C1 836 CHAPTER 26 Capacitance and Dielectrics · PDF file · 2013-08-15second capacitor? ssM llw s&'$& Figure 25-32 shows a circuit section of four air-filled ......

of the device is to be 1.00 E, what must be the separation betweenthe plates? (b) Could this capacitor actually be constructed?*$ A parallel-plate capacitor has circular plates of 8.20 cmradius and 1.30 mm separation. (a) Calculate the capacitance.(b) What charge will appear on the plates if a potential differ-ence of \20 V is applied? ssm

o$ The plates of a spherical capacitor have radii 38.0 mmand 40.0 mm. (a) Calculate the capacitance. (b) What must bethe plate area of a parallel-plate capacitor with the same plateseparation and capacitance?ep What is the capacitance of a drop that results when twomercury spheres, each of radius R - 2.00 ffiffi, merge?

sec" trS-4 Capacitors in Parallel and in Series*$ In Fig. 25-28,find the equivalent capacitance of the combina-tion.Assume that Ct is 10.0 FF,Czis 5.fi) p,E, and C: is 4.00 pEo$ In Fig. 25-29, flnd theequivalent capacitance of thecombination. Assume that C1

- 10.0 pcF, Cz: 5.00 g,F, andC3 - 4.00 g,F. rlw*'f;S How many 1.00 p,F ca-pacitors must be connected inparallel to store a charge of1.00 C with a potential of 110V across the capacitors?e'E t Each of the unchargedcapacitors in Fig. 25-30 has acapacitance of 25.0 pF. A po-tential difference of V - 4200V is established when theswitch is closed. How manycoulombs of charge then passthrough meter A?ss'EA In Fig. 25-3I, the bat-tery has a potential differenceof V : 10.0 V and the five ca-pacitors each have a capaci-tance of 10.0 p,F, What is thecharge on (u) capacitor I and(b) cap acrtor 2?*otS In Fig. 25-29, a potentialdifference of V - 100.0 V is ap-plied across a capacitor arrange-ment with capacitances Cr :10.0 pF, Cz: 5.00 pE, and Cz:4.00 pF, If capacitor 3 undergoeselectrical breakdown so that itbecomes equivalent to conduct-ing wire, what is the increase in(a) the charge on capacitor Iand (b) the potential difference across capacitor 1?

sen4 Two parallel-plate capacitors,6.0 pF each, are connectedin parallel to a 10 V battery. One of the capacitors is thensqueezed so that its plate separation is 50.0% of its initial value.Because of the squeezing, (a) how much additional charge istransferred to the capacitors by the battery and (b) what is theincrease in the total charge stored on the capacitors?

Ftffi. A5-AS Problems 8 and 36.

F$G" AS-AS Problems 9,13, and 34.

F!ffi. A5-3S Problem 11.

FNG. eS-St Problem12.

FBffi" A5-SS ProblemlT.

Problems

o*t$ A 100 pF capacitor is charged to a potential differenceof 50 V, and the charging battery is disconnected. The capa-citor is then connected in parallel with a second (initiallyuncharged) capacitor. If the potential difference across thefirst capacitor drops to 35 V what is the capacitance of thissecond capacitor? ssM llws&'$& Figure 25-32 shows a circuit section of four air-filledcapacitors that is connected to a larger circuit. The graphbelow the section shows the electric potenti al V(x) as a func-tion of position x along the lower part of the section, throughcapacitor 4. Similarly, the graph above the section shows theelectric potential V(*) as a function of position x along theupper part of the section, through capacitors L, 2, and 3.Capacitor 3 has a capacitance of 0.80 p,F. Wh at are the capaci-tances of (a) capacitor 1 and (b) cap acttor 2?

FfiG" fr5-34 Problem 16.

**tT In Fig. 25-33, a 20JV battery is connected acrosscapacitors of capacitances Cr : Ce : 3.00 g,F and Ct : Cs :2.00C2 : 2.00C4- 4.00 p.F. Whatare (a) the equiv-alent capacitanceC"o of the capaci-tors and (b) thecharge s l-ored byC"q? What are (c)Vl and (d) h ofcapacitor 1 ,,(") V,and (f ) ez of ca-pacitor 2, and (g)

IV-

V3 and (h) q, of capacitor 3?

**tS Plot I in Fig. 25-34a gives the charge q that can be

(J

$:-r

V,v(v)

(a)

Filffi- A$-$4 Problem 18.

(b)

ffi,pt#ir ffS I Capacitance

stored on capacitor I versus the electric potential I/ set upacross it.The vertical scale is set by q, = 16.0 p"C,and the hori-zontal scale is set by V, -2.0 V. Plots 2 and 3 arc similar plotsfor capacitors 2 and 3, respectively. Figure 25-34b shows a cir-cuit with those three capacitors and a 6.0 V battery. What isthe charge stored on capacrtor 2 in that circuit?**S# In Fig. 25-35, the capaci-tances arc Ct: 1.0 g,F and Cz:3.0pr,E, and both capacitors are chargedto a potential difference of V : 100V but with opposite polarity asshown. Switches Sr and 52 are nowclosed. (u) What is now the poten- ff*ffi- #S"#Stial difference between points aand b? What now is the charge on capacitor (b)

equilibrium. Then the switch isthrown to the right. When equi-librium is again reached, how much charge is on capacrtor 1?

e*ffS In Fig. 25-37, two parallel-plate capacitors (with avbetween the plates) are connected to a battery. Capacitor t has aplate area of 1 .5 cr# and an electric field (between its plates) ofmagnitude 2000 V/m. Capacitor 2 has aplate area of 0.70 cm2 and an electricfield of magnitude 1500 V/m. What isthe total charge on the two capacitors?p.p#,# Figure 25-38 represents twoair-filled cylindrical capacitors con-nected in series across a battery withpotential V - 10 V. Capacitor I hasan inner plate radius of 5.0 mm, ?router plate radius of 1-.5 cm, and a Vlength of 5.0 cm. Capacitor 2has an in-ner plate radius of 2.5 mm, zfr outerplate radius of 1.0 cm, and a length of9.0 cm. The outer plate of capacitor 2is a conducting organic membrane thatcan be stretched, and the capacitor can be inflated to increase theplate separation. If the outer plate radius is increased to 2.5 cmby inflation, (a) how many electrons move through point P and(b) do they move toward or away from the battery?se#S In Fig. 25-39, the battery has potential difference V -9.0 Y Cz: 3.0 pF, C+ : 4.0 p.F, and all the capacitors areinitially uncharged.When switch S is closed, a total charge of12 p,C passes through point a and a total charge of 8.0 p,Cpasses through point b. What are (u) Cr and (b) C3?

,F$ffi. ffiS-ffi# Problem23.

s*R4 Figure 25-40 shows a varrable "air gap" capacitor formanual tuning. Alternate plates are connected together; onegroup of plates is fixed in position, and the other group is ca-

pable of rotation. Considera capacitor of n - B platesof alternating polarity, eachplate having area A - I.25cm2 and separated from ad- *""

jacent plates by distance d: 3.40 mm. What is themaximum capacitance ofthe device?

FEffi. #S-4S Problem24.

Problem 19.

1 and (.) 2?

ss#S The capacitors in Fig. 25-4IThe capacitances are Cr : 4.0 pcF,

12 p,F, and the battery's potentialWhen switch S is closed, how manyelectrons travel through (a) pointa, (b) point b, (r) point c, and (d) r,point d? In the figure, do the elec- v

trons travel up or down through (e)point b and (f ) point c?

oo*ff$ Figure 25-42 displays aI2.0 V battery and 3 unchargedcapacitors of capacitances Ct :a.00 g,E, Cz: 6.00 p,F, and Cz :3.00 pE The switch is thrown to theleft side until capacitor I is fullycharged. Then the switch is thrownto the right. What is the finalcharge on (a) capacitor 1, (b) ca-pacitor 2, and (c) capacitor 3?

are initially uncharged.Cz: 8.0 pF, and Cz:difference is V-12V.

F$ffi, tr$-4$ Problem25.

F$ffi. trS"S# Problem}}.

FEffi" AS-S?Problem2I.

ffi$ffi" trs-sffiProblem22.

F'8ffi. #S-4tr Problem26.**wffP Figure 25-43 showsa I2.0Y battery and fouruncharged capacitors of ca-pacitances C1 : 1.00 pF, Cz: 2.00 p,F, Cz:3.00 pE,and C+:4.00 pF, If onlyswitch 51 is closed, what isthe charge on (a) capacitorI, (b) capacitor 2, (.)capacit or 3, and (d) capaci-tor 4? If both switches areclosed, what is the chargeon (") capacitor I, (f ) ca-pacitor 2, (g) capacitor 3, and (h) cap acrtor 4?

owoffS Capacitor 3 in Fig. 25-44a is a variable capacitor (rtscapacitance C3 can be varied).Figure 25-44b gives the electricpotential Vy across capacitor 1 versus C3. The horizontal scaleis set by Cr, -- I2.0 pF. Electric potentialVy approaches an as-ymptote of 10 V as C3 + oo. What are (a) the electric potential7 across the battery, (b) Cr, and (c) C2?

0Cz GtF)(b)

FBffi, nS*44 Problem28.

10

B

.6;4

(a)

828 C H A P T E R 2 6 • Capacitance and Dielectrics

62. A 10.0-!F capacitor is charged to 15.0 V. It is nextconnected in series with an uncharged 5.00-!F capacitor.The series combination is finally connected across a 50.0-Vbattery, as diagrammed in Figure P26.62. Find the newpotential differences across the 5-!F and 10-!F capacitors.

sent the potential difference. (c) Find the direction andmagnitude of the force exerted on the dielectric, assuminga constant potential difference "V. Ignore friction. (d) Obtain a numerical value for the force assuming that! # 5.00 cm, "V # 2 000 V, d # 2.00 mm, and the dielec-tric is glass ($ # 4.50). (Suggestion: The system can be con-sidered as two capacitors connected in parallel.)

65. A capacitor is constructed from two square plates of sides !and separation d, as suggested in Figure P26.64. You mayassume that d is much less than !. The plates carry charges% Q 0 and & Q 0. A block of metal has a width !, a length !,and a thickness slightly less than d. It is inserted a distancex into the capacitor. The charges on the plates are notdisturbed as the block slides in. In a static situation, ametal prevents an electric field from penetrating inside it.The metal can be thought of as a perfect dielectric, with$ : '. (a) Calculate the stored energy as a function of x.(b) Find the direction and magnitude of the force thatacts on the metallic block. (c) The area of the advancingfront face of the block is essentially equal to !d. Consider-ing the force on the block as acting on this face, find thestress (force per area) on it. (d) For comparison, expressthe energy density in the electric field between the capaci-tor plates in terms of Q 0 , !, d , and (0.

66. When considering the energy supply for an automobile,the energy per unit mass of the energy source is an impor-tant parameter. Using the following data, compare theenergy per unit mass ( J/kg) for gasoline, lead–acid batter-ies, and capacitors. (The ampere A will be introduced inthe next chapter as the SI unit of electric current.1 A # 1 C/s.)Gasoline: 126 000 Btu/gal; density # 670 kg/m3.Lead–acid battery: 12.0 V; 100 A ) h; mass # 16.0 kg.Capacitor: potential difference at full charge # 12.0 V;capacitance # 0.100 F; mass # 0.100 kg.

An isolated capacitor of unknown capacitance has beencharged to a potential difference of 100 V. When thecharged capacitor is then connected in parallel to anuncharged 10.0-!F capacitor, the potential differenceacross the combination is 30.0 V. Calculate the unknowncapacitance.

68. To repair a power supply for a stereo amplifier, an elec-tronics technician needs a 100-!F capacitor capable ofwithstanding a potential difference of 90 V between theplates. The only available supply is a box of five 100-!Fcapacitors, each having a maximum voltage capability of50 V. Can the technician substitute a combination of thesecapacitors that has the proper electrical characteristics? Ifso, what will be the maximum voltage across any of thecapacitors used? (Suggestion: The technician may not haveto use all the capacitors in the box.)

A parallel-plate capacitor of plate separation d is chargedto a potential difference "V0. A dielectric slab of thicknessd and dielectric constant $ is introduced between theplates while the battery remains connected to the plates.(a) Show that the ratio of energy stored after the dielectricis introduced to the energy stored in the empty capacitoris U/U0 # $. Give a physical explanation for this increasein stored energy. (b) What happens to the charge on thecapacitor? (Note that this situation is not the same as in

69.

67.

d/2

!/2

d

!

1!2!

3!

Figure P26.61

5.00 Fµ

50.0 V

"Vi = 15.0 V

–+10.0 Fµ

Figure P26.62

xd

!

!

Figure P26.64 Problems 64 and 65.

63. (a) Two spheres have radii a and b and their centers are adistance d apart. Show that the capacitance of this system is

provided that d is large compared with a and b. (Suggestion:Because the spheres are far apart, assume that the poten-tial of each equals the sum of the potentials due to eachsphere, and when calculating those potentials assume thatV # keQ /r applies.) (b) Show that as d approaches infinitythe above result reduces to that of two spherical capacitorsin series.

64. A capacitor is constructed from two square plates of sides !and separation d. A material of dielectric constant $is inserted a distance x into the capacitor, as shown inFigure P26.64. Assume that d is much smaller than x.(a) Find the equivalent capacitance of the device. (b) Cal-culate the energy stored in the capacitor, letting "V repre-

C #4*(0

1a

%1b

&2d

frihmpte,r. ff5 I Capacitance

the plates are separated by 2.0 mm. The capacitor will fail(short out and burn up) if the electric fleld between the platesexceeds 200 kN/C. What is the maximum energy that can bestored in the capacitor?**4,ffi Figure 25-48 shows aparallel-plate capacitor with aplate area A - 5.56 cmz andseparation d : 5.56 mm. The lefthalf of the gap is filled with mate-rial of dielectric constant rc1 :7.00; the right half is filled withmaterial of dielectric constant rc2: IZ.0.What is the capacitance?os4$ Figure 25-49 shows a parallel-plate capacitor with a plate area A -7.89 cm2 and plate separation d - 4.62mm. The top half of the gap is filled withmaterial of dielectric constant Kr : 11.0;the bottom half is fllled with material ofdielectric constant xz : I2.0.What is the capacitance? A /m w,$.ffi Figure 25-50 shows aparallel-plate capacitor of platearea A - 10.5 cmz and plateseparation 2d : I .I2 mm. Theleft half of the gap is filled withmaterial of dielectric constantKL: 2I.0; the top of the righthalf is fllled with material ofdielectric constant K2 : 42.0; the bottom of the right half isfilled with material of dielectric constant rc3 : 58.0. What is thecapacitance?

:sffi,fi," ffiS*,ffi Dielectrics and Gauss' Law8S$ A parallel-plate capacitor has a capacrtance of 100 pF, aplate area of 100 cmz, and a mica dielectric (^ : 5.4) com-pletely filling the space between the plates. At 50 V potentialdifference, calculate (a) the electric fleld magnitude E in themica, (b) the magnitude of the free charge on the plates, and(c) the magnitude of the induced surface charge on the mica.ssM

s$g In Sample Problem 25-7, suppose that the batteryremains connected while the dielectric slab is being intro-duced. Calculate (u) the capacitance, (b) the charge on thecapacitor plates, (c) the electric field in the gap, and (d) theelectric field in the slab, after the slab is in place.**SS The space between two concentric conducting spheri-cal shells of radii b - I.70 cm and a - I.20 cm is fllled witha substance of dielectric constant rc : 23.5. A potential differ-ence V - 73.0 V is applied across the inner and outer shells.Determine (u) the capacitance of the device, (b) the freecharge q on the inner shell, and (.) the charge q' inducedalong the surface of the inner shell.s*.S4 Two parallel plates of area 100 cm2 are given chargesof equal magnitudes 8.9 x l0-7 C but opposite signs. Theelectric field within the dielectric material filling the spacebetween the plates is 1.4 X 106 V/m. (a) Calculate the dielec-tric constant of the material. (b) Determine the magnitude ofthe charge induced on each dielectric surf ace.ssSS A parallel-plate capacrtor has plates of area 0.I2 m2

and a separation of 1 .2 cm. A battery charges the plates toa potential difference of I20 V and is then disconnected.A dielectric slab of thickness 4.0 mm and dielectric constant4.8 is then placed symmetrically between the plates. (a) Whatis the capacitance before the slab is inserted? (b) What is thecapacitance with the slab in place? What is the free charge q(c) before and (d) after the slab is inserted? What is the mag-nitude of the electric field (e) in the space between the platesand dielectric and (f ) in the dielectric itself? (g) With the slabin place, what is the potential difference across the plates?(h) How much external work is involved in inserting the slab?

Additional ProblemsSS The chocolate crumb mystery. This story begins withProblem 56 in Chapter 23. As part of the investigation of thebiscuit factory explosion, the electric potentials of the workerswere measured as they emptied sacks of chocolate crumb pow-der into the loading bin, stirring up a cloud of the powderaround themselves. Each worker had an electric potential ofabout 7.0 kV relative to the ground, which was taken as zeropotential. (u) Assuming that each worker was effectively acapacitor with a typic al capacitance of 200 pF, flnd the energystored in that effectiv e capacitor. If a single spark between theworker and any conducting object connected to the groundneutraltzed the worker, that energy would be transferred to thespark. According to measurements, a spark that could ignite acloud of chocolate crumb powder, and thus set off an explosion,had to have an energy of at least 150 mJ. (b) Could a spark froma worker have set off an explosion in the cloud of powder inthe loading bin? (The story continues with Problem 56 inChapter 26.)

capacltor Z \Lz: b.UU tlt'), ' T Cand capacitor 3 (Cz : 8.00 I $

p.F) connected to a I2.0 V # ;:

battery. When switch S is Ff;ffi. #S-S$ proble m 57 .

closed so as to connectuncharged capacitor 4 (C+:6.00 pF), (u) how much chargepasses through point P from the battery and (b) how muchcharge shows up on capacit or 4? (c) Explain the discrepancyin those two results.S TWo air-filled, parallel-plate capacitors are to be con-nected to a 10 V battery, first individually, then in series, andthen in parallel. In those arrangements, the energy stored inthe capacitors turns out to be, listed least to greatest:75 p.J,100 f,cJ, 300 p.J, and 400 p.J. Of the two capacrtors, what is the(a) smaller and (b) greater capacitance?SW Two parallel-plate capacitors, 6.0 pF each, are connectedin series to a 10 V battery. One of the capacitors is thensqueezed so that its plate separation is halved. Because of thesqueezing, (a) how much additionalcharge is transferred to the capaci-tors by the battery and (b) what isthe increase in the total chargestored on the capacrtors (the chargeon the positive plate of one capaci-tor plus the charge on the positiveplate of the other capacitor)?S# In Fig. 25-52, V - IzV, Ct:Cs : Ce : 6.0 frF, and Cz: Cz :

Figure 25-5I shows ca- I

itor I (Ct : 8.00 lrF),rrgure L)-)r SIrOWS Ca- tsl

itor I (ct:8.00 lrF), t J c4capacitor 2 (Cz:6.00 pF), V T,1

Ffiffi. #S-4ffi Problem 48.

TldIlld

]L

T2dI

'F.$ffi" #S-4#Problem 49.

F$ffi. trS-S# Problem 50.

Ff;ffi. HS-ffi# Problem 60.















TldIlld

]L

T2dI


















TldIlld

]L

T2dI




838 C H A P T E R 2 6 Capacitance and Dielectrics

76. Determine the effective capacitance of the combinationshown in Figure P26.76. (Hint: Consider the symmetryinvolved!)

pacitors are disconnected from the battery and fromeach other. They are then connected positive plate tonegative plate and negative plate to positive plate. Cal-culate the resulting charge on each capacitor.

73. The inner conductor of a coaxial cable has a radius of0.800 mm, and the outer conductor’s inside radius is3.00 mm. The space between the conductors is filledwith polyethylene, which has a dielectric constant of2.30 and a dielectric strength of 18.0 ! 106 V/m. Whatis the maximum potential difference that this cable canwithstand?

74. You are optimizing coaxial cable design for a majormanufacturer. Show that for a given outer conductor ra-dius b, maximum potential difference capability is at-tained when the radius of the inner conductor is

where e is the base of natural logarithms.75. Calculate the equivalent capacitance between the points

a and b in Figure P26.75. Note that this is not a simpleseries or parallel combination. (Hint: Assume a poten-tial difference "V between points a and b. Write expres-sions for "Vab in terms of the charges and capacitancesfor the various possible pathways from a to b, and re-quire conservation of charge for those capacitor platesthat are connected to each other.)

a # b/e

72. Capacitors and are chargedas a parallel combination across a 250-V battery. The ca-

C2 # 2.00 $FC1 # 6.00 $F

71. A vertical parallel-plate capacitor is half filled with a di-electric for which the dielectric constant is 2.00 (Fig.P26.71a). When this capacitor is positioned horizon-tally, what fraction of it should be filled with the samedielectric (Fig. P26.71b) so that the two capacitors haveequal capacitance?

of thickness d and dielectric constant % is introducedbetween the plates while the battery remains connected to theplates. (a) Show that the ratio of energy stored after thedielectric is introduced to the energy stored in theempty capacitor is Give a physical explana-tion for this increase in stored energy. (b) What hap-pens to the charge on the capacitor? (Note that this sit-uation is not the same as Example 26.7, in which thebattery was removed from the circuit before the dielec-tric was introduced.)

70. A parallel-plate capacitor with plates of area A and plateseparation d has the region between the plates filledwith two dielectric materials as in Figure P26.70. As-sume that and that (a) Determine thecapacitance and (b) show that when %1 # %2 # % yourresult becomes the same as that for a capacitor contain-ing a single dielectric, C # %&0A/d.

d V W.d V L

U/U0 # %.

C

C

3C

2C

2C

a

b2.00 µF

4.00 µF

2.00 µF 4.00 µF8.00 µF

µ

µ µ

µ

µ

(b)(a)

d!1

!2

LW

Figure P26.70

Figure P26.71

Figure P26.76

Figure P26.75


76. Determine the effective capacitance of the combinationshown in Figure P26.76. (Hint: Consider the symmetryinvolved!)

pacitors are disconnected from the battery and fromeach other. They are then connected positive plate tonegative plate and negative plate to positive plate. Cal-culate the resulting charge on each capacitor.

73. The inner conductor of a coaxial cable has a radius of0.800 mm, and the outer conductor’s inside radius is3.00 mm. The space between the conductors is filledwith polyethylene, which has a dielectric constant of2.30 and a dielectric strength of 18.0 ! 106 V/m. Whatis the maximum potential difference that this cable canwithstand?

74. You are optimizing coaxial cable design for a majormanufacturer. Show that for a given outer conductor ra-dius b, maximum potential difference capability is at-tained when the radius of the inner conductor is

where e is the base of natural logarithms.75. Calculate the equivalent capacitance between the points

a and b in Figure P26.75. Note that this is not a simpleseries or parallel combination. (Hint: Assume a poten-tial difference "V between points a and b. Write expres-sions for "Vab in terms of the charges and capacitancesfor the various possible pathways from a to b, and re-quire conservation of charge for those capacitor platesthat are connected to each other.)

a # b/e

72. Capacitors and are chargedas a parallel combination across a 250-V battery. The ca-

C2 # 2.00 $FC1 # 6.00 $F

71. A vertical parallel-plate capacitor is half filled with a di-electric for which the dielectric constant is 2.00 (Fig.P26.71a). When this capacitor is positioned horizon-tally, what fraction of it should be filled with the samedielectric (Fig. P26.71b) so that the two capacitors haveequal capacitance?

of thickness d and dielectric constant % is introducedbetween the plates while the battery remains connected to theplates. (a) Show that the ratio of energy stored after thedielectric is introduced to the energy stored in theempty capacitor is Give a physical explana-tion for this increase in stored energy. (b) What hap-pens to the charge on the capacitor? (Note that this sit-uation is not the same as Example 26.7, in which thebattery was removed from the circuit before the dielec-tric was introduced.)

70. A parallel-plate capacitor with plates of area A and plateseparation d has the region between the plates filledwith two dielectric materials as in Figure P26.70. As-sume that and that (a) Determine thecapacitance and (b) show that when %1 # %2 # % yourresult becomes the same as that for a capacitor contain-ing a single dielectric, C # %&0A/d.

d V W.d V L

U/U0 # %.

C

C

3C

2C

2C

a

b2.00 µF

4.00 µF

2.00 µF 4.00 µF8.00 µF

µ

µ µ

µ

µ

(b)(a)

d!1

!2

LW

Figure P26.70

Figure P26.71

Figure P26.76

Figure P26.75


fine wire along its axis. Suppose that the internal diame-ter of the cylinder is 2.50 cm and that the wire along theaxis has a diameter of 0.200 mm. If the dielectricstrength of the gas between the central wire and thecylinder is 1.20 ! 106 V/m, calculate the maximumvoltage that can be applied between the wire and thecylinder before breakdown occurs in the gas.

53. The general form of Gauss’s law describes how a chargecreates an electric field in a material, as well as in a vac-uum. It is

where is the permittivity of the material. (a) A sheet with charge Q uniformly distributed over its area A is surrounded by a dielectric. Show that thesheet creates a uniform electric field with magnitude

at nearby points. (b) Two large sheets ofarea A carrying opposite charges of equal magnitude Qare a small distance d apart. Show that they create a uni-form electric field of magnitude betweenthem. (c) Assume that the negative plate is at zero po-tential. Show that the positive plate is at a potentialQd /A". (d) Show that the capacitance of the pair ofplates is

ADDITIONAL PROBLEMS54. For the system of capacitors shown in Figure P26.54,

find (a) the equivalent capacitance of the system, (b) the potential difference across each capacitor, (c) the charge on each capacitor, and (d) the total energy stored by the group.

A"/d # $A"0/d .

E # Q /A"

E # Q /2A"

" # $"0

! E ! dA #q"

56. A 2.00-nF parallel-plate capacitor is charged to an initialpotential difference and then isolated. Thedielectric material between the plates is mica ($ #5.00). (a) How much work is required to withdraw themica sheet? (b) What is the potential difference of thecapacitor after the mica is withdrawn?

57. A parallel-plate capacitor is constructed using a dielec-tric material whose dielectric constant is 3.00 and whosedielectric strength is 2.00 ! 108 V/m. The desired ca-pacitance is 0.250 %F, and the capacitor must withstanda maximum potential difference of 4 000 V. Find theminimum area of the capacitor plates.

58. A parallel-plate capacitor is constructed using threedielectric materials, as shown in Figure P26.58. You mayassume that ! d. (a) Find an expression for the ca-pacitance of the device in terms of the plate area A andd , $1 , $2 , and $3 . (b) Calculate the capacitance usingthe values cm2, mm, $1 # 4.90, $2 #5.60, and $3 # 2.10.

d # 2.00A # 1.00

W

&Vi # 100 V

60. (a) Two spheres have radii a and b and their centers area distance d apart. Show that the capacitance of this sys-tem is

provided that d is large compared with a and b. (Hint:Because the spheres are far apart, assume that the

C "4'"0

1a

(1b

)2d

59. A conducting slab of thickness d and area A is insertedinto the space between the plates of a parallel-plate ca-pacitor with spacing s and surface area A, as shown inFigure P26.59. The slab is not necessarily halfway be-tween the capacitor plates. What is the capacitance ofthe system?

55. Consider two long, parallel, and oppositely chargedwires of radius d with their centers separated by a dis-tance D. Assuming the charge is distributed uniformlyon the surface of each wire, show that the capacitanceper unit length of this pair of wires is

C!

#'"0

ln# D ) dd $

A

A

ds

dd/2

!/2

!

!2

!3

!1

!

!!

4.00 µF2.00 µF

6.00 µF3.00 µF

90.0 V

µ µ

µ µ

Figure P26.54

Figure P26.58

Figure P26.59

WEB

C1 836 CHAPTER 26 Capacitance and Dielectrics · PDF file · 2013-08-15second capacitor? ssM llw s&'$& Figure 25-32 shows a circuit section of four air-filled ......

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