Electricity and Magnetism Capacitance Capacitors in Series and Parallel

Home , Capacitance

Lana Sheridan

De Anza College

Jan 30, 2018 Last time

• Van de Graaﬀ generator

• capacitors

• capacitance

• capacitors of diﬀerent shapes Warm Up Question

True or false: A component (ﬁxed) capacitor has a capacitance, even when storing no charge.

(A) true (B) false Warm Up Question

True or false: A component (ﬁxed) capacitor has a capacitance, even when storing no charge.

(A) true ← (B) false Overview

• Parallel plate capacitors

• capacitors of diﬀerent shapes

• Circuits and circuit diagrams

• Capacitors in series and parallel Capacitance Capacitors with diﬀerent construction will have diﬀerent values of C.

For example, 0A for a parallel plate capacitor: C = d . for a cylinderical capacitor of length L, inner radius a and outer radius b: L C = 2π 0 ln(b/a)

for a spherical capacitor of inner radius a and outer radius b: ab C = 4π 0 b − a

for an isolated charged sphere of radius R:

C = 4π0R Q C = |∆V | Assuming the field inside the capacitor is uniform throughout, it is given by the expression for the field inside infinite planes of charge: σ E = 0 Replace Q = σA and using we have: σA C = |∆V |

Parallel Plate Capacitor Back to the parallel plate capacitor:

A C = 0 d

Let’s justify why this expression should hold. Parallel Plate Capacitor Back to the parallel plate capacitor:

A C = 0 d

Let’s justify why this expression should hold.

Q C = |∆V | Assuming the field inside the capacitor is uniform throughout, it is given by the expression for the field inside infinite planes of charge: σ E = 0 Replace Q = σA and using we have: σA C = |∆V | Parallel Plate Capacitor

Also: ∆V = − E · ds = −Ed Z so: Q σA C = = |∆V | Ed Using our value for E: σA C = (σ/0)d

Conﬁrms that A C = 0 d 780 Chapter 26 Capacitance and Dielectrics

Substituting this result into Equation 26.1, we find that the capacitance is Q Q C 5 5 DV Qd/P0A

Capacitance of parallel plates P0A C 5 (26.3) d That is, the capacitance of a parallel-plate capacitor is proportional to the area of its plates and inversely proportional to the plate separation. Let’s consider how the geometry of these conductors influences the capacity of the pair of plates to store charge. As a capacitor is being charged by a battery, electrons flow into the negative plate and out of the positive plate. If the capacitor plates are large, the accumulated charges are able to distribute themselves over a substantial area and the amount of charge that can be stored on a plate for a given potential difference increases as the plate area is increased. Therefore, it is reasonable that the capacitance is proportional to the plate area A as in Equation 26.3. Now consider the region that separates the plates. Imagine moving the plates closer together. Consider the situation before any charges have had a chance to move in response to this change. Because no charges have moved, the electric field between the plates has the same value but extends over a shorter distance. There- fore, the magnitude of the potential difference between the plates DV 5 Ed (Eq. 25.6) is smaller. The difference between this new capacitor voltage and the terminal voltage of the battery appears as a potential difference across the wires connecting the battery to the capacitor, resulting in an electric field in the wires that drives more charge onto the plates and increases the potential difference between the B plates. When the potential difference between the plates again matches that of the Key battery, the flow of charge stops. Therefore, moving the plates closer together causes the chargeCylindrical on the capacitor Capacitor to increase. If d is increased, the charge decreases. As a result, the inverse relationship between C and d in Equation 26.3 is reasonable. Movable plate Q uick Quiz For26.2 a Manycylinderical computer keyboardcapacitor buttons of length are constructed`, much of greatercapacitors than inner Insulator as shown inradius Figurea 26.3.and When outer a key radius is pushedb: down, the soft insulator between Fixed plate the movable plate and the fixed plate is compressed. When the key is pressed, what happens to the capacitance? (a) It increases.` (b) It decreases.` (c) It changes Figure 26.3 (Quick Quiz 26.2) C = 2π = One type of computer keyboard in a way you cannot determine because the0 electric circuit connected to the key- ln(b/a) 2ke ln(b/a) button. board button may cause a change in DV.

ϪQ Example 26.1 The Cylindrical Capacitor b a A solid cylindrical conductor of radius a and charge Q Q is coaxial with a cylindrical shell of negligible thick- ness, radius b . a, and charge 2Q (Fig. 26.4a). Find the b capacitance of this cylindrical capacitor if its length ᐉ a is ,. Q ϪQ r SOLUTIO N

Conceptualize Recall that any pair of conductors qualifies as a capacitor, so the system described in this example therefore qualifies. Figure 26.4b helps visual- Gaussian ize the electric field between the conductors. We expect surface the capacitance to depend only on geometric factors, a b which, in this case, are a, b, and ,. Figure 26.4 (Example 26.1) (a) A cylindrical capacitor consists of a solid cylindrical conductor of radius a and length , sur- Categorize Because of the cylindrical symmetry of the rounded by a coaxial cylindrical shell of radius b. (b) End view. system, we can use results from previous studies of cylin- The electric field lines are radial. The dashed line represents the drical systems to find the capacitance. end of a cylindrical gaussian surface of radius r and length ,. Let λ = Q/` be the charge per unit length. a |∆V | = Va − Vb = − E · ds Zb a 2k λ = − e dr r Zb b = 2k λ ln e a Capacitance, Q Q C = = ∆V b 2ke λ ln a 2π0` C = b ln a

Cylindrical Capacitor Idea: First ﬁnd ∆V across the plates, assuming charge Q, then evaluate Q/|∆V |. Capacitance, Q Q C = = ∆V b 2ke λ ln a 2π0` C = b ln a

Cylindrical Capacitor Idea: First ﬁnd ∆V across the plates, assuming charge Q, then evaluate Q/|∆V |.

Let λ = Q/` be the charge per unit length. a |∆V | = Va − Vb = − E · ds Zb a 2k λ = − e dr r Zb b = 2k λ ln e a Cylindrical Capacitor Idea: First ﬁnd ∆V across the plates, assuming charge Q, then evaluate Q/|∆V |.

Let λ = Q/` be the charge per unit length. a |∆V | = Va − Vb = − E · ds Zb a 2k λ = − e dr r Zb b = 2k λ ln e a Capacitance, Q Q C = = ∆V b 2ke λ ln a 2π0` C = b ln a Spherical Capacitor 782 Chapter 26 Capacitance and Dielectrics For a spherical capacitor of inner radius a and outer radius b: ab ab C = 4π0 = b − a ke (b − a) ▸ 26.2 continued Categorize Because of the spherical symmetry of the sys- ϪQ tem, we can use results from previous studies of spherical Figure 26.5 (Example 26.2) systems to find the capacitance. A spherical capacitor consists of an inner sphere of radius a sur- ϩQ a Analyze As shown in Chapter 24, the direction of the rounded by a concentric spherical electric field outside a spherically symmetric charge shell of radius b. The electric field b distribution is radial and its magnitude is given by the between the spheres is directed expression E 5 k Q /r 2. In this case, this result applies to radially outward when the inner e sphere is positively charged. the field between the spheres (a , r , b). b S Write an expression for the potential difference between 2 52 ? S Vb Va 3 E d s the two conductors from Equation 25.3: a b b dr 1 b Apply the result of Example 24.3 for the electric field V 2 V 52 E dr 52k Q 5 k Q b a 3 r e 3 2 e r outside a spherically symmetric charge distribution a a r a S S c d and note that E is parallel to d s along a radial line: 1 1 a 2 b (1) V 2 V 5 k Q 2 5 k Q b a e b a e ab a b Q Q ab Substitute the absolute value of DV into Equation 26.1: C 5 5 5 (26.6) DV Vb 2 Va ke b 2 a 0 0 1 2 Finalize The capacitance depends on a and b as expected. The potential difference between the spheres in Equation (1) is negative because Q is positive and b . a. Therefore, in Equation 26.6, when we take the absolute value, we change a 2 b to b 2 a. The result is a positive number.

WHAT IF ? If the radius b of the outer sphere approaches infinity, what does the capacitance become? Answer In Equation 26.6, we let b S `: ab ab a C 5 lim 5 5 5 4pP0a b S ` ke b 2 a ke b ke Notice that this expression is the same as Equation1 26.2,2 the capacitance1 2 of an isolated spherical conductor.

26.3 Combinations of Capacitors Two or more capacitors often are combined in electric circuits. We can calculate the equivalent capacitance of certain combinations using methods described in Capacitor this section. Throughout this section, we assume the capacitors to be combined are symbol initially uncharged. In studying electric circuits, we use a simplified pictorial representation called a Battery ϩ circuit diagram. Such a diagram uses circuit symbols to represent various circuit symbol Ϫ elements. The circuit symbols are connected by straight lines that represent the wires between the circuit elements. The circuit symbols for capacitors, batteries, and switches as well as the color codes used for them in this text are given in Fig- Switch Open ure 26.6. The symbol for the capacitor reflects the geometry of the most common symbol model for a capacitor, a pair of parallel plates. The positive terminal of the battery Closed is at the higher potential and is represented in the circuit symbol by the longer line. Figure 26.6 Circuit symbols for capacitors, batteries, and switches. Parallel Combination Notice that capacitors are in Two capacitors connected as shown in Figure 26.7a are known as a parallel combi- blue, batteries are in green, and switches are in red. The closed nation of capacitors. Figure 26.7b shows a circuit diagram for this combination of switch can carry current, whereas capacitors. The left plates of the capacitors are connected to the positive terminal of the open one cannot. the battery by a conducting wire and are therefore both at the same electric potential Capacitance, Q C = ∆V ab C = ke (b − a)

Spherical Capacitor

a |∆V | = Va − Vb = − E · ds Zb a k Q = − e dr r 2 Zb 1 1 = k Q − e a b b − a = k Q e ab Spherical Capacitor

a |∆V | = Va − Vb = − E · ds Zb a k Q = − e dr r 2 Zb 1 1 = k Q − e a b b − a = k Q e ab

Capacitance, Q C = ∆V ab C = ke (b − a) Capacitance of Isolated Conductors

Isolated conductors on their own (not part of a pair) can also be said to have a capacitance.

The “other plate” is taken to be inﬁnitely far away.

The capacitance is found by dividing the charge on the conductor by it’s electric potential, taking V ( ) = 0

∞ Capacitance of and Isolated Spherical Conductor

For an isolated charged sphere of radius R: R C = 4π0R = ke

Two ways to argue this,

(1) set a = R and take b → in C = ab , or ke (b−a)

ke Q (2) recall that the potential of∞ a sphere of charge Q is V = R Circuits

Circuits consist of electrical components connected by wires.

Some types of components: batteries, resistors, capacitors, lightbulbs, LEDs, diodes, inductors, transistors, chips, etc.

The wires in circuits can be thought of as channels for an electric ﬁeld that distributes charge to (or charge ﬂow through) the components. halliday_c25_656-681v2.qxd 23-11-2009 14:32 Page 658

Circuits HALLIDAY REVISED

The diﬀerent elements can be combined together in various ways to make complete circuits: paths for charge to ﬂow from one terminal of a battery or power supply to the other. 658 CHAPTER 25 CAPACITANCE

l Terminal C h h l C + V B – –+ B S S Terminal (a) (b) This circuit is said to be incomplete while the switch is open. Fig. 25-4 (a) Battery B, switch S, and plates h and l of capacitor C, connected in a circuit. (b) A schematic diagram with the circuit elements represented by their symbols. that maintains a certain potential difference between its terminals (points at which charge can enter or leave the battery) by means of internal electrochemi- cal reactions in which electric forces can move internal charge. In Fig. 25-4a,a battery B,a switch S,an uncharged capacitor C,and inter- connecting wires form a circuit. The same circuit is shown in the schematic diagram of Fig. 25-4b,in which the symbols for a battery,a switch,and a capacitor represent those devices. The battery maintains potential difference V between its terminals. The terminal of higher potential is labeled ϩ and is often called the positive terminal; the terminal of lower potential is labeled Ϫ and is often called the negative terminal. The circuit shown in Figs. 25-4a and b is said to be incomplete because switch S is open; that is, the switch does not electrically connect the wires attached to it.When the switch is closed, electrically connecting those wires, the circuit is complete and charge can then flow through the switch and the wires. As we discussed in Chapter 21, the charge that can flow through a conductor, such as a wire, is that of electrons. When the circuit of Fig. 25-4 is completed, electrons are driven through the wires by an electric field that the battery sets up in the wires. The field drives electrons from capacitor plate h to the positive terminal of the battery; thus, plate h,losing electrons,becomes positively charged. The field drives just as many electrons from the negative terminal of the battery to capacitor plate l;thus,plate l,gaining electrons,becomes nega- tively charged just as much as plate h,losing electrons,becomes positively charged. Initially, when the plates are uncharged, the potential difference between them is zero. As the plates become oppositely charged, that potential difference increases until it equals the potential difference V between the terminals of the battery. Then plate h and the positive terminal of the battery are at the same potential, and there is no longer an electric field in the wire between them. Similarly, plate l and the negative terminal reach the same potential, and there is then no electric field in the wire between them. Thus, with the field zero, there is no further drive of electrons. The capacitor is then said to be fully charged, with a potential difference V and charge q that are related by Eq. 25-1. In this book we assume that during the charging of a capacitor and after- ward, charge cannot pass from one plate to the other across the gap separating them. Also, we assume that a capacitor can retain (or store) charge indeﬁnitely, until it is put into a circuit where it can be discharged.

CHECKPOINT 1 Does the capacitance C of a capacitor increase,decrease,or remain the same (a) when the charge q on it is doubled and (b) when the potential difference V across it is tripled? 782 Chapter 26 Capacitance and Dielectrics

▸ 26.2 continued Categorize Because of the spherical symmetry of the sys- ϪQ tem, we can use results from previous studies of spherical Figure 26.5 (Example 26.2) systems to find the capacitance. A spherical capacitor consists of an inner sphere of radius a sur- ϩQ a Analyze As shown in Chapter 24, the direction of the rounded by a concentric spherical electric field outside a spherically symmetric charge shell of radius b. The electric field b distribution is radial and its magnitude is given by the between the spheres is directed expression E 5 k Q /r 2. In this case, this result applies to radially outward when the inner e sphere is positively charged. 820 the field Cbetweenhapter 27 the C spheresurrent and ( aR esistance, r , b). b S Write an expression for the potential difference between 2 52 ? S 782 Chapter 26 Capacitance and DielectricsTable 27.3 CriticalVb TemperaturesVa 3 E d s a the two conductors from Equation 25.3: for Various Superconductors

▸ 26.2 continued Material Tc (K) b b dr 1 b ApplyCategorize the result Because of Example of the spherical 24.3 forsymmetry the electric ofHgBa the sys 2fieldCa- 2Cu 3O8 2 52134 52 Ϫ5Q Vb Va 3 Er dr ke Q 3 2 ke Q Tl—Ba—Ca—Cu—O 125a a r r a outsidetem, wea sphericallycan use results symmetric from previous charge studies distribution of spherical Figure 26.5 (Example 26.2) systems to findS the capacitance. S Bi—Sr—Ca—Cu—O 105 c d and note that E is parallel to d s along a radial line: A spherical capacitor consists of 1 1 a 2 b YBa2Cu3O7 an inner sphere of radius a sur- 92 ϩQ a (1) Vb 2 Va 5 ke Q 2 5 ke Q Analyze As shown in Chapter 24, the directionNb of Ge the rounded by a concentric spherical23.2 782 Chapter 26 Capacitance and Dielectrics 3 b a ab electric field outside a spherically symmetric Nb chargeSn shell of radius b. The electric field18.05 3 a b b distribution is radial and its magnitude is givenNb by the between the spheres is directed 9.46

Courtesy of IBM Research Laboratory Q Q ▸ 26.2 continued 2 radially outward when the inner ab Substituteexpression the E 5absolute keQ /r . Invalue this case,of D Vthis into result Equation appliesPb to 26.1: 5 5 7.18 5 sphere is positivelyC charged. Ϫ (26.6) A small permanentCategorize magnet levi- Because of the spherical, symmetry, of the sys- DV V 2 QV k b 2 a the fieldtem, we between can use the results spheres from previous (a studiesr b). of spherical Hg b4.15 a e tated above a disk of the super- Figure 26.5 (Example 26.2) systems to find the capacitance. Sn A spherical capacitor consists of b 3.72 conductor YBa Cu O , which is in S ϩQ a Write2 3 an7 expression for the potential difference betweenAl an inner sphere ofV radius2 Va sur52- E ? d Ss 0 1.19 0 1 2 liquid nitrogenFinalize at 77Analyze TheK. Ascapacitance shown in Chapter depends 24, the direction on a of and the b roundedas expected. by a concentricb sphericalThea potential3 difference between the spheres in Equation the twoelectric conductors field outside from a spherically Equation symmetric 25.3: charge Zn shell of radius b. The electric field a 0.88 b (1) is negativedistribution because is radial and Q its is magnitude positive is given and by b the. a. betweenTherefore, the spheres isin directed Equation 26.6, when we take the absolute value, we change expression E 5 k Q /r 2. In this case, this result applies to radially outward when the inner b b b e sphere is positively charged. dr 1 a 2Apply b tothe bthe field2 result a.between The of the Example resultspheres ( ais ,24.3 ar ,positive bfor). the electric number field. V 2 V 52 E dr 52k Q 5 k Q Today, thousands of superconductorsb a are known,3 r and e as3 Table2 27.3e rillustrates, b a a r a outside a spherically symmetric charge distribution S WHAT WriteIF ? an expressionIf Sthe radius for the potential b of the difference outer between sphere approaches2 52 ?infinity,S what does the capacitance become? and note that E is parallelthe to criticald Ss along temperatures a radial line: V bof Vrecentlya 3 E discoveredd s superconductors are substantiallyc d the two conductors from Equation 25.3: a 1 1 a 2 b higher than initially thought possible.(1) V b Two2 Va 5 kindske Q of2 superconductors5 ke Q are recog- Answer In Equation 26.6, we let b `: b b b a ab nized. The Smore recently identified ones are dressentially1 b ceramics with high criti- Apply the result of Example 24.3 for the electric field V 2 V 52 E dr 52k Q 5 k Q b aab 3 r abe 3 r 2 a ae r b outside a spherically symmetriccal temperatures,charge distribution whereas superconductinga Q a materialsQ sucha ab as those observed by S S D C 5 lim 5 5 5 c4pd P0a 980 SubstituteChapterand note the32 that absolute EInductance is parallel value to d s of along V ainto radial Equation line: S26.1: C 5 5 5 (26.6) Kamerlingh-Onnes areb `metals.ke b 2 If a room-temperature1ke 1b ka e2 b superconductor is ever iden- (1) Vb 2 Va 5 ke Q DV2 5Vbke2Q Va ke b 2 a tified, its effect on technology could beb tremendous.a ab Notice that this expression is the same as Equation 26.2, thea capacitanceb of an isolated spherical conductor. The value of T is sensitive1Q to chemicalQ2 1 2ab composition,0 0 1 pressure,2 and molecular Substitute the absolute value of DV into Equation 26.1:c C 5 5 5 (26.6) ▸ 32.5 continuedFinalize The capacitance depends on a and b as expected.D The 2potential difference2 between the spheres in Equation structure. Copper, silver, andV gold,Vb whichVa k eareb aexcellent conductors, do not exhibit (1) is negative because Q is positive and b . a. Therefore, in Equation 26.6, when we take the absolute value, we change superconductivity. 0 0 1 2 SOLUTI O N a 2 bFinalize to b 2 The a. Thecapacitance result depends is a positive on a and number b as expected.. The potential difference between the spheres in Equation (1) is negative because Q is positiveOne and truly b . a . remarkableTherefore, in Equation feature 26.6, whenof superconductors we take the absolute value, is wethat change once a current is set up WHAT IF ? If the radius b of the outer sphere approaches infinity, what does the capacitance become? Conceptualize Bea sure 2 b to byou 2 a. Thecan result identify isin a positivethem, the number it two 26.3persists. coils without inCombinations the any situation applied andpotential understand ofdifference Capacitors that(because a changing R 5 0). currentSteady cur in -one coil induces a currentAnswerWHA T In inIF Equation? theIf the second radius 26.6, rentsb of coil.we the let outerhave b S sphere been`: approaches observed infinity, to whatpersist does the in capacitance superconducting become? loops for several years with Answer In Equation 26.6,no we letapparent b S `: decay! ab ab a TwoC 5 orlimab more ab capacitorsa5 5often5 4arepP0a combined in electric circuits. We can calculate Categorize We will determine the result using conceptsb S ` discussed in this section, so we categorize this example as a An importantC 5 lim andk 5eusefulb 2 a5 application5k4epPb0a k eof superconductivity is in the development b S ` k b 2 a k b k substitution problem. of superconductingthe equivalente magnets,e capacitancee in which the of magnitudes certain combinations of the magnetic using field methods are described in NoticeNotice that that this this expression expression is theis the same same as Equation as Equation 26.2, the1 capacitance26.2,2 the ofcapacitance an1 2isolated spherical of an conductor.isolated spherical conductor. Capacitor approximatelythis section. ten1 times2 Throughout1 2 greater than this those section, produced we assume by the the best capacitors normal elec to- be combined are Circuit component symbols NB Use Equation 30.17symbol to express the magnetictromagnets. initiallyfield Such in the superconductinguncharged. B 5m magnets i are being considered as a means of 0 , interior of the base solenoid: storing energy. In Superconductingstudying electric magnets circuits, are we currently use a simplified used in medical pictorial magnetic representation called a Combinations of Capacitors resonance26.3circuit imaging, diagram. or MRI, Suchunits, awhich diagramF produce uses high-quality circuit symbols images toof internalrepresent various circuit Battery ϩ 26.3 Combinations NofH CapacitorsBH N H BA NBNH Find the mutual inductance, notingorgans thatTwo the withoutor more magnetic capacitors the need often for are excessivecombinedM 5 in exposure electric 5circuits. of patientsWe can5 calculate mto0 x-rays orA other harm- battery symbol∆V Ϫ the equivalentelements. capacitance The of certain circuit combinations symbolsi using are methods connectedi described in by , straight lines that represent the flux F through the handle’s coilful caused radiation. by the mag- BH Capacitor this Twosection.wires or Throughout more between capacitors this section, the often we circuit assume are thecombined elements. capacitors toin be Theelectric combined circuit circuits. are symbols We can forcalculate capacitors, batteries, netic fieldThe ofdirection the base of thesymbol coil is BA: initiallytheand uncharged. equivalent switches capacitance as well as of the certain color combinations codes used using for methods them in described this text in are given in Fig- effective flowCapacitor of positive Inthis studying section. electric Throughout circuits, we use this a simplified section, pictorial we assume representation the capacitors called a to be combined are circuit diagram. Such a diagram uses circuit symbols to represent various circuit Wirelesscapacitorcharge charging is clockwise.Switch Csymbol is usedBattery inOpen ϩa number ofinitially otherure 26.6.“cordless”uncharged. The devices.symbol Onefor the significant capacitor example reflects is the inductivegeometry charging of the most common symbol Ϫ 27.6elements. Electrical The circuit symbols Power are connected by straight lines that represent the used by some manufacturerssymbol of electricwires cars betweenInmodel that studying theavoids for circuit electric a direct elements.capacitor, circuits, metal-to-metal The circuita we pair use symbols ofa simplified parallel forcontact capacitors, pictorialbetweenplates. batteries, Therepresentation the positivecar and thecalledterminal charg a - of the battery I and switches as well as the color codes used for them in this text are given in Fig- ing apparatus. Battery Closedϩ In typicalcircuit iselectric at diagram. the circuits, higher Such energypotential a diagram TET and isuses transferred iscircuit represented symbols by electrical toin represent the transmission circuit various symbol circuitfrom by the longer line. Switch Open ure elements.26.6. The symbol The for circuit the capacitor symbols reflects are the connected geometry of the by most straight common lines that represent the switch Ssymbolsymbol Ϫ a sourcemodel forsuch a capacitor, as a battery a pair of parallelto some plates. device The positive such terminal as a oflightbulb the battery or a radio receiver. Figure 26.6 CircuitClosed symbolsLet’s for is determine atwires the higher between potential an expression the and circuitis represented that elements. inwill the allowcircuit The symbol us circuit to by calculate the symbols longer line. the for capacitors,rate of this batteries, energy capacitors, batteries, and switches. b Figure 26.6 Circuitc symbols fortransfer. and First,Parallel switches consider as Combination well the as simple the color circuit codes in used Figure for 27.11,them inwhere this textenergy are isgiven delivered in Fig - ϩ Notice thatcapacitors,Switch capacitors batteries,Open andare switches. in Parallelure 26.6.Combination The symbol for the capacitor reflects the geometry of the most common Notice that capacitors are in to a resistor.Two (Resistors capacitors are designatedconnected by as the shown circuit in symbol Figure 26.7a are.) Because known theas a parallel combi- ⌬V R Two capacitors connected as shown in Figure 26.7a are known as a parallel combi- resistorblue, batteriesblue, Rsymbol batteries are arein ingreen, green, and and model for a capacitor, a pair of parallel plates. The positive terminal of the battery Ϫ connectingnation of capacitors.wires also Figure have 26.7b resistance,shows a circuit somediagram energyfor this combination is delivered of to the wires and switches areswitches in are red. in red. TheClosed The closed32.5 isOscillations atnation the higher of capacitors.potential in and Figurean is represented LC 26.7b Circuit in shows the circuit a circuit symbol diagram by the longer for line.this combination of a switch can carry current,d whereassome capacitors. to the Theresistor. left plates Unless of the capacitors noted areotherwise, connected to we the shallpositive assume terminal of the resistance of the switch canthe carryopen one current, cannot. whereas the batterycapacitors. by a conducting The wire andleft are plates therefore of both the at the capacitors same electric potential are connected to the positive terminal of ϩ Figure 26.6 Circuit symbolsWhen for a capacitor is connected to an inductor as illustrated in Figure 32.10, the C the open one cannot. wires is smallthe comparedbattery by with a conducting the resistance wire of andthe circuitare therefore element bothso that at the the energy same electric potential Ϫ capacitors, batteries,L and switches. Parallel Combination inductorQ L combinationdelivered to isthe an wires LC iscircuit. negligible. If the capacitor is initially charged and the switch is max Notice that capacitors are in Two capacitors connected as shown in Figure 26.7a are known as a parallel combi- blue, batteries are in green,then and Imagine closed, followingboth the acurrent positive in quantity the circuit of charge and Qthe moving charge clockwise on the aroundcapacitor the oscil - Figure 27.11 A circuit consist- nation of capacitors. Figure 26.7b shows a circuit diagram for this combination of switches are in red. The closedlatecircuit between in Figure maximum 27.11 from positive point a and through negative the battery values. and If resistor the resistance back to point of the a. cir- ing of a resistorswitch of canresistance carry current, R whereas capacitors. The left plates of the capacitors are connected to the positive terminal of S We identify the entire circuit as our system. As the charge moves from a to b through and a batterythe having open onea potential cannot. cuit is zero,the batteryno energy by a conducting is transformed wire and to are internal therefore energy. both at the In same the electricfollowing potential analysis, difference DV across its terminals. the battery, the electric potential energy of the system increases by an amount Q DV Figure 32.10 A simple LC cir- the resistance in the circuit is neglected. We also assume an idealized situation in cuit. The capacitor has an initial which energy is not radiated away from the circuit. This radiation mechanism is charge Q max, and the switch is discussed in Chapter 34. open for t , 0 and then closed at Assume the capacitor has an initial charge Q (the maximum charge) and t 5 0. max the switch is open for t , 0 and then closed at t 5 0. Let’s investigate what happens from an energy viewpoint. When the capacitor is fully charged, the energy U in the circuit is stored in 2 the capacitor’s electric field and is equal to Q max/2C (Eq. 26.11). At this time, the current in the circuit is zero; therefore, no energy is stored in the inductor. After the switch is closed, the rate at which charges leave or enter the capacitor plates (which is also the rate at which the charge on the capacitor changes) is equal to the current in the circuit. After the switch is closed and the capacitor begins to discharge, the energy stored in its electric field decreases. The capacitor’s discharge represents a current in the circuit, and some energy is now stored in the magnetic field of the inductor. Therefore, energy is transferred from the electric field of the capacitor to the magnetic field of the inductor. When the capacitor is fully discharged, it stores no energy. At this time, the current reaches its maximum value and all the energy in the circuit is stored in the inductor. The current continues in the same direction, decreasing in magnitude, with the capacitor eventually becoming fully charged again but with the polarity of its plates now opposite the initial polarity. This process is followed by another discharge until

the circuit returns to its original state of maximum charge Q max and the plate polarity shown in Figure 32.10. The energy continues to oscillate between inductor and capacitor. The oscillations of the LC circuit are an electromagnetic analog to the mechanical oscillations of the particle in simple harmonic motion studied in Chapter 15. Much of what was discussed there is applicable to LC oscillations. For example, we investigated the effect of driving a mechanical oscillator with an external force, 782 Chapter 26 Capacitance and Dielectrics

b S Write an expression for the potential difference between 2 52 ? S Vb Va 3 E d s the two conductors from Equation 25.3: a

b b dr 1 b Apply the result of Example 24.3 for the electric field V 2 V 52 E dr 52k Q 5 k Q b a 3 r e 3 2 e r outside a spherically symmetric charge distribution a a r a S S c d and note that E is parallel to d s along a radial line: 1 1 a 2 b (1) V 2 V 5 k Q 2 5 k Q b a e b a e ab a b Q Q ab Substitute the absolute value of DV into Equation 26.1: C 5 5 5 (26.6) DV Vb 2 Va ke b 2 a 0 0 1 2 Finalize The capacitance depends on a and b as expected. The potential difference between the spheres in Equation (1) is negative because Q is positive and b . a. Therefore, in Equation 26.6, when we take the absolute value, we change a 2 b to b 2 a. The result is a positive number.

26.3 Combinations of Capacitors Two or more capacitors often are combined in electric circuits. We can calculate the equivalent capacitance of certain combinations using methods described in CapacitorCircuits: Batteries this section. Throughout this section, we assume the capacitors to be combined are symbol initially uncharged. In studying electric circuits, we use a simplified pictorial representation called a Battery ϩ circuit diagram. Such a diagram uses circuit symbols to represent various circuit symbol Ϫ elements. The circuit symbols are connected by straight lines that represent the wires between the circuit elements. The circuit symbols for capacitors, batteries, Batteries cause a potential difference between two partsand of the switches as well as the color codes used for them in this text are given in Fig- circuit. Switch Open ure 26.6. The symbol for the capacitor reflects the geometry of the most common symbolThis can drive a charge flow. (Current is the rate of flow of charge.) model for a capacitor, a pair of parallel plates. The positive terminal of the battery Closed is at the higher potential and is represented in the circuit symbol by the longer line. Figure 26.6 Circuit symbols for capacitors, batteries, and switches. Parallel Combination Notice that capacitors are in Two capacitors connected as shown in Figure 26.7a are known as a parallel combi- blue, batteries are in green, and switches are in red. The closed nation of capacitors. Figure 26.7b shows a circuit diagram for this combination of switch can carry current, whereas capacitors. The left plates of the capacitors are connected to the positive terminal of the open one cannot. the battery by a conducting wire and are therefore both at the same electric potential Series and Parallel

Series Parallel When components are When components are connected one after the other connected side-by-side on along a single path, they are diﬀerent paths, they are connected in series. connected in parallel.

R1 V V R1 R2

R2 What would be the capacitance of this equivalent capacitor?

halliday_c25_656-681v2.qxd 23-11-2009 14:32 Page 663

HALLIDAY REVISED

PART 3 25-4 CAPACITORS IN PARALLEL AND IN SERIES 663 halliday_c25_656-681v2.qxd 23-11-2009 14:32 Page 663 Terminal When a potential difference V is appliedHALLIDAY across several REVISED capacitors connected in parallel, that potential difference V is applied across each capacitor.The total charge +q +q +q + 3 2 1 q stored on the capacitors is the sum of the charges stored on all the capacitors. B V V V V – –q –q –q Capacitors in Parallel 3 C3 2 C2 1 C1

When we analyze a circuit of capacitors in parallel,Capacitors we in can parallel simplify all have it the withsame potential(a) PART differenceTerminal 3 across them. this mental replacement:25-4 CAPACITORS IN PARALLEL AND IN SERIES 663 Parallel capacitors and Three capacitors in parallel: their equivalent have Equivalent circuit: the same V (“par-V”). Capacitors connected in parallel can be replaced with an equivalentTerminal capacitor that When a potential difference V is appliedhas across the sameseveral total capacitorscharge q connectedand the same in potential difference V as the actual parallel, that potential difference V is applied across each capacitor.The total charge +q +q +q +q capacitors. + 3 2 1 + q stored on the capacitors is the sum of the charges stored on all the capacitors. B V V V V B – V – –q –q –q –q 3 C3 2 C2 1 C1 Ceq (You might remember this result with the nonsense word “par-V,” which is close (b) When we analyze a circuit of capacitors in parallel, we can simplify it with (a) Terminal to “party,” to mean “capacitors in parallel have the same V.”) Figure 25-8b shows this mental replacement: Parallel capacitors andFig. 25-8 (a) Three capacitors connected We could replace all three capacitors in the circuit with one the equivalent capacitor (with equivalent capacitance Ceq) that has replaced the in parallel to battery B.The battery main- equivalent capacitance.their equivalent The current have and potential difference in the three capacitors (with actual capacitances C1, C2,and C3) of Fig. 25-8a. tains potential difference V across its termi- rest of the circuitthe is unchanged same V (“par-V”). by this. Capacitors connected in parallel can be replacedTo derive with anan equivalentexpression capacitor for Ceq inthat Fig. 25-8b,we first use Eq.25-1 to find the nals and thus across each capacitor. (b) The has the same total charge q and the samecharge potential on differenceeach actual V ascapacitor: the actual equivalent capacitor, with capacitance C , +q eq capacitors. + replaces the parallel combination. q ϭ C V, q ϭ C V,andqB ϭ VC V. 1 1 2 2 3– 3 –q Ceq The total charge on the parallel combination of Fig. 25-8a is then (You might remember this result with the nonsense word “par-V,” which is close (b) to “party,” to mean “capacitors in parallel have the same Vq.”)ϭ Figureq ϩ q 25-8ϩ qb showsϭ (C ϩ C ϩ C )V. 1 2 3 1 Fig.2 25-83 (a) Three capacitors connected the equivalent capacitor (with equivalent capacitance C ) that has replaced the The equivalent capacitance,eq with the same totalin charge parallel q toand battery applied B.The potential battery main- three capacitors (with actual capacitances C , C ,and C ) of Fig. 25-8a. difference1 2V as the3 combination, is then tains potential difference V across its termi- To derive an expression for Ceq in Fig. 25-8b,we first use Eq.25-1 to find the nals and thus across each capacitor. (b) The charge on each actual capacitor: q equivalent capacitor, with capacitance C , C ϭ ϭ C ϩ C ϩ C , eq eq V 1 2 replaces3 the parallel combination. q1 ϭ C1V, q2 ϭ C2V,andq3 ϭ C3V. a result that we can easily extend to any number n of capacitors, as The total charge on the parallel combination of Fig. 25-8a is then n q ϭ q1 ϩ q2 ϩ q3 ϭ (C1 ϩ C2 ϩ C3)V. Ceq ϭ ͚ Cj (n capacitors in parallel). (25-19) The equivalent capacitance, with the same total charge q and appliedjϭ1 potential difference V as the combination, is then Thus, to find the equivalent capacitance of a parallel combination, we simply add q the individual capacitances. C ϭ ϭ C ϩ C ϩ C , eq V 1 2 3 a result that we can easily extend to anyCapacitors number n inof Seriescapacitors, as n Figure 25-9a shows three capacitors connected in series to battery B.This description Ceq ϭ ͚ Cj has(n capacitorslittle to do in parallel).with how the capacitors(25-19) are drawn. Rather,“in series” means that the jϭ1 capacitors are wired serially, one after the other, and that a potential difference V is Thus, to find the equivalent capacitanceapplied of a acrossparallel the combination, two ends of the we simplyseries. (In add Fig. 25-9a,this potential difference V is the individual capacitances. maintained by battery B.) The potential differences that then exist across the capacitors in the series produce identical charges q on them.

Capacitors in Series When a potential difference V is applied across several capacitors connected in series, the capacitors have identical charge q.The sum of the potential differences Figure 25-9a shows three capacitors connectedacross all in theseries capacitorsto battery is equal B.This to the description applied potential difference V. has little to do with how the capacitors are drawn. Rather,“in series” means that the capacitors are wired serially, one after the other, and that a potential difference V is applied across the two ends of the series. (InWe Fig. can 25-9 explaina,this how potential the capacitors difference end V is up with identical charge by following maintained by battery B.) The potentiala differenceschain reaction that ofthen events, exist across in which the capaci- the charging of each capacitor causes the tors in the series produce identical chargescharging q on them. of the next capacitor. We start with capacitor 3 and work upward to capacitor 1. When the battery is ﬁrst connected to the series of capacitors, it

When a potential difference V is applied across several capacitors connected in series, the capacitors have identical charge q.The sum of the potential differences across all the capacitors is equal to the applied potential difference V.

We can explain how the capacitors end up with identical charge by following a chain reaction of events, in which the charging of each capacitor causes the charging of the next capacitor. We start with capacitor 3 and work upward to capacitor 1. When the battery is ﬁrst connected to the series of capacitors, it halliday_c25_656-681v2.qxd 23-11-2009 14:32 Page 663

HALLIDAY REVISED

When we analyze a circuit of capacitors in parallel,Capacitors we in can parallel simplify all have it the withsame potential(a) PART differenceTerminal 3 across them. this mental replacement:25-4 CAPACITORS IN PARALLEL AND IN SERIES 663 Parallel capacitors and Three capacitors in parallel: their equivalent have Equivalent circuit: the same V (“par-V”). Capacitors connected in parallel can be replaced with an equivalentTerminal capacitor that When a potential difference V is appliedhas across the sameseveral total capacitorscharge q connectedand the same in potential difference V as the actual parallel, that potential difference V is applied across each capacitor.The total charge +q +q +q +q capacitors. + 3 2 1 + q stored on the capacitors is the sum of the charges stored on all the capacitors. B V V V V B – V – –q –q –q –q 3 C3 2 C2 1 C1 Ceq (You might remember this result with the nonsense word “par-V,” which is close (b) When we analyze a circuit of capacitors in parallel, we can simplify it with (a) Terminal to “party,” to mean “capacitors in parallel have the same V.”) Figure 25-8b shows this mental replacement: Parallel capacitors andFig. 25-8 (a) Three capacitors connected We could replace all three capacitors in the circuit with one the equivalent capacitor (with equivalent capacitance Ceq) that has replaced the in parallel to battery B.The battery main- equivalent capacitance.their equivalent The current have and potential difference in the three capacitors (with actual capacitances C1, C2,and C3) of Fig. 25-8a. tains potential difference V across its termi- rest of the circuitthe is unchanged same V (“par-V”). by this. Capacitors connected in parallel can be replacedTo derive with anan equivalentexpression capacitor for Ceq inthat Fig. 25-8b,we first use Eq.25-1 to find the nals and thus across each capacitor. (b) The has the same total charge q and the samecharge potential on differenceeach actual V ascapacitor: the actual What would be the capacitance of this equivalentequivalent capacitor? capacitor, with capacitance C , +q eq capacitors. + replaces the parallel combination. q ϭ C V, q ϭ C V,andqB ϭ VC V. 1 1 2 2 3– 3 –q Ceq The total charge on the parallel combination of Fig. 25-8a is then (You might remember this result with the nonsense word “par-V,” which is close (b) to “party,” to mean “capacitors in parallel have the same Vq.”)ϭ Figureq ϩ q 25-8ϩ qb showsϭ (C ϩ C ϩ C )V. 1 2 3 1 Fig.2 25-83 (a) Three capacitors connected the equivalent capacitor (with equivalent capacitance C ) that has replaced the The equivalent capacitance,eq with the same totalin charge parallel q toand battery applied B.The potential battery main- three capacitors (with actual capacitances C , C ,and C ) of Fig. 25-8a. difference1 2V as the3 combination, is then tains potential difference V across its termi- To derive an expression for Ceq in Fig. 25-8b,we first use Eq.25-1 to find the nals and thus across each capacitor. (b) The charge on each actual capacitor: q equivalent capacitor, with capacitance C , C ϭ ϭ C ϩ C ϩ C , eq eq V 1 2 replaces3 the parallel combination. q1 ϭ C1V, q2 ϭ C2V,andq3 ϭ C3V. a result that we can easily extend to any number n of capacitors, as The total charge on the parallel combination of Fig. 25-8a is then n q ϭ q1 ϩ q2 ϩ q3 ϭ (C1 ϩ C2 ϩ C3)V. Ceq ϭ ͚ Cj (n capacitors in parallel). (25-19) The equivalent capacitance, with the same total charge q and appliedjϭ1 potential difference V as the combination, is then Thus, to find the equivalent capacitance of a parallel combination, we simply add q the individual capacitances. C ϭ ϭ C ϩ C ϩ C , eq V 1 2 3 a result that we can easily extend to anyCapacitors number n inof Seriescapacitors, as n Figure 25-9a shows three capacitors connected in series to battery B.This description Ceq ϭ ͚ Cj has(n capacitorslittle to do in parallel).with how the capacitors(25-19) are drawn. Rather,“in series” means that the jϭ1 capacitors are wired serially, one after the other, and that a potential difference V is Thus, to find the equivalent capacitanceapplied of a acrossparallel the combination, two ends of the we simplyseries. (In add Fig. 25-9a,this potential difference V is the individual capacitances. maintained by battery B.) The potential differences that then exist across the capacitors in the series produce identical charges q on them.

Capacitors in parallel all have the same potential diﬀerence across them.

∆V1 = ∆V2 = ∆V3 = ∆V

The total charge on the three capacitors is the sum of the charge on each.

qnet = q1 + q2 + q3

where q1 = C1 ∆V .

Capacitance is C = q/(∆V ): q C = net eq ∆V So in general, for any number n of capacitors in parallel, the eﬀective capacitance of them all together is:

n Ceq = C1 + C2 + ... + Cn = Ci i=1 X

Capacitors in Parallel

Equivalent capacitance: q C = net eq ∆V q q q = 1 + 2 + 3 ∆V ∆V ∆V

= C1 + C2 + C3 Capacitors in Parallel

Equivalent capacitance: q C = net eq ∆V q q q = 1 + 2 + 3 ∆V ∆V ∆V

= C1 + C2 + C3

So in general, for any number n of capacitors in parallel, the eﬀective capacitance of them all together is:

n Ceq = C1 + C2 + ... + Cn = Ci i=1 X halliday_c25_656-681v2.qxd 23-11-2009 14:32 Page 664

HALLIDAY REVISED

664 CHAPTER 25 CAPACITANCE

Terminal produces charge Ϫq on the bottom plate of capacitor 3. That charge then repels negative charge from the top plate of capacitor 3 (leaving it with charge ϩq).The halliday_c25_656-681v2.qxd 23-11-2009 14:32 Page 664+q repelled negative charge moves to the bottom plate of capacitor 2 (giving it V 1 charge Ϫq). That charge on the bottom plate of capacitor 2 then repels negative –q C1 HALLIDAY REVISED charge from the top plate of capacitor 2 (leaving it with charge ϩq) to the bottom +q + plate of capacitor 1 (giving it charge Ϫq). Finally, the charge on the bottom plate V Capacitors in Series B – V 2 of capacitor 1 helps move negative charge from the top plate of capacitor 1 to the –q C2 Capacitors in series all store the same charge. battery, leaving that top plate with charge ϩq. 664 CHAPTER 25 CAPACITANCE +q Here are two important points about capacitors in series: Three capacitors in series: V3 –q C3 1. When charge is shifted from one capacitor to another in a series of capacitors, Terminal produces charge Ϫq on the bottom plateit ofcan capacitor move along3. That only charge one then route, repels such as from capacitor 3 to capacitor 2 in Terminalnegative charge from the top plate of capacitorFig. 25-9 3 (leavinga.If there it with are additionalcharge ϩq).The routes,the capacitors are not in series. +q Series capacitors and Equivalent circuit:(arepelled) negative charge moves to the bottom plate of capacitor 2 (giving it V1 2. The battery directly produces charges on only the two plates to which it is charge Ϫtheiq). Thatr equivale chargent onhave the bottom plate of capacitor 2 then repels negative –q C1 charge fromthe thesam tope q plate (“se rofi-q” capacitor). 2 (leavingconnected it with (the charge bottom ϩq )plate to the of bottom capacitor 3 and the top plate of capacitor 1 in +q +q Fig. 25-9a). Charges that are produced on the other plates are due merely to + + plate of capacitor 1 (giving it charge Ϫq). Finally, the charge on the bottom plate V B – V 2 B V of capacitor 1 helps move negative chargethe from shifting the top of plate charge of capacitor already there.1 to the For example, in Fig. 25-9a,the part of the –q C – –q 2 battery,Ceqleaving that top plate with chargecircuit ϩq. enclosed by dashed lines is electrically isolated from the rest of the +q Here are two important points aboutcircuit. capacitors Thus, in series: the net charge of that part cannot be changed by the battery— V3 (b) its charge can only be redistributed. –q C3 1. When charge is shifted from one capacitor to another in a series of capacitors, Fig. 25-9 (ita )can Three move capacitors along only con- one route, suchWhen as from we capacitoranalyze a3 circuitto capacitor of capacitors 2 in in series, we can simplify it with this nected in series to battery B.The battery Terminal Series capacitors and Fig. 25-9a.If there are additional routes,themental replacement: capacitors are not in series. maintains potential difference V between (a) their equivalent have 2. The battery directly produces charges on only the two plates to which it is the top and bottom plates of the series the same q (“seri-q”). connected (the bottom plate of capacitor 3 and the top plate of capacitor 1 in combination. (b)The equivalent capacitor, Capacitors that are connected in series can be replaced with an equivalent capacitor that +q Fig. 25-9a). Charges that are produced on the other plates are due merely to + with capacitance C ,replaces the series has the same charge q and the same total potential difference V as the actual series capacitors. B V the shiftingeq of charge already there. For example, in Fig. 25-9a,the part of the – –q combination. Ceq circuit enclosed by dashed lines is electrically isolated from the rest of the circuit. Thus, the net charge of that(You part mightcannot remember be changed this by thewith battery— the nonsense word “seri-q” to mean “capacitors (b) its charge can only be redistributed. in series have the same q.”) Figure 25-9b shows the equivalent capacitor (with Fig. 25-9 (a) Three capacitors con- When we analyze a circuit of capacitors in series, we can simplifyC it with this nected in series to battery B.The battery equivalent capacitance eq) that has replaced the three actual capacitors mental replacement: maintains potential difference V between (with actual capacitances C1, C2,and C3) of Fig. 25-9a. the top and bottom plates of the series To derive an expression for Ceq in Fig. 25-9b,we first use Eq.25-1 to find the combination. (b)The equivalent capacitor, Capacitors that are connected in series canpotential be replaced difference with an equivalent of each capacitor actual that capacitor: with capacitance Ceq,replaces the series has the same charge q and the same total potential difference V as the actual series capacitors. combination. q q q V ϭ , V ϭ , and V ϭ . 1 C 2 C 3 C (You might remember this with the nonsense word “seri-q” to mean1 “capacitors2 3 in series have the same q.”) Figure 25-9Theb totalshows potential the equivalent difference capacitor V due (with to the battery is the sum of these three equivalent capacitance Ceq) that haspotential replaced differences.Thus, the three actual capacitors (with actual capacitances C1, C2,and C3) of Fig. 25-9a. To derive an expression for Ceq in Fig. 25-9b,we first use Eq.25-1 to find the 1 1 1 V ϭ V1 ϩ V2 ϩ V3 ϭ q ϩ ϩ . potential difference of each actual capacitor: ΂ C1 C2 C3 ΃ q Theq equivalent capacitanceq is then V1 ϭ , V2 ϭ , and V3 ϭ . C1 C2 C3 q 1 The total potential difference V due to the battery is the sumCeq ofϭ theseϭ three , V 1/C ϩ 1/C ϩ 1/C potential differences.Thus, 1 2 3 1 1 1 V ϭ V1 ϩ V2 ϩ V3 ϭ q ϩ ϩ . 1 1 1 1 or ΂ C1 C2 C3 ΃ ϭ ϩ ϩ . Ceq C1 C2 C3 CHECKPOINTThe equivalent 3 capacitance is then We can easily extend this to any number n of capacitors as A battery of potential V stores charge qq 1 C ϭ ϭ , on a combination of two identicaleq ca- V 1/C1 ϩ 1/C2 ϩ 1/C3 1 n 1 pacitors.What are the potential differ- ϭ ͚ (n capacitors in series). (25-20) ence across and the charge on either ca- Ceq jϭ1 Cj pacitor if the capacitors are (a) in1 1 1 1 or ϭ Usingϩ Eq.ϩ 25-20. you can show that the equivalent capacitance of a series of parallel and (b) in series? Ceq C1 C2 C3 CHECKPOINT 3 capacitances is always less than the least capacitance in the series. We can easily extend this to any number n of capacitors as A battery of potential V stores charge q on a combination of two identical ca- 1 n 1 pacitors.What are the potential differ- ϭ ͚ (n capacitors in series). (25-20) ence across and the charge on either ca- Ceq jϭ1 Cj pacitor if the capacitors are (a) in parallel and (b) in series? Using Eq. 25-20 you can show that the equivalent capacitance of a series of capacitances is always less than the least capacitance in the series. Summary

• parallel plate capacitors • capacitors of diﬀerent shapes • circuits, circuit diagrams • capacitors in parallel

Homework Serway & Jewett: • PREVIOUS: Ch 26, onward from page 799. Problems: 1, 5, 7, 11, 51 • NEW: Ch 26. Problems: 13