Capacitance and the Storage of Electric Energy

CAPACITANCE and The Storage of Electric EnergyElectric Energy

PARALLEL PLATEPARALLEL PLATE CAPACITORS

Outline Objectives

1. Capacitors and Capacitance2 Th C bi i f

At the end of this chapter, you should be able to:

2. The Combination of Capacitors3. Energy Storage in

1. Explain how electrostatic energy is stored;2. Define capacitance;3. Energy Storage in

Capacitors4. Capacitors and

3. Define a dielectric and explain how dielectrics affect the energy stored in a capacitor; andDielectrics capacitor; and4. Solve problems involving capacitors.

Chapter TwoChapter TwoChapter TwoChapter Two

The First of ThreeThe First of ThreeThe First of ThreeThe First of ThreeThe Capacitor is one of the three simple circuit elements that can be connected with wires to form an electric circuit!

Capacitors have variety of uses, ranging y g gfrom: radio fine tuner circuits to camera flashes to defibrillatorsflashes to defibrillators

ceceCapacitor ◦ is a system composed

itanc

itanc

y pof two conductors (plates) with equal and opposite charges

apac

apac

pp gon them!

ΔV

of C

of C

ΔV ◦ exists between the

plates because of the f ld b h

tion

tion field between them

Question???:

efin

itef

init Question???:◦ What determines the

amount of charge Q th l t t

Experiment shows that the amount Q is

proportional to ΔV!

1. D

1. D

on the plates at a given V?

ee D fi iti S ta

nce

tanc

e Definition of C Unit Some

Notes

apac

itap

acit

Amount of Charge Stored in a given

PotentialThe Farad Always positive

of C

aof

Ca Potential

Does not actually

ion

oio

n o

C = Q/ΔV 1 F = 1C/1V

ydepend on the

charge or potential, but on the geometry.

efin

itef

init

The Farad is a large unit!

C is Proportionality

Constant

I. D

eI.

De

Initially there are no charges at the plates.

k?k?After making the connection, charging happens.

Wor

kW

ork

Let us focus on the negative plate◦ Charging Stops if the wire plate

tors

Wto

rs W Charging Stops if the wire, plate,

and terminal are all at the same potential!

paci

tpa

cit The positive plate also

experiences a similar phenomenon!

w C

aw

Ca

In the final configuration the potential difference between the plates is the same as the

How

How

the plates is the same as the battery!

A h Q l A capacitor stores charge Q at a potential difference ΔV. If the voltage applied by a battery to the capacitor is doubled to 2ΔV, battery to the capacitor is doubled to 2ΔV,

(a) the capacitance falls to half its initial value

2.1

2.1

( ) pand the charge remains the same (b) the capacitance and the charge both fall to half their initial values

oint

2oi

nt 2 to half their initial values

(c) the capacitance and the charge both double

eckp

oec

kpo (d) the capacitance remains the same and the charge doubles.

Che

Che

tors

tors Parallel Plate Capacitors

Th C i f PP i ap

acit

apac

it ◦ The Capacitance of PP is related only to the area A of the plates and the separation

of C

aof

Ca the plates and the separation

distance d between them!

ance

an

ce

paci

tapa

cita

he C

ahe

Ca

1. T

h1.

Th

Many computer keyboard buttons are constructed of capacitors, as shown in the Figure. When a key is pushed down, the soft insulator between the movable plate and the fixed plate is compressed. When the key

2.2

2.2

p yis pressed, the capacitance

(a) increases

oint

2oi

nt 2 (a) increases,

(b) decreases, or (c) changes in a way that we

eckp

oec

kpo ( ) g y

cannot determine because the complicated electric circuit connected to the

Che

Che keyboard button may cause

a change in ΔV.

A parallel-plate capacitor with air between h l h A 2 00 10 4 2 d the plates has an area A = 2.00 x10-4 m2 and

a plate separation d = 1.00 mm.Find its capacitance.

Ans: 1.77 pF

1 1m

ple

mpl

e Ex

amEx

am

ors

ors Cylindrical Capacitors (Coaxial)

acito

acito

Cap

a C

apa

L is length of the conductors, b is outer radius and a is inner radius

ce o

f ce

of

Spherical Capacitors (Concentric)

itanc

itanc

Spherical Capacitors (Concentric)

apac

apac a is inner radius, b is outer radius. If b→ ∞ then we

have an isolated conductor with a new capacitance

1. C

a1.

Ca

ss 1. An isolated spherical capacitor has a capacitance of 1F Calculate the radius of

cito

rci

tor capacitance of 1F. Calculate the radius of

the spherical capacitor.

Cap

acC

apac

2. You bought a 1-m coaxial cable for your TVR If it’s indicated that the outer radius is

her

Che

r C TVR. If it s indicated that the outer radius is

2mm and the capacitance is 2μF, find the inner radius of the coaxial cable.

: Oth

: Oth

inner radius of the coaxial cable.

3 Your laboratory instructor asked you to

mpl

em

ple 3. Your laboratory instructor asked you to create a spherical capacitor with capacitance 12 pF. The instructor gave you

Exam

Exam

p p g ya solid sphere of radius 2.4mm, what should be the diameter of the shell enclosure?

nts

nts As mentioned

li i h em

enem

en earlier in the chapter, capacitors

d i l i

uit

Ele

uit

Ele are used in electric

circuits.

Circ

uC

ircu

In circuit analysis, we

rs a

s rs

as study pictorial

representations of

acito

rac

itor

circuits known as circuit diagrams

Cap

aC

apa

2 Combinations of Capacitors2 Combinations of Capacitors2. Combinations of Capacitors2. Combinations of Capacitors

There are two types It’s series if the ypof circuit element combination:

elements are connected from end-

A. Series

to-end.

B. Parallel It’s parallel if the elements are connected at connected at common ends.

With parallel: the capacitors are at a common potential!

The equivalent capacitance is Ceq

ion

ion

common potential! capacitance is Ceqwhich is the sum of

the individual capacitances

bina

tibi

nati

Com

bC

omb

lel C

lel C

Para

lPa

ral

2.1

P2.

1 P

The individual charges can be found by QN = CNΔV

With series: the same charge Q is stored among all the capacitors!

ononis stored among all the capacitors!

The equivalent capacitance can be found by taking the reciprocal of the

natio

natio

y g psum of the reciprocal capacitances.

ombi

nom

bin

es C

oes

Co

The individual

Seri

eSe

rie voltage across each

capacitor can be found by

2.2

S2.

2 S

ΔVN = Q/CN

t ed

el!

el!

ival

ent

s st

ore

V = 18 Volts

Para

llePa

ralle

e eq

uiha

rges

ies

ies--

PPfin

d th

d th

e c

or!

: Ser

i: S

eri

rcui

t, f

ce a

ndap

acito

Vab = 15 Volts

mpl

em

ple

each

ci

acita

ncea

ch c

a ab

Exam

Exam In e

capa

in e

0)0)Find the equivalent capacitance. Note that

h i h h i10

/10

10/1

0 each capacitor has the same capacitance.ee

t, 1

eet,

12

she

2 sh

e5 5

(1/2

(1/2

IZ #

5IZ

#5

QU

IQ

UI

gygyEn

erg

Ener

gel

d E

eld

Etic

Fie

tic F

ieos

tat

osta

tec

tro

ectr

o3.

El

3. E

l

gygyEn

erg

Ener

gel

d E

eld

Etic

Fie

tic F

ieos

tat

osta

tec

tro

ectr

o3.

El

3. E

l

gygyFigure to the right

shows the linear En

erg

Ener

g shows the linear relationship between Q and ΔV.

eld

Eel

d E

The Energy U can be computed by taking

tic F

ietic

Fie computed by taking

the area under the curve!

osta

tos

tat

The energy density uE(Energy/Volume)

ectr

oec

tro (Energy/Volume)

◦ This is the energy that is stored in an electric

3. E

l3.

El field, regardless of the

configuration!

You have three capacitors and a battery. In hi h f h f ll i bi i f h which of the following combinations of the

three capacitors will the maximum possible b d h h bi i i energy be stored when the combination is

attached to the battery?

2.3

2.3

(a) series (b) parallel

oint

2oi

nt 2 (c) Both combinations will store the same

amount of energy.

eckp

oec

kpo gy

Ans: (b) for a given voltage capacitances add up

Che

Che

Ans: (b) for a given voltage, capacitances add up when in parallel and U = ½ C(ΔV)2

You charge a parallel-plate capacitor, remove it from the battery, and prevent the wires from the battery, and prevent the wires connected to the plates from touching each other. When you pull the plates apart to a larger separation do the following quantities increase separation, do the following quantities increase, decrease, or stay the same?

(a) C;

2.4

2.4

( ) ;(b) Q; (c) E between the plates;

oint

2oi

nt 2 (d)ΔV

(e) Energy stored (U) in the capacitor.

eckp

oec

kpo

Ans: C decreases, Q stays the same, E remains constant, V increases, U increases because U =

Che

Che

constant, V increases, U increases because U ½ QΔV

ors

ors Dielectrics are

insulators!pa

cito

paci

toinsulators!◦ Some examples of

dielectrics are air, paper, wax rubber and glass

Cap

Cap

wax, rubber, and glass.◦ Characterized by the

dielectric constant κ(>1) which modifies

and

a

nd (>1) which modifies

the permittivity of free space ε0!

ctri

csct

rics “When dielectrics

occupy the space

iele

cie

lec occupy the space

between the plates of a capacitor, the capacitance

4. D

4. D

capacitance increases!”

WARNING!er

y!er

y!◦ Before analyzing and solving for the effect of the dielectric on electrical

properties such as charge and potential… we have to ask…

Batt

eBa

tte

“IS THE CAPACITOR CONNECTED TO A BATTERY?”

f the

f t

he

CASE 1:If it was connected then removed before dielectric was inserted!

ct o

f ct

of If it was connected then removed before dielectric was inserted!

• The charge on the capacitor remain the same

CASE 2:

Effe

c E

ffec CASE 2:

If it remains connected when the dielectric was connected• The voltage across the capacitor remains the same

The

T

he

o a

o a

c in

toc

into

lect

rile

ctri

itor!

itor!

a di

ea

die

capa

cica

paci

rtin

g rt

ing

ged

cge

d c

Inse

rIn

ser

char

gch

arg

When the dielectric is

o a

o a

inserted, the charge remains the same, voltage

drops by:

c in

toc

into

lect

rile

ctri

itor!

itor!

If voltage drops, then capacitance increases

a di

ea

die

capa

cica

paci

pby:

rtin

g rt

ing

ged

cge

d c

Inse

rIn

ser

char

gch

arg

or is

or

is

When the dielectric is inserted, the voltage

capa

cito

capa

cito remains the same, to

accomplish this battery must supply additional

h h

le t

he c

le t

he c

ery

ery

charge so charge increases by:

tric

whi

tric

whi

he b

atte

he b

atte

If charge increases on

die

lect

die

lect

ed t

o th

ed t

o th

gthe plates,

capacitance also increases by:

ing

the

ing

the

onne

cte

onne

cte

Inse

rti

Inse

rti

still

co

still

co

1. A capacitor is to be constructed by making circular parallel plates (radius 1cm) and with separation of parallel plates (radius 1cm) and with separation of 2.5mm with a paper dielectric between the plates (κ= 3.7). Find the capacitance of this capacitor.

2. A 10nF capacitor is charged with a 12V battery. After fully charging the capacitor the battery was After fully charging the capacitor, the battery was removed and a dielectric (κ = 2) was inserted between the plates of the capacitor. Find the f ll i ( ) i i i l d fi l h (b) fi l l

s:s:

following: (a) initial and final charges, (b) final voltage, (c) final capacitance

mpl

em

ple

3. A 4.7nF capacitor is charged with a 9V battery. Then a glass dielectric was inserted between the

Exam

Exam plates of the capacitor. Find the following: (a) initial

and final charges, (b) final voltage, (c) final capacitance

CHAPT

This Chapter is divided into two parts:TER TH

parts:

CIRCUIT ELEMENTSH

REE: D

1.Resistors2. Batteries3. Combination of Resistors

DIRECT C

Resistors

DC CIRCUITS1 AnalysisCU

RREN

1. Analysis2. RC Circuits

NT CIRCCU

ITS

CHAPTER OBJECTIVES

1. Define steady state currents and its relation to ya material’s resistance.2. Relate voltage, current and resistance of

i tresistors.3. Differentiate a real from an ideal battery and monitor the energies of the circuital parametersmonitor the energies of the circuital parameters4. Compute for the equivalent resistance of a network of resistors5. Analyze Direct Current Circuits using Kirchhoff’s Rules.6 Analyze the behavior of RC Circuits6. Analyze the behavior of RC Circuits.

1. Electric Current

CI

2. Resistance and Resistors: Ohm’s LawRCU

IT3. EMF Sources:

Real and Ideal B tt i

T ELEMBatteries

4. Combination of ResistorsM

ENT

5. Energy in Electric CircuitsTS

ELECTRIC CURRENTI h f fl f h Is the rate of flow of charges per unit time

SI Unit: Ampere (A) after Andre Marie Amperep

1 A = 1C/1s

The direction of current is the direction of flow of positive charges

Many types:1. Electron in Hydrogen

Atom2. Electron Beam in Cathode

Ray Tubes in TV’s3. Electricity in Wires

ELECTRIC CURRENT AND THE ELECTRICFIELD

Since the I

direction of current is the direction of Eflow of positive charges: OHM’s LAW:

Current is related to the Electric The direction of the electric field is in the

Current is related to the Electric Field via the conductivity σ of the and cross sectional area A of the material:field is in the

same direction as the electric current!current!

ELECTRICAL RESISTANCE

I

High V Low VWhen a current passes through a material it

E

material it encounters a potential drop!

This potential drop is related to the current via the current via the Ohm’s Relation!

R is the

OhmicNon

Oh iR is the proportionality constant called “RESISTANCE OF

Ohmic

RESISTANCE OF THE MATERIAL”

ELECTRICAL RESISTANCE ANDRESISTIVITY

The Unit for Electrical R i i h Oh ( ) Resistance is the Ohm (Ω) after Georg Ohm

J lik i ρ is the resistivity of the Just like capacitance, resistance is not dependent on V or I, it’s dependent on the geometry and the kind

ρ s t e es st v ty o t e material in (Ω -m), L is the

length (m) and A(m2) is the cross sectional area of the material!the geometry and the kind

of material we have.

Examples:Examples:

A Nichrome wire ( ρ=10-6 Ω-m) has a radius of 0.65mm. What length of wire is needed to obtain a resistance of 2.0 Ω)?

RESISTANCE AND RESISTORS

Resistors are devices Schematic Symbol for Resistors

that provide resistance in a circuit.

Resistors and the resistance they carry resistance they carry have many purposes and applications in a variety circuitsvariety circuits.

(ELECTROMOTIVE FORCE)EMF SOURCES

An emf source is a device th t l t th that elevates the potential of a charge across its terminal. Positive Terminal

The potential gain is the emf of the battery!

It serves as the source or the pump of current in the circuit!

A very good example of emf source is the b !battery! +

Negative Terminal

REAL AND IDEAL BATTERIESTh di i i The distinction between an ideal and real battery is in their yterminal voltages (TV).

Ideal Battery:

Real Battery: (Ideal Battery plus a small y pinternal resistance r)

IDEAL REAL

BATTERIES…

Thus, the TV of

Rate and Energy Stored:

,a real battery is always less than an EMFan EMF.

Malfunctioning

Rate:1 Ah = 3600 C

Malfunctioning batteries have very large internal T l E internal resistances.

Total Energy StoredW = Qξξ

ANALYZING A BASIC CIRCUIT

ENERGY IN ELECTRIC CIRCUITS

The Unit of Power is Watts.

Power delivered by Power delivered by a battery ideal real

Powered dissipated i t

If V and R are givenacross a resistor If I and R are given

g

Example: Example: 1. A 12-Ω resistor carries a current of 3 A. Find the power dissipated in this resistor.

2. A wire of resistance 5 Ω carries a current of 3A for 6s.(a) How much power is put into the wire? (45W)

(b) How much thermal energy is produced? (270 J)

EXAMPLE:An 11-Ω resistor is connected across a battery of yemf 6V and internal resistance 1 Ω.

Find the followingFind the following(a) The current(b) The terminal voltage of the batteryg y(c) The power delivered by the emf source(d) The power delivered to the external resistor

Th di i t d b th b tt ’ i t l (e) The power dissipated by the battery’s internal resistance

(f) If the battery is rated at 150 A•h, how much energy does it store?

COMBINATIONS OF RESISTORS

Resistors are also known as “Loads”

SERIES PARALLEL

RESISTORS IN SERIES oYou can replace R1and R2 with a single 2 gresistor with a resistance Req.

oFor series connection, the

i h Req

current is the same across each capacitor but there is a potential drop across each resistor!

RESISTORS IN PARALLEL You can replace R1 and R2 with a resistor with R2 with a resistor with resistance Req.

For parallel connection the voltage across each resistor is the same R resistor is the same but the current splits along the junctions.

Req

EXAMPLE:

For the circuit that appears below find the following:a) I, I1 and I2b) Rb) Reqc) Voltage drop across each resistor

1. Kirchhoff’s Rules

C

Rules

2 RC IRCUI

2. RC Circuits

IT ANAALYSIIS

KIRCHHOFF’S RULES I2

Junction Rule:I3I1

All currents in and all currents out the junction are equal I4

I4 = I1+I2+I3

Iin = Iout

Loop Rule:

4

Current I Loop directionLoop Rule:In a single loop, all voltage gain is equal to all voltage drop

Current, I Loop direction

DROP GAIN

all voltage dropVgain = VdropIt is important to take note of the loop

Loop Directionnote of the loop direction GAIN DROP

ANALYSIS OF CIRCUITS1 Si l L 2 M l il Ci i1. Single Loop

Find the current in this

2. Multiloop Circuits

Find the currents I I Find the current in this circuit

Find the currents I1, I2, and I3.

MORE KIRCHHOFF’SFind all the currents through junction bg j

RC CIRCUITSC i i d Contains a resistor and a capacitor.I flows in a single direction but its direction but its magnitude varies with time.RC Circuit “charges” and g“discharges”

For charging: we put in g g pthe maximum amount of charge possible in the capacitor over a time constantconstant

For discharging: we drain the charge until it’s value the charge until it s value is negligible!

CHARGING RCW h h We assume that the capacitor is initially uncharged.g

Charge will increase in the capacitor, however, current decreases.

Charge in the capacitor at some time later, will reach its maximum value of Q = Cξ when the current I equals

Qf is the maximum charge that can be stored in a capacitor

the current I equals zero. I0 is the initial current in

the circuit

DISCHARGING RCDischarge happens Discharge happens because when the switch is closed at t = 0, there is a potential drop across

the resistor, meaning there is current in it.

Af i h After some time, the charge on the capacitor is reduced, hence the current is also reduced! current is also reduced! (Why is this happening?)

This happens again and pp gagain, until at some time, the charge and the current are both negligible hence

Qo is the initial charge that is stored in a capacitor

I0 is the initial current in negligible hence “discharged”

I0 is the initial current in the circuit

EXAMPLES:1. An uncharged capacitor and a resistor are connected in series to a battery. If ξ =12 0 V C = 5 00 μF and R = 12.0 V, C 5.00 μF, and R 8.00 x 105 Ω, find the time constant of the circuit, the maximum charge on the

it d th capacitor, and the maximum current in the circuit.

2. Consider a capacitor of capacitance C that is being discharged through a discharged through a resistor of resistance R, as shown in the figure. After how many time constants is th h g th it the charge on the capacitor one-fourth its initial value?