Fundamentals of Electrodynamics
Problem 52. D eterm ine th e capacitance of the capacitor
bank shown in Fig. 18a if CY = 1.5 jjlF, C2 = 2 jiF, C3 = 3 |iF , C4 = 4 fiF, and C5 = 2 p,F. W h at energy is stored in
the bank if the voltage applied to it is 500 V?
Given: Cx = 1.5 pF, C2 = 2 pF, C3 = 3 pF, C4 = 4 pF, and C5 = 2 pF are th e capacitances of the capacitors, and U = 500 V is the voltage across the bank.
96 Ch. II. Fundam entals of E lectrodynam ics
F in d : the capacitance C of the bank and th e energy W of the electric field of th e capacitor bank.
So lution. In the problem , it is convenient to express the capacitances of the capacitors in m icrofarads. The capacitors
m Cl +° — l h C2 lb) +o- C, (C) + o- Ct -o - F ig. 18
C2, C 3, and C4 are connected in series. They can be replaced by a single capacitor having an eq u iv alent capacitance (Fig. 186). For series-connected capacitors, we have
Cl c t ^ Ca ^ Ct ’
Cf = T li7 ;^ " 3 iI f ,+ 4 7 F ’ C1 = °-9 2 M'F -
The capacitors C5 and Cl are connected in p arallel. There fore, th e ir equ iv alent capacitance is
Cn =
c5
+ Cn C „ = 2 pF + 0.92 pF = 2.92 pF ~ 3 pF. As a resu lt, we o b tain th e cap acito rs Ci and ClY in series (Fig. 18c).The capacitance of the bank can be found from the form ula _ L _ J L _ L
c = - e r & T 7 '
The energy stored in the bank can be determ ined from the form ula
W .
cm
2 , w
l x i c r 6 F x 2 5 x l 0 4 V 2
1.3
x
10"3 J. Answer. The cap acitance of th e bank is 1 pF and the ener gy is ap p ro x im ately 1.3 X 10"3 J.Problem 53. An electron flies from point A to point fi, the p o ten tia l difference between these points being 100 V.
§ 9. E lectric Field 97 W hat is th e v elo city acquired by th e electron a t po in t B if th e velocity a t point A is zero? The charge-to-m ass ra tio for electrons is 1.76 x 1011 C/kg.
Given: U = 100 V is the p o ten tia l difference between p oints A and B , e / m e = 1.76 X 1011 C/kg is th e charge-to- m ass ra tio for th e electron.
Find: th e v elocity v of th e electron a t p o int B .
Solution. The work done by th e field in disp lacing th e electron is A = e U . T his work is converted in to th e k in etic
energy of th e electron 2?k = m ev2l2. From the energy con serv atio n law , we have e U = m ev2/2, whence
v = 1 /2 X 100 V x 1.76 x 1011 C /kg = 5.9 x 106 m/s. Answer. The v elo city of th e electron a t p o int B is ap proxim ately 5900 km/s.
Questions and Problems
Conservation of electric charge. Coulomb's law
9 .1 . A drop of oil has an electric charge of —3.2 X
10”19 C. D eterm ine th e num ber of excess electrons on th e drop.
9 .2 . Three electrons are m issing from an e le c tro sta tic a lly charged body. D eterm ine th e m ag n itud e and sign of the charge on th e body.
9 .3 . Can an electric charge of one sign be obtained as a resu lt of electro static charging by friction?
9 .4 . U nder w hat conditions can a brass rod be charged? 9 .5 . An electric charge on a conducting sphere has to be divided into three equal p arts. How can th is be done?
9 .6 . W hy is a m etal ch ain th a t reaches th e ground fixed to a lo rry for tran sp o rtin g petrol?
9 .7 . Can a positive charge be obtained on an electroscope using a n eg ativ ely charged ebonite rod?
9 .8 . W h at w ill happen to th e surface charge d en sity on a m etal sheet rolled in to a cylind er?
98 Ch. II. Fundam entals of Electrodynam ics
9 .9 . An elder ball is tied to a silk th read. W h at w ill h a p pen when an e le c tro sta tic a lly charged rod is brought close to it?
9.10. W ill th e force of in teractio n between two p o int charges change if the m agnitude of each charge and the sep aratio n between charges are halved?
9.1 1. Two id en tical conducting spheres bearing elec tric charges of 3.2 X 10-19 and —3.2 X 10-19 C are brought in co n tact. W h at are th e new values of charge on the spheres? How m any electrons have passed from one sphere to anoth er?
9 .12. W h at is the force of in te rac tio n betw een two p o in t charges of 12 nC each separated in vacuum by 3 cm? By w hat factor w ill th e force of in teractio n be reduced if the charges are placed in w ater?
9.13. Two id entical point charges in glycerol 9.0 cm a p a rt in te ra c t w ith a force of 1.3 X 10“5 N. D eterm ine th e m ag n itu d e s of the charges.
9.14. W h a t is the force of in te rac tio n between two 1-C charges 1 m a p a rt (a) in vacuum and (b) in kerosene?
9.15. At w hat distance would two 1-C charges in te ra c t in vacuum w ith a force of 1 N?
9 .16. Using T able 14, answer the follow ing questions: (1) W h at is the ra tio of the forces of in te ra c tio n between electric charges in vacuum and in m ica? (2) In w hat m edia is the force of in te rac tio n between charges equal to half th e force of th eir in te rac tio n in vacuum ? (3) To w hat m edium should charges be transferred from vacuum so th a t th e ir in te rac tio n is reduced by a factor of 81?
9.17. W ill the e lectro static in teractio n force between two charges change if they are set in ice instead of vacuum ?
9.18. Two m ercury drops on glass of mass 20 g each and 4.0 cm a p a rt receive charges of —6.0 x 10~8 and 2.0 x
10‘ 7 C. In w hat d irectio n and w ith w hat acceleration w ill they s ta rt to move? W ill th e m otion be uniform ly accelerated? G ra v ita tio n a l forces should be neglected.
9.19. W ith w h at force is an electron m oving in an o rbit of rad iu s 5.0 x 10"11 m a ttra c te d to a helium nucleus? The charge of the nucleus is 3.2 X 10“19 C. The orb it should be assumed to be circular.
9.20. Two little spheres having a m ass of 1 g each and suspended in vacuum from the same point on silk threads
§ 9. Electric Field 99 have equal n egative charges. T hey repel each other and move a p a rt to 12 cm, form ing an angle of 22°. D eterm ine the num ber of electrons on each ball and the tension in the threads. Show th e forces acting on the balls on a diagram . W ill the tension in the th reads change if the charges in te ra c t in zero-gravity?
9.21. Two point charges of 5.0 x 10~9 and 1.5 x 10-8 C are 4.0 cm a p a rt in vacuum . D eterm ine th e force w ith which these charges w ill act on a th ird charge of 1.0 X 10”9C, located a t th e m idpoint of th e line connecting the charges.
9.2 2 . A charge of 1.57 x 1 0 '8 C is transferred to a m etal sphere of rad iu s 5 cm. W hat is the surface charge density on the sphere?
9.23. The surface charge d en sity of a conducting sphere is 5 x 10"5 C /m 2. D eterm ine the m agnitude of the charge on the sphere if its rad iu s is 8 cm.
9 .2 4 . Two electric charges, one of which is tw ice the o th er, in te ra c t at a distance of 0.60 m in vacuum w ith a force of 2.0 m N. C alculate the m agnitudes of charges. A t w hat d istan ce in kerosene w ill the in te rac tio n between the charges be th e sam e?
9.25. W h a t is the ra tio of th e elec tro sta tic force of re p u l sion between two electron s to th e ir g ra v ita tio n a l a ttra c tio n ?
9.26 . D eterm ine th e p e rm ittiv ity of kerosene if two equal like charges in te ra c t in vacuum w ith th e sam e force at a dis tance of 0.283 m as in kerosene a t 0.20 m. A ssum ing th a t th e force of in te rac tio n in kerosene is 3.0 X 10”2 N, deter m ine th e m agnitudes of th e charges.
Electric field strength
9.27. Do th e electric field vector and the vector of the force exerted on a charge by an electric field alw ays have the same directions?
9.28. W hy is a m etal cap som etim es p u t on a vacuum tube?
9.29. Can electric charges be separated on (a) a conductor, (b) a dielectric?
9.30. A cy lin drical conductor is attach ed to a conical conductor w ith the same base area. W h at can be said about th e electric field stre n g th near various p oints on the surface of th e re s u lta n t conductor?
100 Ch. II. Fundam entals of Electrodynam ics
9.31 . W hy is a b all usu ally fixed to the rod of an electro scope?
9.32. Two electric charges of the same m agnitude and sign produce an electric field. W h at is the field stre n g th at the m id p o in t of the stra ig h t line connecting th e conductors?
9.33 . The electric field stren g th at a given p o int is 300 V /m . W hat does this m ean?
9.34. D eterm ine the electric field stre n g th a t a p o int w here a force of 5 mN acts on a charge of 0.7 X 10~6 C.
9.3 5. W h at is the force exerted by an electric field on a charge of 3.2 X 10-8 C at a point where the field stren g th is 500 V /m ?
9.36. The electric field stren g th near the surface of the E a rth before a stroke of lig h tn in g is 2 X 105 V /m . W hat is the force exerted by the field on an electron?
9.37. An electric field is produced in vacuum by a point charge of 7.5 X 10‘ 8 C. D eterm ine the field stre n g th 15 cm away from th e charge.
9.38. A point charge of 2.2 nC produces a field w ith a stre n g th of 2.5 kV/m a t 6.0 cm from th e charge. D eterm ine
th e p e rm ittiv ity of th e m edium .
9.39. A vessel of kerosene contains a conducting sphere w ith a n egligibly sm all size and a charge of 8.0 x 10~8 C. D eterm ine th e electric field stren g th 5.0 cm from the charge. W hat w ill be the change in th e field stre n g th a t th is point if the kerosene is le t out of the vessel?
9 .4 0 . The electric field stre n g th 5.0 cm from a charge is 1.5 x 105 V /m . W hat is the electric field stre n g th 10.0 cm from th e charge? D eterm ine th e m agnitude of th e charge.
9.41. How m any excess electrons are contained on a dust p a rticle acted upon by a force of 2.4 X 10"10 N in an electric field of stren g th 1.5 x 105 V /m ?
9.42. An electric field is produced by a p o int charge. W h at are the loci of th e p o in ts where the m agnitude of the electric field stren g th is the sam e?
9.43. W ill an electric field rem ain uniform if a conducting sphere is placed in it?
9.44. Two id en tical poin t charges of 1.0 X 10“8 and 2.0 x 10‘ 8 C are in vacuum 20 cm a p a rt and produce an electric field. D eterm ine th e field stren g th a t th e m idpoint of th e line connecting the charges. W h at w ill the electric field stren g th be if th e charges are u nlike?
§ 9. Electric Field 101 9.45. Two conducting spheres having diam eters of 10.0 and 4.0 cm are 120.0 cm a p a rt and bear charges of 3.0
x
10“® and 2.0 X 10"6 C respectively. D eterm ine the electric held stre n g th a t the m idp o in t O of the s tra ig h t line con necting th e spheres (Fig. 19).9.46. Two like electric charges of 7.0 X 10~8 C each are a t points A and B (Fig. 20). D eterm ine the electric field
&
60 60
0 F ig. 19
s tre n g th a t point O which is the apex of the rig h t angle AOjB; A O = BO = 5.0 cm.
9.47. D eterm ine th e electric field stre n g th of the field at a point three rad ii from the surface of a charged conducting sphere. The surface charge d en sity on th e sphere is 1.6 X 10“7 C/m 2.
9.48. A conducting sphere of rad iu s R is e le c tro sta tic a lly charged to a surface density a. W h at w ill be the field stre n g th (a) at th e centre of th e sphere, (b) a t h a lf th e rad iu s of the sphere, (c) on th e surface of th e sphere?
9.49. A drop bearing a charge of 2 X 10~8 C is in eq ui lib riu m in a uniform electric field of stren g th 49 V /m . D eter m ine th e mass of the drop.
9.50. W ith w hat acceleration w ill a 10-g b all w ith charge 1.0 X 10“5 C fall in the electric field of the E a rth ? The elec tric field stre n g th near the surface of the E a rth is 130 V /m .
9.51 . A th in conducting rin g of rad iu s R has an electric charge Q. D eterm ine the electric field stren g th a t th e cen tre O of the ring and a t point A (Fig. 21).
9.52. W hy is th e u n ifo rm ity of th e electric field of a charged p a ra lle l-p la te cap acito r violated if an uncharged
102 Ch. II. Fundam entals of Electrodynam ics
Electric potential. Potential difference. Work done in an electric field
9.53. C onsidering the E a rth to be a sphere of rad iu s 6400 km , determ ine its electric charge and p o ten tia l if the electric field stren g th produced by the E a rth near the surface is 130 V/m .
9 .54. The work done in displacing a 2.0 x 10~8-C charge from infinity to a point in a field is 1.13 x 10~4 J . W hat is the electric p oten tial at th is point?
9.55. D eterm ine the p o ten tial difference between two points in a field if a work of 3.0 x 10~5 J has to be done in m oving an 8.0 X 10"7-C charge between these points.
9.56. An electric field moves a positive charge of 3.0 x
10"7C between two points w ith p o ten tials 200 V and 1200 V. W hat is th e work done by the field in th is case?
9.57. An electric field is produced by a point charge of 4 X 10~8 C. W h at is the electric p o ten tial 6 cm away from th is charge? W h at work m ust be done against the field in order to b rin g a positive charge of 1 C from infinity to th is point?
9.58. Two point electric charges of 1.0 X 10"5 and 6.0 X 10"6 C are 20 cm a p a rt in air. D eterm ine the electric po ten tia l a t the m idp o int of the stra ig h t line connecting the charges.
9.59. A 1.6 X 10“7-C charge is moved 3.0 cm along a field line in a uniform electric field having a stre n g th of 5.0 X 103V/m . D eterm ine the work done and the p o ten tia l differ ence for two points between which the charge is moved.
9 .6 0. As an electron passes between two points in an electric field, its velocity increases from 2.0 X 106 to 3.0 X 107 m /s. W hat is the p o ten tia l difference between these points? W h at is the increase in the k in etic energy of the electron?
9.61. Two 8-nC charges are located at two apexes of an e q u ila te ral trian g le having a side of 6.0 cm. D eterm ine the field stre n g th and the p o ten tia l a t the th ird apex.
9.62. Two conducting spheres of rad ii 2.0 and 3.0 cm are charged to 30 and 40 V respectively. W hat w ill be the elec tric p o ten tia l of the spheres after they have been connected by a w ire? Assume th a t the sep aration between the spheres is large compared to th eir radii.
§ 9. E lectric Field 403 9.63. An electron having a v elocity of 6.0 X 407 m /s flies into a p a ra lle l-p la te air capacitor m idw ay between th e plates. W h at m ust the voltage across the p lates be for th e d eviation of th e electron to be m axim um ? The p lates are
40.0 cm long and 3.0 cm ap art.
9.64. An electron w ith a v elocity of 1.6 x 106 m /s flies into a uniform electric field of stren g th 90 V /m and moves along a field line u n til it comes to a h a lt. How long and how far does it fly in the field? Assume th a t the electron mass is 9.0 X 40~31 kg.
Capacitance. Capacitors. The electrical energy in a capacitor
9.65. Can the p o ten tial of a charged conductor be changed w ith o u t changing its charge?
9.66. A p arallel-p late air cap acitor of capacitance C is imm ersed in a m edium w ith a p e rm ittiv ity of 2. W h at will the capacitance of th e capacito r be?
9.67. The capacitance of a spherical conductor is propor tion al to its radius. W hat m ust the radiu s of a conducting sphere be for its capacitance in vacuum to be 1 F? W hat is the ra tio between the radius of such a sphere and the E a rth ’s radius?
9 .68 . A p a ra lle l-p la te air capacitor is connected to a con- stan t-v o ltag e source. W hat w ill be the change in the cap aci tance and the energy of th e capacitor, the voltage across it, and th e charge on the cap acito r plates if th e sep aration between the plates is reduced?
9.69. D eterm ine the capacitance of an isolated conducting sphere of rad iu s 5.0 cm im m ersed in kerosene. E xpress the answer in farads, m icrofarads, and picofarads.
9.70. A ssum ing the E a rth to be a sphere of rad iu s 6400 km , determ ine its capacitance.
9.71. An isolated conducting sphere whose capacitance is 5.0 pF is e le c tro sta tic a lly charged to a p o ten tial of 4570 V. D eterm ine the rad iu s of the sphere and the surface charge density on it.
9.72. W hy do electroly tic capacitors have a large capa citance?
104 _________ Ch. II. Fundam entals of Electrodynam ics
10“8 C each are arranged a long way ap art. The capacitance of the larger sphere is 2.2 X 10-11 F, and th a t of the sm aller
sphere is 5.6 X 10"12 F. W hat are th eir potentials? W hat
w ill happen when they are connected by a conductor?
9 .74. C alculate the capacitance of a p a ra lle l-p la te capaci to r m ade from tin foil plates 15 cm2 in area w ith a m ica dielectric layer 0.8 mm thick. The p e rm ittiv ity of mica is e = 6.
9>75. A p a ra lle l-p la te air capacitor consists of two plates 100 cm* each in area. W hen a charge of 6.0 X 10~9 C is tra n s
ferred to one of the plates, the capacito r is charged to a vol tage of 120 V. D eterm ine the separation between the plates. 9.76. The plates of a p a ra lle l-p la te air cap acitor have an area of 62.3 cm 2 each and th eir sep aratio n is 5 mm. D eterm ine the charge of the cap acito r if the p o ten tia l difference across its plates is 60 V.
9.77. The separation between the p lates of a parallel- plate air capacitor is 0.1 cm and the area of each plate is 200 cm2. The p o ten tial difference across the plates is 600 V. W hat charge is stored in the capacitor? W h at will be the change in the voltage if the space between the plates is filled w ith m ica, whose p e rm ittiv ity is 6?
9.78. A p a ra lle l-p la te air capacitor w ith a p late separa tion of 1.5 mm is charged to a voltage of 150 V. How far ap a rt should the p lates be moved in order to increase the voltage to 600 V?
9.79. D eterm ine the capacitance of a cap acitor in which nine mica plates having a thickness of 0.12 mm and an area of 12.56 cm 2 are interleaved between tin foil plates. The p e rm ittiv ity e of mica is 6.
9.80. D eterm ine the energy of a p a ra lle l-p la te paper ca pacitor w ith a plate area of 600 cm2. The charge on the capacitor is 2 X 10-7 C and the dielectric is wax paper
2.0 mm thick.
9.81. Three capacitors whose capacitances are 4, 2, and 6 fxF are connected to form a bank and to a co nstant-voltage source of 200 V. D eterm ine the capacitance and the energy of the bank when the capacitors are connected (1) in series and (2) in parallel.
9.82. D eterm ine the capacitance of a capacitor bank connected as shown in Fig. 22 if Cx = 1.2 pF and C2 = C3 - 0.6. pF.
§ 9. Electric Field 105 9.83. D eterm ine the capacitance of a bank of capacitors connected as shown in Fig. 23. The capacitance of all the capacitors is the same and equal to 0.6 pF. D eterm ine the electric charge stored in th e bank if a v o ltag e of 100 V is applied to it.
9.84. C apacitors are connected as shown in Fig. 24. Given: Cx = C2 = 2 pF, C3 = C4 = C5 = 6 pF. D eterm ine
F ig . 22
the p o ten tial difference between points A and B if the energy stored in the bank is 1.35 X 1 0 '1 J.
9.85. An uncharged 100-pF capacitor is connected in parallel to a 50-pF capacitor charged to a voltage of 300 V.
Cf <-/ <-2 r —
II
---IH
— ii
— 3 C2 c,HI
—II
—II—
HI
—IH
G G G
i!>
o F ig . 24 F ig . 25W h at is the voltage across the capacitors? W h at is the charge d istrib u tio n between them ?
9.86. A fter a capacitor having an unknown capacitance and charged to a voltage of 600 V is connected in parallel to an uncharged 5-pF cap acito r, the voltage across the bank drops to 100 V. W h at is the capacitance of the first capacitor?
9.87. Two capacitors having different capacitances are connected in p arallel and to a constant-voltage source. W hich of the capacitors w ill accum ulate more energy?