Chapter 20Chapter 20

Electric EnergyElectric Energy

andand

CapacitanceCapacitance

1 Electric Potential 1 Electric Potential EnergyEnergy

The electrostatic force is a The electrostatic force is a conservative (conservative (=“path independent”=“path independent”) ) forceforce

It is possible to define an electrical It is possible to define an electrical potential energy function with this potential energy function with this forceforce

Work done by a conservative force is Work done by a conservative force is equal to the negative of the change in equal to the negative of the change in potential energypotential energy

Work and Potential EnergyWork and Potential Energy

There is a uniform There is a uniform field between the field between the two platestwo plates

As the positive As the positive charge moves from A charge moves from A to B, work is doneto B, work is done

WWABAB==F dF d==q E dq E d

ΔΔPEPE =- =-WW ABAB=-=-q E dq E d only for a uniform only for a uniform

fieldfield

E=F/q

Potential Difference Potential Difference (=“Voltage Drop”)(=“Voltage Drop”)

The potential difference between The potential difference between points A and B is defined as the points A and B is defined as the change in the potential energy change in the potential energy (final value minus initial value) of a (final value minus initial value) of a charge charge qq moved from A to B divided moved from A to B divided by the size of the chargeby the size of the charge ΔΔVV = = VVBB – – VVAA = Δ = ΔPEPE / /qq

Potential difference is Potential difference is not not the same the same as potential energyas potential energy

Potential Difference, cont.Potential Difference, cont. Another way to relate the energy and the Another way to relate the energy and the

potential difference: potential difference: ΔΔPEPE = = qq Δ ΔVV Both electric potential energy and potential Both electric potential energy and potential

difference are difference are scalarscalar quantities quantities Units of potential differenceUnits of potential difference

V = J/C (Volt=V = J/C (Volt=Joule/Coulomb)Joule/Coulomb) A special case occurs when there is a A special case occurs when there is a

uniform electric fielduniform electric field VVBB – – VVAA= -= -EdEd

Gives more information about units: Gives more information about units: N/C = V/mN/C = V/m

ΔΔPEPE = = qq Δ ΔV=-qEdV=-qEd

Energy and Charge Energy and Charge MovementsMovements

A A positivepositive charge charge gainsgains electrical electrical potential energy when it is moved in a potential energy when it is moved in a direction opposite the electric fielddirection opposite the electric field

If a charge is released in the electric field, If a charge is released in the electric field, it experiences a force and accelerates, it experiences a force and accelerates, gaining kinetic energygaining kinetic energy As it gains kinetic energy, it loses an equal As it gains kinetic energy, it loses an equal

amount of electrical potential energyamount of electrical potential energy A A negativenegative charge charge losesloses electrical electrical

potential energy when it moves in the potential energy when it moves in the direction opposite the electric fielddirection opposite the electric field

Energy and Charge Energy and Charge Movements, contMovements, cont

When the electric field is When the electric field is directed downward, point directed downward, point B is at a lower potential B is at a lower potential than point Athan point A

A positive test charge A positive test charge that moves from A to B that moves from A to B loses electric potential loses electric potential energyenergy

It will gain the same It will gain the same amount of kinetic energy amount of kinetic energy as it loses potential as it loses potential energyenergy

ΔΔPEPE =- =-WW ABAB=-=-q E dq E d

VVBB – – VVAA= -= -EdEd

Summary of Positive Summary of Positive Charge Movements and Charge Movements and EnergyEnergy

When a positive charge is placed in When a positive charge is placed in an electric fieldan electric field It moves in the direction of the fieldIt moves in the direction of the field It moves from a point of higher It moves from a point of higher

potential to a point of lower potentialpotential to a point of lower potential Its electrical potential energy Its electrical potential energy

decreasesdecreases Its kinetic energy increasesIts kinetic energy increases

conservation law

Summary of Negative Summary of Negative Charge Movements and Charge Movements and EnergyEnergy

When a negative charge is placed in When a negative charge is placed in an electric fieldan electric field It moves opposite to the direction of It moves opposite to the direction of

the fieldthe field It moves from a point of lower potential It moves from a point of lower potential

to a point of higher potentialto a point of higher potential Its electrical potential energy Its electrical potential energy

decreasesdecreases Its kinetic energy increasesIts kinetic energy increases

linear accelerator linear accelerator

Stanford linear Stanford linear accelerator accelerator center (SLAC)center (SLAC)

Tunnel of SLAC

Positron-electron project (PEP)

Example: A proton moves from rest in an electric field of Example: A proton moves from rest in an electric field of

8.08.0101044 V/m along the +V/m along the +xx axis for 50 cm. Find a) the axis for 50 cm. Find a) the change in in the electric potential, b) the change in the change in in the electric potential, b) the change in the electrical potential energy, and c) the speed after it has electrical potential energy, and c) the speed after it has moved 50 cm. moved 50 cm.

a) a) VV=-=-EdEd=-(8.0=-(8.0101044 V/m)(0.50 m)=-4.0 V/m)(0.50 m)=-4.0101044 V V

b) b) PEPE==qq VV=(1.6=(1.61010-19-19 C)(-4.0 C)(-4.0 101044 V)=-6.4 V)=-6.4 1010-15-15 JJ KEi+PEi=KEf+PEf, KEi=0

KEf=PEi-PEf=-PEPE,,

mmppvv22/2=6.4/2=6.41010-15-15 J J

mmpp=1.67=1.671010-15-15 kg kg

s/m108.2kg1067.1

)J104.6(2 627

15

v

2 Electric Potential of a 2 Electric Potential of a Point ChargePoint Charge

The point of zero electric potential The point of zero electric potential is taken to be at an infinite is taken to be at an infinite distance from the chargedistance from the charge

The potential created by a point The potential created by a point charge charge qq at any distance at any distance rr from the from the charge ischarge is

r

qkV e if r, V=0 and if r=0, V

Electric Potential of a Electric Potential of a Point ChargePoint Charge

b b

a b e e e2a aa b

q q qV V Edl k dl k k

r r r

br

Potential Difference between points a and bPotential Difference between points a and b

The point of zero electric potential is taken to be at an The point of zero electric potential is taken to be at an infinite distance from the charge:infinite distance from the charge:

r

qkV e

V decreases as 1/r, and, as a consequence, E decreases 1/r2.

Electric Potential of an Electric Potential of an electric Dipoleelectric Dipole

-q

+q

Electric Potential of Electric Potential of Multiple Point ChargesMultiple Point Charges

Superposition principle appliesSuperposition principle applies The total electric potential at some The total electric potential at some

point P due to several point point P due to several point charges is the charges is the algebraicalgebraic sum of the sum of the electric potentials due to the electric potentials due to the individual chargesindividual charges The algebraic sum is used because The algebraic sum is used because

potentials are scalar quantitiespotentials are scalar quantities

Electrical Potential Energy Electrical Potential Energy of Two Chargesof Two Charges

VV11 is the electric is the electric potential due to potential due to qq11 at at some point some point PP11

The work required to The work required to bring bring qq22 from infinity to from infinity to PP11 without acceleration without acceleration is is qq22EE11dd==qq22VV11

This work is equal to This work is equal to the potential energy of the potential energy of the two particle systemthe two particle system

r

qqkVqPE 21

e12

Notes About Electric Notes About Electric Potential Energy of Two Potential Energy of Two ChargesCharges

If the charges have the If the charges have the samesame sign, sign, PEPE is is positivepositive Positive work must be done to force the two Positive work must be done to force the two

charges near one anothercharges near one another The like charges would repelThe like charges would repel

If the charges have If the charges have oppositeopposite signs, signs, PEPE is is negativenegative The force would be attractiveThe force would be attractive Work must be done to hold backWork must be done to hold back the unlike the unlike

charges from accelerating as they are charges from accelerating as they are brought close togetherbrought close together

Example: Finding the Electric Example: Finding the Electric Potential at Point P (apply Potential at Point P (apply VV==kkeeqq//rr).).

5.0 C -2.0 C

V1060.3)m0.4()m0.3(

)C100.2()C/Nm1099.8(

,V1012.1m0.4

C100.5)C/Nm1099.8(

3

22

6229

2

46

2291

V

V

Superposition: Vp=V1+V2

Vp=1.12104 V+(-3.60103 V)=7.6103 V

e

qV k

r

Problem Solving with Problem Solving with Electric Potential (Point Electric Potential (Point Charges)Charges)

Remember that potential is a scalar Remember that potential is a scalar quantityquantity So no components to worry aboutSo no components to worry about

Use the superposition principle when you Use the superposition principle when you have multiple chargeshave multiple charges Take the algebraic sumTake the algebraic sum

Keep track of signKeep track of sign The potential is positive if the charge is The potential is positive if the charge is

positive and negative if the charge is negativepositive and negative if the charge is negative Use the basic equation Use the basic equation VV = = kkeeqq//rr

3 Potentials and Charged 3 Potentials and Charged ConductorsConductors

WW =- =-PEPE= -= -qq((VVBB – – VVAA)) , no work is , no work is required to move a charge between two required to move a charge between two points that are at the same electric points that are at the same electric potential potential WW=0 when =0 when VVAA==VVBB

All points on the surface of a charged All points on the surface of a charged conductor in electrostatic equilibrium conductor in electrostatic equilibrium are at the same potentialare at the same potential

Therefore, the electric potential is a Therefore, the electric potential is a constant everywhere on the surface of a constant everywhere on the surface of a charged conductor in equilibriumcharged conductor in equilibrium

Overview: Conductors in Overview: Conductors in EquilibriumEquilibrium

The conductor has an The conductor has an excess of positive chargeexcess of positive charge

All of the charge resides at All of the charge resides at the surfacethe surface

EE = 0 inside the conductor = 0 inside the conductor The electric field just The electric field just

outside the conductor is outside the conductor is perpendicular to the surfaceperpendicular to the surface

The potential is a constant The potential is a constant everywhere on the surface everywhere on the surface of the conductor of the conductor

The potential everywhere The potential everywhere inside the conductor is inside the conductor is constant and equal to its constant and equal to its value at the surfacevalue at the surface

The Electron VoltThe Electron Volt

The electron volt (eV) is defined as the The electron volt (eV) is defined as the energy that an electron (or proton) energy that an electron (or proton) gains when accelerated through a gains when accelerated through a potential difference of 1 Vpotential difference of 1 V Electrons in normal atoms have energies of Electrons in normal atoms have energies of

10’s of eV10’s of eV Excited electrons have energies of 1000’s of Excited electrons have energies of 1000’s of

eVeV High energy gamma rays have energies of High energy gamma rays have energies of

millions of eVmillions of eV 1 V=1 J/C 1 V=1 J/C 1 eV = 1.6 x 10 1 eV = 1.6 x 10-19-19 J J

4 Equipotential Surfaces4 Equipotential Surfaces

An An equipotential surfaceequipotential surface is a is a surface on which all points are at surface on which all points are at the same potentialthe same potential No work is required to move a charge No work is required to move a charge

at a constant speed on an at a constant speed on an equipotential surfaceequipotential surface

The electric field at every point on an The electric field at every point on an equipotential surface is perpendicular equipotential surface is perpendicular to the surfaceto the surface

Equipotentials and Electric Equipotentials and Electric Fields Lines (Positive Fields Lines (Positive Charge):Charge):

The equipotentials The equipotentials for a point charge for a point charge are a family of are a family of spheres centered on spheres centered on the point chargethe point charge

The field lines are The field lines are perpendicular to the perpendicular to the electric potential at electric potential at all pointsall points

Equipotentials and Electric Equipotentials and Electric Fields Lines (Dipole):Fields Lines (Dipole):

Equipotential lines Equipotential lines are shown in blueare shown in blue

Electric field lines Electric field lines are shown in are shown in orange orange

The field lines are The field lines are perpendicular to perpendicular to the equipotential the equipotential lines at all pointslines at all points

5 Applications – 5 Applications – Electrostatic PrecipitatorElectrostatic Precipitator

It is used to remove It is used to remove particulate matter particulate matter from combustion from combustion gasesgases

Reduces air pollutionReduces air pollution Can eliminate Can eliminate

approximately 90% approximately 90% by mass of the ash by mass of the ash and dust from smokeand dust from smoke

Negative

How does it work?How does it work? High voltage (4-100 kV) is maintained High voltage (4-100 kV) is maintained

between the coil wire and the grounded wallbetween the coil wire and the grounded wall The electric field at the wire causes The electric field at the wire causes

discharges, i.e., ions (charged oxygen discharges, i.e., ions (charged oxygen atoms) are formedatoms) are formed

The negative ions and electrons move to the The negative ions and electrons move to the positively biased wallpositively biased wall

On their way the ions and electrons ionize On their way the ions and electrons ionize dirt particles due to collisionsdirt particles due to collisions

Most of the dirt particles become negatively Most of the dirt particles become negatively charged and are attracted to the wall as charged and are attracted to the wall as well – cleaning effect well – cleaning effect

Electrostatic Air CleanerElectrostatic Air Cleaner

Used in homes to relieve the Used in homes to relieve the discomfort of allergy sufferersdiscomfort of allergy sufferers

It uses many of the same It uses many of the same principles as the electrostatic principles as the electrostatic precipitatorprecipitator

Application – Xerographic Application – Xerographic CopiersCopiers

The process of xerography is used The process of xerography is used for making photocopiesfor making photocopies

Uses photoconductive materialsUses photoconductive materials A photoconductive material is a poor A photoconductive material is a poor

conductor of electricity in the dark conductor of electricity in the dark but becomes a good electric but becomes a good electric conductor when exposed to lightconductor when exposed to light

The Xerographic ProcessThe Xerographic Process

Application – Laser PrinterApplication – Laser Printer The steps for producing a document on a The steps for producing a document on a

laser printer is similar to the steps in the laser printer is similar to the steps in the xerographic processxerographic process Steps a, c, and d are the sameSteps a, c, and d are the same The major difference is the way the image The major difference is the way the image

forms of the selenium-coated drumforms of the selenium-coated drum A rotating mirror inside the printer causes the beam A rotating mirror inside the printer causes the beam

of the laser to sweep across the selenium-coated of the laser to sweep across the selenium-coated drumdrum

The electrical signals form the desired letter in The electrical signals form the desired letter in positive charges on the selenium-coated drumpositive charges on the selenium-coated drum

Toner is applied and the process continues as in the Toner is applied and the process continues as in the xerographic processxerographic process

6 Capacitance6 Capacitance

A capacitor is a device used in a A capacitor is a device used in a variety of electric circuitsvariety of electric circuits

The The capacitancecapacitance, , CC, of a capacitor , of a capacitor is defined as the ratio of the is defined as the ratio of the magnitude of the charge on either magnitude of the charge on either conductor (plate) to the magnitude conductor (plate) to the magnitude of the potential difference of the potential difference between the conductors (plates)between the conductors (plates)

Capacitance, contCapacitance, cont

Units: Farad (F)Units: Farad (F) 1 F = 1 C / V1 F = 1 C / V A Farad is very largeA Farad is very large

Often will see µF or pFOften will see µF or pF

V

Q

V

QC

V=V and means voltage drop

7 Parallel-Plate Capacitor7 Parallel-Plate Capacitor

The capacitance of a device The capacitance of a device depends on the geometric depends on the geometric arrangement of the conductorsarrangement of the conductors

For a parallel-plate capacitor For a parallel-plate capacitor whose plates are separated by air:whose plates are separated by air:

d

AC o

Permittivity of the free space

Applications of Capacitors Applications of Capacitors – Camera Flash– Camera Flash

The flash attachment on a camera The flash attachment on a camera uses a capacitoruses a capacitor A battery is used to charge the A battery is used to charge the

capacitorcapacitor The energy stored in the capacitor is The energy stored in the capacitor is

released when the button is pushed to released when the button is pushed to take a picturetake a picture

The charge is delivered very quickly, The charge is delivered very quickly, illuminating the subject when more illuminating the subject when more light is neededlight is needed

Applications of Capacitors Applications of Capacitors -- Computers-- Computers

Computers use Computers use capacitors in many capacitors in many waysways Some keyboards use Some keyboards use

capacitors at the capacitors at the bases of the keysbases of the keys

When the key is When the key is pressed, the capacitor pressed, the capacitor spacing decreases spacing decreases and the capacitance and the capacitance increasesincreases

The key is recognized The key is recognized by the change in by the change in capacitancecapacitance

8 Capacitors in Circuits8 Capacitors in Circuits

QQ11==CC11VVabab, , QQ22==CC22VVabab

  The total charge supplied by the The total charge supplied by the

source:source:

QQtotaltotal==QQ11++QQ22==VVabab((CC11++CC22))

Equivalent capacitanceEquivalent capacitance C Ceqeq

CCeqeq=C=C11+C+C22    

Q1 and Q2 are not

necessarily equal but Vab is the same.

parallel connection

Capacitors in ParallelCapacitors in Parallel

The total charge is The total charge is equal to the sum of equal to the sum of the charges on the the charges on the capacitorscapacitors QQtotaltotal = = QQ1 1 + + QQ22

The potential The potential difference across the difference across the capacitors is the samecapacitors is the same And each is equal to And each is equal to

the voltage of the the voltage of the batterybattery

Capacitors in Parallel, finalCapacitors in Parallel, final

CCeqeq = = CC11 + + CC22

The equivalent capacitance of The equivalent capacitance of a parallel combination of a parallel combination of capacitors is greater than any capacitors is greater than any of the individual capacitorsof the individual capacitors

VV11==QQ//CC11, , VV22==QQ//CC22

  

VV==VV11++VV22==

21

11

CCQ

21eq

111

CCC

In a series connection the magnitude of charge on all plates is the same!

Equivalent capacitanceEquivalent capacitance C Ceqeq

More About Capacitors in More About Capacitors in SeriesSeries

An equivalent An equivalent capacitor can be capacitor can be found that found that performs the same performs the same function as the function as the series combinationseries combination

The potential The potential differences add up differences add up to the battery to the battery voltagevoltage

Capacitors in Series, contCapacitors in Series, cont

The equivalent capacitance of a The equivalent capacitance of a series combination is always less series combination is always less than any individual capacitor in the than any individual capacitor in the combinationcombination

21eq

21

111

CCC

VVV

Problem-Solving StrategyProblem-Solving Strategy

Be careful with the choice of unitsBe careful with the choice of units When two or more unequal capacitors When two or more unequal capacitors

are connected are connected in seriesin series, they carry the , they carry the same charge, but the potential same charge, but the potential differences across them are not the samedifferences across them are not the same The capacitances add as The capacitances add as

reciprocals and the equivalent reciprocals and the equivalent capacitance is always less than capacitance is always less than the smallest individual capacitorthe smallest individual capacitor

Problem-Solving Strategy, Problem-Solving Strategy, contcont

When two or more capacitors are When two or more capacitors are connected connected in parallelin parallel, the potential , the potential differences across them are the samedifferences across them are the same The charge on each capacitor is The charge on each capacitor is

proportional to its capacitanceproportional to its capacitance The capacitors add directly to The capacitors add directly to

give the equivalent capacitancegive the equivalent capacitance

Problem-Solving Strategy, Problem-Solving Strategy, finalfinal

A complicated circuit can often be A complicated circuit can often be reduced to one equivalent capacitorreduced to one equivalent capacitor Replace capacitors in series or parallel with Replace capacitors in series or parallel with

their equivalenttheir equivalent Redraw the circuit and continueRedraw the circuit and continue

To find the charge on, or the potential To find the charge on, or the potential difference across, one of the capacitors, difference across, one of the capacitors, start with your final equivalent capacitor start with your final equivalent capacitor and work back through the circuit and work back through the circuit reductionsreductions

Example:Example:

Step 1: Step 1:

CCpp==CC11++CC22

CCpp=0.10 =0.10 F+0.20 F+0.20 FF

CCpp =0.30 =0.30 FF

Step 1

Step

2

Step 2:Step 2:

1/1/CCss=1/=1/CC33+1/+1/CCpp  

F20.0FF60.0

FF60.0

p3s

CC

CCC p3

9 Energy Stored in a 9 Energy Stored in a CapacitorCapacitor

Average voltage during charging:Average voltage during charging:

Since Since VVfinalfinal is the applied voltage, we write is the applied voltage, we write VVaa==VV/2./2.

Energy stored (=work done by the battery): Energy stored (=work done by the battery): 

22finalinitialfinal

a

VVVV

2a 2

1

2

1CVQVQVW

0

A plot of voltage vs. A plot of voltage vs. charge of a charge of a capacitor is a capacitor is a straight line with straight line with slope 1/slope 1/CC. The area . The area under the line under the line equals equals QVQV/2=Energy /2=Energy stored.stored.

V

ApplicationsApplications

DefibrillatorsDefibrillators When fibrillation occurs, the heart produces When fibrillation occurs, the heart produces

a rapid, irregular pattern of beatsa rapid, irregular pattern of beats A fast discharge of electrical energy through A fast discharge of electrical energy through

the heart can return the organ to its normal the heart can return the organ to its normal beat patternbeat pattern

In general, capacitors act as energy In general, capacitors act as energy reservoirs that can slowly charged and reservoirs that can slowly charged and then discharged quickly to provide large then discharged quickly to provide large amounts of energy in a short pulseamounts of energy in a short pulse

10 Capacitors with 10 Capacitors with DielectricsDielectrics

A A dielectricdielectric is an insulating material is an insulating material that, when placed between the that, when placed between the plates of a capacitor, increases the plates of a capacitor, increases the capacitancecapacitance Dielectrics include rubber, plastic, or Dielectrics include rubber, plastic, or

waxed paperwaxed paper CC = = κCκCoo = = κεκεoo((AA//dd))

The capacitance is multiplied by the The capacitance is multiplied by the factor factor κκ when the dielectric completely when the dielectric completely fills the region between the platesfills the region between the plates

(a)(a) Electric field lines inside Electric field lines inside an empty capacitoran empty capacitor

(b)(b) The electric field The electric field produces polarizationproduces polarization

(c)(c) The resulting positive The resulting positive and negative surface and negative surface charges on the dielectric charges on the dielectric reduce the electric field reduce the electric field within the dielectricwithin the dielectric

E0

E=E0/or V=V0/

+Q0-Q0

Reasoning:

Dielectric constant

Capacitance in presence of a Capacitance in presence of a dielectric:dielectric:

0 0 00

0

0

Q Q QC C

V V / V

AC

d

Since >1, the dielectric enhances the capacitance of the capacitor!

Capacitors with DielectricsCapacitors with Dielectrics

The value of The value of depends on the nature of depends on the nature of the dielectric material, as the table below the dielectric material, as the table below indicates:indicates:

(at 300 K)

Dielectric StrengthDielectric Strength

For any given plate separation, For any given plate separation, there is a maximum electric field there is a maximum electric field that can be produced in the that can be produced in the dielectric before it breaks down dielectric before it breaks down and begins to conductand begins to conduct

This maximum electric field is This maximum electric field is called the called the dielectric strengthdielectric strength

Capacitors DesignsCapacitors Designs

(a)(a) Paper capacitorPaper capacitor(b)(b) High-voltage oil capacitorHigh-voltage oil capacitor(c)(c) Electrolytic capacitorElectrolytic capacitor