Chapter 16Chapter 16

Electrical EnergyElectrical Energy

And And

CapacitanceCapacitance

General Physics

ReviewReview

Electric Flux Electric Flux through area A: through area A: = E = EAA i.e. i.e. E A cos E A cos

Each field line represents one unit of fluxEach field line represents one unit of fluxDensity of field lines = #/A = EDensity of field lines = #/A = E

The term flux comes from analogy with The term flux comes from analogy with fluids:fluids:It is the ‘flow rate of … through an area’It is the ‘flow rate of … through an area’While the velocity in a stream changes, While the velocity in a stream changes,

the total flow rate has to remain constant the total flow rate has to remain constant (conservation of water)(conservation of water)

General Physics

ReviewReview

Gauss’ Law:Gauss’ Law:For a closed surface A enclosing charge Q,For a closed surface A enclosing charge Q,

AA = Q/ = Q/00 kkee = 1/4 = 1/400

Flux begins at (+) and ends at (−) chargeFlux begins at (+) and ends at (−) charge(+) is like a water tap, (−) is like the drain(+) is like a water tap, (−) is like the drain

It is a prescription of how to draw field It is a prescription of how to draw field lines.lines.

It can be used to calculate E (symmetry)It can be used to calculate E (symmetry)It is a mathematical statement It is a mathematical statement

of Coulomb’s law (at least half of it)of Coulomb’s law (at least half of it)

General Physics

In the “flowing water” In the “flowing water” analogy,analogy,

what represents the electric what represents the electric field?field? 25% 25%25%25%

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1.1. Water velocityWater velocity

2.2. Cross-sectionalCross-sectionalarea of streamarea of stream

3.3. Flow rateFlow rate

4.4. Flow rate per Flow rate per areaarea

General Physics

Which best Which best represents the represents the

charge distribution charge distribution on a capacitor?on a capacitor?

33% 33%33%

45

(using principles of electrostatic equilibrium)

+ + + + + + + + + +

− − − − − − − − − −

1.1.

2.2.

3.3.

++++ + + + + + ++++

−−−− − − − − − −−−−

+ + + + ++++ + + + +

− − − − −−−− − − − −

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General Physics

Electrical EnergyElectrical Energy

Sections 1 – 4Sections 1 – 4

General Physics

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

General Physics

Work and Potential EnergyWork and Potential Energy

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

As the charge moves from A As the charge moves from A to B, work is done on itto B, work is done on it

The change in potential The change in potential energy between points A and energy between points A and B isB is

ΔPE = -W = -qEΔPE = -W = -qEx x xx Only for a Only for a uniform electric uniform electric

fieldfield

xqEW

xqEW

xFW

x

cos

cos

EX16.1

General Physics

Potential DifferencePotential Difference

In general, the potential In general, the potential difference between points difference between points A and B is defined as the A and B is defined as the change in the potential change in the potential energy (final value minus energy (final value minus initial value) of a charge q initial value) of a charge q moved from A to B divided moved from A to B divided by the size of the chargeby the size of the charge ΔV = VΔV = VBB – V – VAA = ΔPE / q = ΔPE / q

Potential difference is Potential difference is notnot the same as potential the same as potential energyenergy

General Physics

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: 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 difference1 V = 1 J/C1 V = 1 J/C

A special case occurs when there is a A special case occurs when there is a uniform electric fielduniform electric field V = VV = VBB – V – VAA= -E= -Exx xx

Gives more information about units of E-field: N/C = Gives more information about units of E-field: N/C = V/mV/m

General Physics

Energy and Charge Energy and Charge MovementsMovements

A positive charge gains electrical A positive charge gains electrical potential energy when it is moved in potential energy when it is moved in a direction opposite the electric fielda direction opposite the electric field

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

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

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

General Physics

Comparison of Positive Comparison of Positive Charge to Mass MovementsCharge to Mass Movements

General Physics

Summary of Positive Charge Summary of Positive Charge Movements and EnergyMovements and Energy

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

point of lower potentialpoint of lower potential Its electrical potential energy decreasesIts electrical potential energy decreases Its kinetic energy increasesIts kinetic energy increases Obeys conservation of energy relationship:Obeys conservation of energy relationship:

EX16.2a / EX16.3

General Physics

Summary of Negative Charge Summary of Negative Charge Movements and EnergyMovements and Energy

When a negative charge is placed in an When a negative charge is placed in an electric fieldelectric field It moves opposite to the direction of the fieldIt moves opposite to the direction of the field It moves from a point of lower potential to a It moves from a point of lower potential to a

point of higher potentialpoint of higher potential Its electrical potential energy decreasesIts electrical potential energy decreases Its kinetic energy increasesIts kinetic energy increases Obeys conservation of energy relationship:Obeys conservation of energy relationship:

EX16.2b

General Physics

Electric Potential of a Point Electric Potential of a Point ChargeCharge

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

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

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

r

qkV e

General Physics

Electric Field and Electric Electric Field and Electric Potential Depend on DistancePotential Depend on Distance

The electric field is The electric field is proportional to 1/rproportional to 1/r22

The electric potential The electric potential is proportional to 1/ris proportional to 1/r

2r

qkE e

r

qkV e

General Physics

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

Draw a diagram of all chargesDraw a diagram of all charges Note the point of interestNote the point of interest

Calculate the distance from each charge to the Calculate the distance from each charge to the point of interestpoint of interest

Use the basic equation V = kUse the basic equation V = keeq/rq/r Include the signInclude the sign The potential is positive if the charge is positive and The potential is positive if the charge is positive and

negative if the charge is negativenegative if the charge is negative Use the superposition principle when you have Use the superposition principle when you have

multiple chargesmultiple charges Take the algebraic sumTake the algebraic sum

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

General Physics

Electric Potential for an Electric Electric Potential for an Electric DipoleDipole

MU30T13-14

r

qkV e

General Physics

Electric Potential of Multiple Electric Potential of Multiple Point ChargesPoint Charges

Superposition principle Superposition principle appliesapplies

The total electric The total electric potential at some point potential at some point P due to several point P due to several point charges is the charges is the algebraicalgebraic sum of the electric sum of the electric potentials due to the potentials due to the individual chargesindividual charges The algebraic sum is used The algebraic sum is used

because potentials are because potentials are scalar quantitiesscalar quantities

EX16.4

General Physics

Electrical Potential Energy of Electrical Potential Energy of Two ChargesTwo Charges

VV11 is the electric potential is the electric potential due to qdue to q11 at some point P at some point P

The work required to bring The work required to bring qq22 from infinity to P without from infinity to P without acceleration is qacceleration is q22VV11

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

r

qqkVqPE 21e12

General Physics

Notes About Electric Potential Notes About Electric Potential Energy of Two ChargesEnergy of Two Charges

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 brought charges from accelerating as they are brought close togetherclose together

Active Figure: Electric Potential and Potential Energy

EX16.5

General Physics

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

chargecharge All of the charge resides at the surfaceAll of the charge resides at the surface E = 0 inside the conductorE = 0 inside the conductor The electric field just outside the The electric field just outside the

conductor is perpendicular to the surfaceconductor is perpendicular to the surface The potential is a constant everywhere The potential is a constant everywhere

on the surface of the conductoron the surface of the conductor W = -q(VW = -q(VBB – V – VAA) = 0, no work is required ) = 0, 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

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

General Physics

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 gains when energy that an electron gains when accelerated through a potential accelerated through a potential difference of 1 Vdifference of 1 V Electrons in normal atoms have energies Electrons in normal atoms have energies

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

of eV or keV’sof eV or keV’s High energy gamma rays have energies of High energy gamma rays have energies of

millions of eV or MeV’smillions of eV or MeV’s 1 eV = 1.6 x 101 eV = 1.6 x 10-19-19 J J

General Physics

An electron and proton are An electron and proton are accelerated through a accelerated through a

potential difference of 1V, -1V potential difference of 1V, -1V respectively,respectively,

have kinetic energies KEhave kinetic energies KEee, KE, KEpp..Which statement is true?Which statement is true?25% 25%25%25%

45

1.1. KEKEee > Ke > Kepp

2.2. KEKEee = KE = KEpp

3.3. KEKEee < Ke < Kepp

4.4. Impossible to tell?Impossible to tell?

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General Physics

Equipotential SurfacesEquipotential Surfaces

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

a constant speed on an equipotential a constant speed on an equipotential surfacesurface

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

General Physics

Equipotentials and Electric Equipotentials and Electric Fields Lines – Positive ChargeFields Lines – Positive Charge

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

The field lines are The field lines are perpendicular to the perpendicular to the electric potential at all electric potential at all pointspoints

General Physics

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 redare shown in red

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