Chapter 16Chapter 16

Electric EnergyElectric Energy

andand

CapacitanceCapacitance

Electric Potential EnergyElectric Potential Energy

The electrostatic force is a The electrostatic force is a conservative forceconservative force

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 Work done by a conservative force is equal to the negative of the is equal to the negative of the change in potential energychange in potential 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 charge As the charge moves from A to B, moves from A to B, work is done in itwork is done in it

W = F d= q E dW = F d= q E d ΔPE = - W = - q E dΔPE = - W = - q E d

only for a uniform only for a uniform fieldfield

Potential DifferencePotential Difference

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 (final change in the potential energy (final value minus initial value) of a charge q value minus initial value) of a charge q moved from A to B divided by the size moved from A to B divided by the size of the chargeof the charge ΔV = VΔV = VBB – V – VAA = ΔPE / q = ΔPE / q

Potential difference is Potential difference is notnot the same as the same as potential energypotential energy

Potential Difference, cont.Potential Difference, cont.

Another way to relate the energy and Another way to relate the energy and the potential difference: the potential difference: ΔPE = q ΔVΔPE = q ΔV

Both electric potential energy and Both electric potential energy and potential difference are potential difference are scalarscalar quantities quantities

Units of potential differenceUnits of potential difference V = J/CV = J/C

A special case occurs when there is a A special case occurs when there is a uniform electric fielduniform electric field VVBB – V – VAA= -Ed= -Ed

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

Energy and Charge Energy and Charge MovementsMovements

A positive charge gains electrical A positive charge gains 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 negative charge loses electrical A negative charge loses 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

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

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

QUICK QUIZ 16.1

If an electron is released from rest in a uniform electric field, the electric potential energy of the charge-field system (a) increases, (b) decreases, or (c) remains the same.

QUICK QUIZ 16.1 ANSWER

(b). The field exerts a force on the electron, causing it to accelerate in the direction opposite to that of the field. In this process, electrical potential energy is converted into kinetic energy of the electron. Note that the electron moves to a region of higher potential, but because the electron has negative charge this corresponds to a decrease in the potential energy of the electron.

Electric Potential of a Point Electric Potential of a Point ChargeCharge

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

The potential created by a point charge The potential created by a point charge q at any distance r from the charge isq at any distance r from the charge is

A potential exists at some point in space A potential exists at some point in space whether or not there is a test charge at whether or not there is a test charge at that pointthat point

r

qkV e

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 qpotential due to q11 at at some point Psome point P11

The work required to The work required to bring qbring q22 from infinity to from infinity to PP11 without acceleration without acceleration is qis q22VV11

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 21e12

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, PE is sign, PE 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, PE is signs, PE is negativenegative The force would be attractiveThe force would be attractive Work must be done to hold back the unlike Work must be done to hold back the unlike

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

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 V = kUse the basic equation V = keeq/rq/r

QUICK QUIZ 16.2If the electric potential at some point is zero, you can conclude that (a) no charges exist in the vicinity of that point, (b) some charges are positive and some are negative, or (c) all charges in the vicinity have the same sign. Choose each correct answer.

QUICK QUIZ 16.2 ANSWEREither (a) or (b), but not both. The

absence of any electrical charges within a finite distance from the point would produce an electric potential of zero at the point. Thus, (a) could be a true statement. If electrical charges exist at finite distances from the point, then (b) must be true. Both positive and negative charges must be present in the vicinity so their contributions to the electrical potential at the observation point may cancel each other.

QUICK QUIZ 16.3A spherical balloon contains a positively charged particle at its center. As the balloon is inflated to a larger volume while the charged particle remains at the center, which of the following changes? (a) the electric potential at the surface of the balloon, (b) the magnitude of the electric field at the surface of the balloon, (c) the electric flux through the balloon.

QUICK QUIZ 16.3 ANSWER

(a) and (b). Both the electric potential and the magnitude of the electric field decrease as the distance from the charged particle increases.

Potentials and Charged Potentials and Charged ConductorsConductors

Since W = -q(VSince W = -q(VBB – V – VAA), no work is required ), no work is required to move a charge between two points to move a charge between two points that are at the same electric potentialthat are at the same electric potential W = 0 when VW = 0 when VAA = V = VBB

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

Conductors in EquilibriumConductors in Equilibrium The conductor has an excess of The conductor has an excess of

positive chargepositive charge All of the charge resides at the All of the charge resides at the

surfacesurface E = 0 inside the conductorE = 0 inside the conductor The electric field just outside The electric field just outside

the conductor is perpendicular the conductor is perpendicular to the surfaceto the surface

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

The potential everywhere inside The potential everywhere inside the conductor is constant and the conductor is constant and equal to its value at the surfaceequal to its value 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 eV = 1.6 x 101 eV = 1.6 x 10-19-19 J J

Equipotential SurfacesEquipotential 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 ChargeCharge

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 -- DipoleFields Lines -- Dipole

Equipotential lines Equipotential lines are shown in blueare shown in blue

Electric field lines Electric field lines are shown in redare shown in red

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

Application – Electrostatic Application – Electrostatic PrecipitatorPrecipitator

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

Application – Electrostatic Application – Electrostatic Air CleanerAir 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

CapacitanceCapacitance

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, C, of a capacitor , C, 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

QC

Parallel-Plate CapacitorParallel-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

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

Capacitors in CircuitsCapacitors in Circuits

A A circuitcircuit is a collection of objects is a collection of objects usually containing a source of usually containing a source of electrical energy (such as a electrical energy (such as a battery) connected to elements battery) connected to elements that convert electrical energy to that convert electrical energy to other formsother forms

A A circuit diagramcircuit diagram can be used to can be used to show the path of the real circuitshow the path of the real circuit

Capacitors in ParallelCapacitors in Parallel When capacitors are first connected in the When capacitors are first connected in the

circuit, electrons are transferred from the circuit, electrons are transferred from the left plates through the battery to the right left plates through the battery to the right plate, leaving the left plate positively plate, leaving the left plate positively charged and the right plate negatively charged and the right plate negatively chargedcharged

The flow of charges ceases when the The flow of charges ceases when the voltage across the capacitors equals that voltage across the capacitors equals that of the batteryof the battery

The capacitors reach their maximum The capacitors reach their maximum charge when the flow of charge ceasescharge when the flow of charge ceases

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 = Q = Q1 1 + Q+ Q22

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

More About Capacitors in More About Capacitors in ParallelParallel

The capacitors can The capacitors can be replaced with be replaced with one capacitor with a one capacitor with a capacitance of Ccapacitance of Ceqeq

The equivalent The equivalent capacitor must have capacitor must have exactly the same exactly the same external effort on external effort on the circuit as the the circuit as the original capacitorsoriginal capacitors

Capacitors in Parallel, finalCapacitors in Parallel, final

CCeqeq = C = C11 + C + C22

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

Capacitors in SeriesCapacitors in Series When a battery is connected to the circuit, When a battery is connected to the circuit,

electrons are transferred from the left plate electrons are transferred from the left plate of Cof C11 to the right plate of C to the right plate of C22 through the through the batterybattery

As this negative charge accumulates on the As this negative charge accumulates on the right plate of Cright plate of C22, an equivalent amount of , an equivalent amount of negative charge is removed from the left negative charge is removed from the left plate of Cplate of C22, leaving it with an excess positive , leaving it with an excess positive chargecharge

All of the right plates gain charges of –Q and All of the right plates gain charges of –Q and all the left plates have charges of +Qall the left plates have charges of +Q

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

C

1

C

1

C

1

VVV

Problem-Solving StrategyProblem-Solving Strategy

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

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

and the equivalent capacitance is and the equivalent capacitance is always less than the smallest individual always less than the smallest individual capacitorcapacitor

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 differences across them are the samesame The charge on each capacitor is The charge on each capacitor is

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

the equivalent capacitancethe 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

A capacitor is designed so that one plate is large and the other is small. If the plates are connected to a battery, (a) the large plate has a greater charge than the small plate, (b) the large plate has less charge than the small plate, or (c) the plates have charges equal in magnitude but opposite in sign.

QUICK QUIZ 16.4

QUICK QUIZ 16.4 ANSWER

(c). The battery moves negative charge from one plate and puts it on the other. The first plate is left with excess positive charge whose magnitude equals that of the negative charge moved to the other plate.

Energy Stored in a Energy Stored in a CapacitorCapacitor

Energy stored = Energy stored = ½ Q ΔV½ Q ΔV From the definition of capacitance, From the definition of capacitance,

this can be rewritten in different this can be rewritten in different formsforms

C2

QVC

2

1VQ

2

1Energy

22

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

QUICK QUIZ 16.5You charge a parallel-plate capacitor, remove it from the battery, and prevent the wires connected to the plates from touching each other. When you pull the plates farther apart, do the following quantities increase, decrease, or stay the same? (a) C; (b) Q; (c) E between the plates; (d) DV; (e) energy stored in the capacitor.

QUICK QUIZ 16.5 ANSWER

(a) C decreases

(b) Q stays the same

(c) E stays the same

(d) DV increases (e) The energy stored increases.

Capacitors with DielectricsCapacitors with Dielectrics

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 C = C = κCκCoo = κε = κεoo(A/d)(A/d)

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

Capacitors with DielectricsCapacitors with Dielectrics

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

QUICK QUIZ 16.6A fully charged parallel-plate capacitor remains connected to a battery while you slide a dielectric between the plates. Do the following quantities increase, decrease, or stay the same? (a) C; (b) Q; (c) E between the plates; (d) DV; (e) energy stored in the capacitor.

QUICK QUIZ 16.6 ANSWER

(a) C increases

(b) Q increases

(c) E stays the same

(d) DV remains the same

(e) The energy stored increases

An Atomic Description of An Atomic Description of DielectricsDielectrics

Polarization occurs when there is a Polarization occurs when there is a separation between the “centers of separation between the “centers of gravity” of its negative charge and gravity” of its negative charge and its positive chargeits positive charge

In a capacitor, the dielectric In a capacitor, the dielectric becomes polarized because it is in becomes polarized because it is in an electric field that exists an electric field that exists between the platesbetween the plates

More Atomic DescriptionMore Atomic Description The presence of the The presence of the

positive charge on positive charge on the dielectric the dielectric effectively reduces effectively reduces some of the negative some of the negative charge on the metalcharge on the metal

This allows more This allows more negative charge on negative charge on the plates for a given the plates for a given applied voltageapplied voltage

The capacitance The capacitance increasesincreases