence no work will be one in moving a charge from P to P, The answer will not change if the path of the charge is change because the work one is inepenent of the path taken.. Refer to Fig. The total potential energy of the arrangement of charges is the sum of the energies of each pair of charges. The potential energy of the system comprising the three charges q 1, q an q is U = W 1 + W + W 1 qq q q q q 4 r r r 1 1 (i) 1 1 Here q 1 = q = q = + 1.6 1-19 C (proton), q = -q = - 1.6 1-19 C (electron), r 1 = 1.5 Å = 1.5 1-1 m, r 1 = r = 1 Å = 1 1-1 m an 1/4 n = 9 1 9 Nm C -. Thus 1 4 q 1 U. joule 4 1 4 q1. ev 4 41.61 1 91 19 1 9 = 19.eV 1. Charge on electron (- e) = - 1.6 1-19 C, charge on proton (e) = 1.6 1-19 C, separation r =.5 Å =.5 1-1 m. If the zero of potential energy is taken to be at infinite separation, the potential energy of the electron-proton system is 1 e U. joule 4 r 1 e. ev 4 r 9 19 91 (1.6 1 ).51 1 7.eV Hence the correct choice is.. If the electron was at rest, 7. ev of the energy will have to be supplie (or 7. 1.6 1-19 J of work will have to be one) to free the electron from the attraction of the proton an remove it to infinity. Since the electron is moving (roun the proton) with a kinetic energy = U/ = 1/ (-
7.) = - 1.6 ev, the electron itself is supplying an energy of 1.6 ev ue to centrifugal action. Hence the minimum amount of work require to free the electron = 7. - 1.6 = 1.6 ev = 1.6 1.6 1-19 =. 1-18 J. Hence the correct choice is ().. The potential energy of electron-proton system at a separation of 1.6 Å = half that at a separation of.5 Å = half of - 7. ev = - 1.6 ev. If the zero of potential energy at a separation of 1.6 Å is taken to be zero (instea of- 1.6 ev), the potential energy of the electron-proton system woul be = - 7. - (- 1.6) = - 1.6 ev, which is choice. 4. Since the potential energy of the system is now - 1.6 ev, the energy supplie by the electron itself is 1.6 ev by virtue of its orbital motion roun the proton. Hence the minimum work to pull the electron from the atom will be zero. 5. The series combination of C an C is equivalent to a capacitance C' given by 1 1 1 C' C C CC C' 1pF C C Or Therefore the circuit reuces to the one shown in Fig. (a). The equivalent capacitance between points A an B is C" = C 1 + C = 1 + 1 = pf The circuit may be further simplifie to that in Fig.. The equivalent capacitance C of the entire network, i.e., between points A an D, is now that of the series combination of C" an C 4. Thus 1 1 1 1 1 4 C C" C 1 or C pf Hence the correct choice is. 6. The series combination of 6 an 1 is equivalent to 4 an the parallel combination of an is also equivalent to 4. Therefore the network can be simplifie as shown in Fig. The parallel combination of 4 an 4 is equivalent to 8 an the series combination of 8 an 4 is equivalent to 8/. Thus the combination in Fig.1 reuces to that in Fig. The series combination of 1 an 8 in Fig. yiels 8/9 as shown in Fig.
(1) () () Now 8/ an 8/9 are in parallel an their equivalent is /9. Therefore, the network finally reuces to that in Fig. 4. Since the total capacitance between A an B is to be (i.e. 1F), we have 1 9 1 C (4) C F. Hence the correct choice is. 7. Charge 1 an C 1 = C 1 V 1 = 1-6 1 = 1-4 C. Charge on C = C V = 4 1-6 5 = 1-4 C. Total charge = 1 + = 4 1-4 C. Total energy before connection is E 1 C V 1 C V 1 1 1 1 1 1 (1) 41 (5) 6 6 = 1.5 1 - J The common potential ifference F after connection is given by C 1 V + C V = or Therefore, total energy after connection is 1 1 E (C1 C )V (C C ) 4 1 (41 ) 6 ( 4) 1 V C C 1 1. 1 J 1 Loss of energy = E 1 - E =.17 1 J. Hence the correct choice is (). 8. Points A, B an C are at the same istance from charge + q; hence electrical potential is the same at these points, i.e. there is no potential ifference between A, B an C. Hence W 1 = W =. 9. The electric potential at the common centre is q q 1 V 4 r 1 4 r q1 q Now 4r 4r 1
1 q r q r V (r1 r ) 1 1 4r1 4r Hence the correct choice is (). 4. If q is charge on the sphere, the electric fiel on its surface is But E q 4 r q. Therefore q = 4 r. Hence 4 r 4r E 4 r Thus the correct choice is (a). 41. The correct choice is 4. The correct choice is. 4. If the raius of a bubble is increase by a factor n, its capacitance is also increase by a factor n, i.e. C' = nc. Since the charge on the bubble remains unchange, we have Or = CV = C' V' CV CV V V' C' nc n Hence the correct choice is. 44. If p is the ensity of a small rop an r its raius, then the mass of each small rop is m = If n such rops coalesce to form a big rop of raius R, then the mass of the big rop is nm = 4 R. Hence R = n 1/ r. Now, the capacitance of a sphere is proportional to its raius. Hence the capacitance of the big rop will be C' = n 1/ C. Hence the correct choice is (). 4 r. 45. If is the charge on each small rop, charge on the big rop is ' = n. Now ' = C'V = = CV. 46. Therefore V' ' C n n 1/ V C' n Hence the correct choice is (). E CV,E' C'V' / 1 1. Therefore, E ' C' V '. = n 1/ (n / ) = n 5/ E C V Hence the correct choice is. 47. The combination is equivalent to(1-1) = 9 capacitors, each of capacitance C connecte in parallel. Hence the correct choice is.
48. The total capacitance across power supply = 8 F. The charge on F capacitor or F capacitor 6 = 9 6 C. So the charge on each 1 F capacitor = C. Therefore, potential ifference 8 across each 1 F capacitor = charge/capacitance = C/1F = volts. 49. Capacitors of capacitances F an F are in parallel an this combination is in series with 1 F capacitor. Thus we have 1 F capacitor in series with 5 F capacitor an the potential ifference across this series combination is 6V. Therefore, the potential ifferences across 5 F capacitor (which consists of a parallel combination of F an F capacitors) is 1 V. Hence the charge on F capacitor = F 1 V = C, which is choice. 5. Capacitors 1 F, F an F are in parallel, their total capacitance is 6 F. Thus, we have three capacitors in series each of capacitance 6 F across the 1 V power supply. So the potential rop across each is 1/ = 4 V. This is also the potential across 1 F capacitor an F capacitor an F capacitor, because they are in parallel. Therefore, charge on F capacitor = F 4V = 8 C. Hence the correct choice is. 51. The istance of the point of intersection of iagonals = sie of the hexagon = a. The potential at this point ue to each charge = 1 q.. Therefore, total potential = 4 a 1 6q. which is choice. 4 a 5. The net electric fiel at the point of intersection of iagonals is zero because the electric fiel at this point ue to equal charges at opposite corners with cancel each other in pairs. 5. If C is the capacitance of the air-fille capacitor, the total charge on its plates, before connection, is = CV. After it is connecte with an uncharge capacitor, let V' be the common potential an 1 be the charge on capacitor C an on the other capacitor 1 = V' C an = V' kc. Also = 1 +. Therefore, CV = V' C + V' kc or V= V' (1 + k) or V V'. Hence the correct choice is (). (1 k) 54. Energy store in the capacitor is 1 1 CV C () = 1 4 C joule Energy appearing as heat in the block is or m c =.1.5 1.4 = 1 J Therefore, 1 4 C = 1 C = 5 1-4 F = 5F 55. The force of repulsion between the two parts is given by
1 q( q) F. 4 r For F to be maximum, F =, i.e. q 1 q( q). q 4 r Since r is fixe, we have q q 1 [q ( - q)] = or 1( - q) + q ( - 1) = or Hence the correct choice is (a). 56. For points on the surface of the sphere or outsie the sphere, a charge sphere behaves as if the charge is concentrate at its center. Therefore, the potential at the surface of the sphere is given by 1 V., which is choice. 4 R 57. At points insie a charge metallic sphere, i.e. for r < R, the potential is zero. Hence the correct 58. choice is (a). V E i where i is a unit vector along the positive x-axis. Hence E at a point whose x- x coorinate is x = 1 m is E (4x )i = -8xi = -8i Vm -1. x The negative sign shows that E is along the negative x-axis. Hence the correct choice is (a). 59. If a symmetrical close surface has n ientical surfaces an a charge is place at its centre, then the flux through each face = n Hence the correct choice is ().. For a cube n = 6. 6. Since a ipole consists of two equal an opposite charges, the net charge of a ipole is zero. Hence the correct choice is (). 61. The correct choice is. 6. Let C be the capacitance of each capacitor. For parallel combination, the net capacitance is C 1 = nc. Also V 1 = V. Therefore, the energy store in the parallel combination is U 1 C V 1 nc V 1 ncv 1 1 1 For series combination, we have C = C/n an V = nv. Therefore, the energy store in the series combination is U 1 C V 1 C (nv) 1 ncv n Hence the correct choice is.
6. There will be no loss of energy if the potential of the spheres is the same i.e. if Or q V 4 r 4 R q. Hence the correct choice is. r R 64. r = (i +j + k) - (i + j - k) = (i + j + k)m. The magnitue of r is r 1 1 4 4 m 1 qq 1 F. 4 r 9 6 6 91 1 1 () Hence the correct choice is (a). 1 N 65. Let the electric fiel be zero at a istance x from charge 4q. Then 1 4q 1 q.. 4 x 4 (r x) or (r - x) - x or x - r/ which is choice (), 66. Initial capacitance C A. When a metal plate of thickness t is introuce, the capacitance 67. 68. becomes A C'. Given C' = 4.5 C ( t) Thus A A 9 t which gives 9( - t) = or t = 7 9 1 1 1 U1 CV, U C(1.V) CV 1.44 U U U 1 U 1 1 which is choice. 1 = (1.44-1) 1 = 44% Thus the correct choice is (). 1 CV. Therefore, U = CV V. Therefore U CVV V 1 1 1 U 1 CV V.11.% 1 Hence the correct choice is ().
1 1 1 1 69. Given (1) C C C an 1 1 1 1 () C C 1 Using () in (1), we have 1 1 1 which gives C = 6 F Hence the correct choice is (). C CC 1 5 7. Given C 1 + C = C C 6 1 or 6(C 1 + C ) = 5 C 1 C or or 6C 6C 1C C 5C C 1 1 1 6C 6C 1C C 1 1 Let C = xc 1. Then, we have 6C 6x C 1xC 1 1 1 or 6x 1x + 6 = which gives x or. Since C > C 1, x is not possible. Hence the correct choice is (a). 71. The system be in equilibrium if the net force on charge q at one vertex ue to charges q at the other two vertices is equal an opposite to the force ue to charge at the centroi. i.e. (here a is the sie of the triangle) which gives q q 4 a a 4 q. Hence the correct choice is. 7. Charge will flow from A to B until their potentials become equal. If charge q flows from A to B, then or q q 4 a 4 b a q q which gives b Hence the correct choice is (). b q a b. Hence charge left on A = - q = - b a a b a b. 7. When any aitional negative charge is given to a hollow spherical shell, the potential on its surface falls, but the potential at each point within the shell also falls by the same amount. Hence the potential ifference between the given surfaces remains unchange. Thus the correct choice is (a).
74. Refer to Fig. Potential at C 1 is 1 1 V1 4 R R Potential at C is 1 1 V 4 R R Work one = q(v 1 - V ) 1 1 1 4 R R R R 1 (1 )( 1) 4 R Hence the correct choice is. 75. Force F = qe. Therefore, acceleration a = qe/m. Now istance move in time t is For electron : For proton: 1 1 qe s at t m s s 1 qe t e 1 me 1 qe t p m p Given s e = s p. Therefore 1/ t1 t t m p or e p 1 e m m t m. Hence the correct choice is. 76. The electric fiel is always perpenicular to the surface of a conuctor. On the surface of a metallic soli sphere, the electric fiel is perpenicular to the surface an irecte towars the centre of the sphere. Hence the correct choice is (). 1 q q q 77. V...upto infinity 4 x x 5x
1 q q q...upto infinity 4 x 4x 5x 1 q 1 1 1 1 1. 1...upto infinity 4 x 4 5 6 1 q ln() ln (1 1) 4 x 4 x Hence the correct choice is (). 78. Since the hypotenuse sie of triangle = a, the net electrostatic energy is 1 q q qq U 4 a a a For U=, we require q q q a a a q which gives q 1 Hence the correct choice is. 79. 1 = CV an = (C) (V) = 4CV. Since the capacitors are connecte in parallel such that the plates of opposite polarity are connecte together, the common potential is V' 4CV CV C C C C 1 1 V Equivalent capacitance C' = C + C = C. Therefore, the final energy of the configuration is 1 1 U' C'V' C V CV, which is choice. 8. The insertion of the ielectric slab ecreases the electric fiel without changing its irection. The electric potential increases as we go from the negative to the positive plate. Hence the correct choice is. 81. Within the plates electric fiels ue to charges 1 an are E an E A A 1 1 As these fiels are in opposite irections an 1 >, the net electric fiel within the plates is E = E 1 - E = 1 ( 1 - ) A 1 Hence V E (1 ) which is choice (). A C
8. When switch S is close, the potential ifference across C 1 an C will become equal to the average of V 1 an V, i.e. ( + )/ = 5 V. Hence the correct choice is. (A / ) K1 AK1 8. We have C1 ( / ) C C (A / ) K A K ( / ) an A K A K ( / ) The capacitors C 1 an C are in parallel an their equivalent capacitance is A C' C1 C (K1 K ) This combination is in series with C. Hence the net capacitance is 1 1 1 C" C' C A (K K ) A K 1 1 1 A (K1 K ) K AK 1 1 1 or C" where K (K K ) K Hence the correct choice is. 1 84. The capacitance of a parallel plane capacitor is given by C = A/. Hence the imensions of L are the same as those of capacitance. L Dimensions of L t imension of C imensions of V time imension of time ch arg e current time Hence the correct choice is (). 85. Since the outer plate of B is free, charge cannot flow from A to B. Hence the correct choice is (a). 86. Electric fiel is the negative graient of potential, i.e. V E x Thus V ecreases as x increases in the irection of the fiel. This implies that V A > V B, which is choice. 87. Potential energy of the system when charge is at O' is
U q q q a a a When charge is shifte to position O', the potential energy will be (see Fig.) q q U (a x) (a x) q(a) q x 1 (a x ) a a q x 1 a a U = U U a a a a 1 q x q q 1 (x ) Hence U x which is choice. 88. Given v x = 1 ms -1. Since the electric fiel is irecte along the y-axis, the acceleration of the boy along the y-irection is a 6 qe 1 1 1ms m 1 y Therefore, the velocity of the boy along the y-axis at time t = 1 s is v y = at = 1 1 = 1 ms -1 Resultant velocity v = v v x y 1 (1) (1) 1 ms Hence the correct choice is. 89. Let be the magnitue of each charge an a the length of each sie of the triangle. The potential energy of the system of two equal charges place at vertex A an B is U (given). This means that U is the work one in bringing a charge from infinity to vertex B. Hence the work one in bringing an ientical charge from infinity to the thir vertex C = work one to overcome the force of repulsion of place at A at a istance a + work one to overcome the force of repulsion of place at B at the same istance a= U + U= U, which is choice. 9. Charge on capacitor plates without the ielectric is = CV= (5 1-6 F) 1 V = 5 1-6 C = 5 C The capacitance after the ielectric is introuce is
A A / C' t t t t K 1 K 5F 1F 41 1 6 Charge on capacitor plates now will be ' = C'V = 1 F 1 V = 1 C Aitional charge transferre = ' = 1 C - 5 C = 5 C, which is choice. C1V1 CV 91. Common potential is V = C C 1 6 6 (41 ) 8 (61 ) 41 61 6 6 Energy lost by 4 F capacitor 1 1 C V C V 1 1 1 1 C 1 (V 1 V ) 1 (4 1 6 ) {(8) (5) } 5V = 7.8 1 - J = 7.8mJ Hence the correct choice is (a). q 9. Given E. Hence the magnitue of the electric fiel at a istance x from charge q is 4 x q q 1 E E' 4 (x) 4 x 4 4 Therefore, the force experience by a similar charge q at a istance x is qe F qe ' 4 Hence the correct choice is (). 9. Refer to Fig. Let 5 be the surface area of each face of the cube. The flux through surfaces ABCD an EFGH is zero because these surfaces are parallel to the electric fiel E ( = 9 ).
Flux through face BFGC is 1 = ES cos = ES. Flux through face AEHD is = ES cos 18 = - ES. Total flux through the cube = 1 + = ES - ES =. Hence the correct choice is (a). 94. The electric flux is given by the surface integral E.s. Here the electric fiel E is ue to all the charges, both insie an outsie the Gaussian surface. Hence the correct choice is. 95. The electric fiel at a point P ue to an infinite long plane sheet carrying a uniform charge ensity a is given by E It is inepenent of the istance of point P from the sheet an is, therefore, uniform. The irection of the electric fiel is away from the sheet an perpenicular to it if is positive an is towars the sheet an perpenicular to it if is negative. Hence E 1 ( j) along -ve j-irection E ( j) along -ve y-irection an E ( j) along vey irection From the superposition principle, the net electric fiel at point P is E = E 1 + E + E ( j) ( j) ( j) j, which is choice. 96. Electric fiel ue to charge - on the shell at a istance r from its center is (for r > R) E 1 4 r irecte towars the centre. Electric fiel ue to charge + at the centre at a istance r is E 4 r
irecte away from the centre. Net electric fiel E (for r > R) = E 1 - E =. For r < R, the electric fiel ue to the shell is zero. In this region, the electric fiel ue to charge + at the centre ecreases as 1/r. Hence the correct graph is (a). 97. Let the charge on the sphere be. Then V 4 R which gives = 4 RV The electric fiel at a istance r is 4 RV RV E 4 r 4 r r thus the correct choice is. 98. If charge is move from C to D along the arc, the potential energy between pairs (q 1, ) an (q 1, q ) will not change as the istance between them remains unchange ( AC = AD). The potential energy of the pair of charges q an will change. Now, istance BC = (8) (6) = 1 cm an BD = 8-6 = cm. Therefore, change in P.E. is U = q 1 4 BD BC = ( 1-6 ) (5 19-6 ) (9 1-9 ) =.6 J, which is choice. 1 1..1 99. If the mile charge is isplace by a istance x, the net force acting it, when it is release, is F 1 q 1 q 4 (L x) 4 (L x) 4q Lx 4 (L x ) qx For x << L, F = = -kx L Where Now q k L T m k So, the correct choice is. 1. The batteries are in opposition as their positive terminals are connecte together. Hence the effective voltage is