1

Chapter 25

Capacitance & Dielectrics

Prof. Raymond Lee,revised 2-4-2011

2

• Capacitors

• Capacitors are devices that store q

• Examples of capacitors’ uses:

• radio receivers

• filters in power supplies

• energy-storing devices in electronic flashes

3

• Definition of capacitance

• Define capacitance C as ratio of |Q| on

either conductor:!V between conductors, or

C = Q/!V (compare Eq. 25-1, p. 657)

• SI capacitance unit is farad (F), where

1 F = 1 C/V (C = coulomb, not capacitance C)

4

• Capacitor’s makeup

• Capacitor consists of 2conductors called plates

• When conductor ischarged, plates carrycharges of = magnitude &opposite signs

• Potential difference !Vexists between platesdue to charge

(compare Fig. 25-2, p. 656)

5

• More about capacitance

• Capacitance C is always a +quantity &

is constant for a given capacitor

• C measures capacitor’s ability to store

charge

• Farad is a very large unit; more typical

are microfarads (µF) & picofarads (pF)

6

• Parallel-plate capacitor

• Each plate is connectedto a battery terminal

• If capacitor is initiallyuncharged, batteryestablishes E-field inconnecting wires

(compareFig. 25-4,p. 658)

7

• Parallel-plate capacitor, 2

• This E-field exerts a force Fe on e(electrons) in wire just outside the plates

• Fe causes e to move onto the –plate

• This continues until plate, wire, & terminalare at same V, & so equilibrium then exists

• Now E-field no longer exists in wire, &movement of e ceases

8

• Parallel-plate capacitor, 3

• Right-hand plate is now –charged (Fig. 25-4)

• Similar process occurs on left-hand plate,with e moving away from it & thus giving ita +charge

• In capacitor’s final (charged) state, !Vacross capacitor plates = !V across batteryterminals

9

• C for isolated sphere

• Assume (1) a solid spherical charged

conductor of radius R & (2) V = 0 at !:

C = Q/!V = Q/(keQ/R) = R/ke = 4"#0R

(Eq. 25-18, p. 661) since ke = 1/(4"#0)

• N.B.: Result is independent of Q & !V

10

• C for parallel plates

• Charge density on plates is ! = Q/A

• A = each plate’s (equal) area

• Q = charge on each plate, equal with

opposite signs

• E-field is uniform between plates & = 0

elsewhere

11

• C for parallel plates, 2

• C $: area A of each plate, &

1/(distance d between plates)

• Now, since between plates we have

E = %/#0 = Q/(#0A) (see Eq. 23-14), then

• C = Q/!V = Q/(Ed) = Q/[Qd/(#0A)]

= #0A/d (Eq. 25-9, p. 660)

12

• Parallel-plate assumptions

• Uniform E-field assumption is valid in centerof plates, but not at their ends

• If plate separation is small w.r.t. plate length,can ignore effects of this non-uniform E-field

(SJ 2004 Fig. 26.3,p. 799)

13

• C for cylindrical capacitor

• From Gauss’ law, E-field

between cylinders is Er = 2ke"/r

• &V = – 'Erdr = –2ke " ln(b/a)

• Then C = Q/!V = l/[2ke ln(b/a)](compare Eq. 25-14, p. 661) aftersubstituting " = Q/l

b

a

(compare Fig. 25-6, p. 660)

14

• C for spherical capacitor

• Potential difference is

where b & a are radii of

outer & inner spheres

• C = Q/!V = ab/(ke(b - a))

(SJ 2008 Ex. 26.3, p. 726)

15

• Circuit symbols

• Circuit diagram is a simplified

picture of actual circuit

• Use circuit symbols to represent

various elements

• Lines represent wires

• Indicate battery’s +terminal by

using the longer line

16

• Capacitors in parallel

• When capacitors first connected, eare moved from L plate, throughbattery to R plate, leaving L plate+charged & right plate –charged

(SJ 2008Fig. 26.7,p. 728)

17

• Capacitors in parallel, 2

• Flow of e ends when !V across capacitors =!V across battery

• Capacitors reach maximum Q when e flow ends

• Total Q = sum of charges on capacitors, orQtotal = Q1 + Q2

• !V across each capacitor is same & = battery !V

18

• Capacitors in parallel, 3

• Can replace capacitors with

1 capacitor having Ceq,

where this equivalent

capacitor has exactly same

external effect on circuit as

do original capacitors

(compareFig. 25-8,p. 663)

19

• Capacitors in parallel, 4

• Ceq = C1 + C2 + … (compare Eq. 25-19, p. 663)

• Equivalent capacitance Ceq of || combination

of capacitors > C of any 1 capacitor

• Essentially, capacitor areas are combined

20

• Capacitors in series

• When battery is connected

to circuit, e are moved from

C1’s L plate to C2’s R plate

through battery

• As –charge accumulates on

C2’s R plate, an = amount

of –charge is removed from

C2’s L plate, leaving it with

excess +charge (SJ 2008 Fig. 26.8, p. 729)

21

• Capacitors in series, 2

• All R plates gain charges that

= –Q & all L plates have

charges that = +Q

• An equivalent capacitor

exists that functions just as

series combination does

• Sum of capacitors’ !Vs =

battery !V

(compareFig. 25-9,p. 664)

22

• Capacitors in series, 3

• Since &Vtot = V1 + V2 + V3 +… = Q/Ceq

& since corresponding Q1=Q2=Q3= ...,

• then

• Ceq of series combination is always

< smallest individual C in the series

(compare Eq. 25-20, p. 664)

23

• C problem-solving hints

• Be careful with units choice:• In SI, C in F, distance in m, & &V in V

• E-fields can be in V/m or N/C

• If connect " 2 capacitors in parallel,&Vs across them are the same

• Q on each capacitor $ its C

• Capacitances add directly to give Ceq

24

• C problem-solving hints, 2

• If connect " 2 capacitors in series, they

all carry same Q, but &Vs across them

aren’t the same

• Add &Vs just as for batteries in series

• Capacitances add as reciprocals

• Ceq is always < smallest individual C

25

• Ceq example

• 1.0-µF & 3.0-µF, 6.0-µF & 2.0-µF capacitors are in ||

• These || combinations are in series with adjacent capacitors

• In turn, these series combinations are in ||, so now cancalculate final Ceq

(SJ 2008Ex. 26.3,p. 730)

26

• Some uses of capacitors

• Defibrillators• When fibrillation occurs, heart produces a rapid,

irregular beat pattern

• A fast discharge of electrical energy through heartcan return it to its normal beat pattern

• In general, capacitors act as energyreservoirs that can be slowly charged & thenquickly discharged ( large amounts of short-duration energy

27

• Capacitor energy

• Consider circuit as a system

• Before switch is closed, energyis stored as chemical PE inbattery

• When switch is closed, energyis transformed from chemical toelectric PE

(SJ 2008Fig. 26.10,

p. 731)

28

• Capacitor energy, 2

• Electric PE is related to separation of

the + & –charges on plates

• Thus can describe a capacitor as a

device that stores both charge & energy

29

• Capacitor energy storage

• Assume capacitor is being charged &,

at some point, has charge q on it

• Work W needed to transfer dq between

plates is dW = &Vdq = (q/C)dq (p. 667)

• Total W required is

(Eq. 25-21, p. 667)

30

• Capacitor energy storage, 2

• Work done in charging capacitor appears aselectric PE U:

• True for capacitor of any geometry

• So energy stored ) as either Q ) or &V )

• In practice, there’s a maximum &V beyondwhich discharge occurs between plates

(Eq. 25-22,p. 667)

31

• Capacitor energy storage, 3

• Consider U as stored in plates’ E-field

• For ||-plate capacitor, U = (1/2)#oAdE 2

• Or write U in terms of energy density(energy/volume) as uE = (1/2)#oE

2

(Eq. 25-25, p. 668) for volume betweencapacitor plates = A*d

• uE equation holds for any E-field, not justthat from capacitors

32

• Capacitor dielectrics

• Dielectric: insulator that, if placedbetween capacitor’s plates, increases C

• Include rubber, plastic, & waxed paper

• For ||-plate capacitor, C = *C0 = *#0A/d(Eq. 25-27, p. 670)

• Multiply C by factor * when dielectriccompletely fills region between plates

33

• Capacitor dielectrics, 2

• In principle, very small d ( very large C

• But in practice, lower limits exist on d• d is limited by electric discharge that could occur

though dielectric separating the plates

• For given d, maximum V that can be appliedto a capacitor without ( discharge dependson material’s dielectric strength

34

• Capacitor dielectrics, 3

• Dielectrics give following advantages:

• Increase in C

• Increase maximum operating V

• Possible mechanical support betweenplates, allowing smaller d without touching,thus increasing C

35

(compare Table 25-1, p. 669)

?

36

Date: 1 Feb 2006

From: Prof John Fontanella

Subject: Re: Dielectric strength of water

Even pure water contains ions since water dissociates into protons and

hydroxyl ions: i.e., pure water is a weak electrolyte. The resulting

conductivity is too high for water to be classified as a dielectric.

Also, the breakdown voltage of most materials is determined by the material’s

defect structure. What usually happens is that trapped or defect electrons are

accelerated. Those electrons eject other electrons, etc. hence the breakdown.

Consequently, dielectric breakdown is only indirectly related to the nature of

the atoms making up the material.

John Fontanella

Physics Department

U. S. Naval Academy

37

• Capacitor types – tubular

• Can interlace metallic foil with

thin paper sheets or Mylar

• Roll layers into a cylinder to

form small-volume capacitor

(SJ 2008Fig. 26.15,

p. 737)

38

• Capacitor types – oil-filled

• Common for high-V

capacitors

• Immerse several

interwoven metallic

plates in silicone oil

(SJ 2008Fig. 26.15,

p. 737)

39

• Capacitor types – electrolytic

• Used to store large Q at relatively low V

• Electrolyte is solution that conductselectricity via ion motion within solution

• If V is applied, then dielectric forms onfoil as thin layer of insulating metal oxide

• Has fixed polarity — if applied V is ofwrong polarity, then oxide layer isremoved & capacitor ( conductor

(SJ 2008Fig. 26.15,

p. 737)

40

• Capacitor types – variable

• Variable capacitors consist of 2interlaced sets of metallic plates

• 1 plate is fixed & other ismovable

• Such capacitors generally have10 pF < C < 500 pF, & are usedin some radio tuning circuits

(SJ 2008Fig. 26.16,

p. 737)

41

Dielectrics – An atomic view

• Molecules that make

up dielectric are

modeled as dipoles

• Molecules are

randomly oriented in

absence of E-field

42

Dielectrics – An atomic view, 2

• Applying an external E-

field produces a torque on

molecules

• Dielectric’s molecules

partially align with E-field

• Molecules’ degree of alignment withE-field depends on |E| & temperature T

• In general, alignment ) as: T+ & |E|)

43

Dielectrics – An atomic view, 3

• If dielectric’s molecules are nonpolar,

then E-field produces some Q separation

• This ( an induced dipole moment

• So net effect is as if molecules were polar

44

Dielectrics – An atomic view, 4

• So external E-field can polarize

dielectric whether its molecules

are polar or nonpolar

• Dielectric’s charged edges act as

2nd pair of plates, producing an

induced E-field opposite to

external E-field’s direction

45

Induced charge & external E-field

• E-field due to plates points to

right & it polarizes dielectric

• Net effect on dielectric is an

induced surface charge that( an induced E-field

• If dielectric were replaced with

conductor, then net E-field

between plates would = 0

46

Geometry of some capacitors