Electric Potential Energy Electric Potential Energy and Electric Potentialand Electric Potential

Chapter 16Chapter 16

Potential EnergyPotential Energy

Potential Energy is stored energy do to Potential Energy is stored energy do to an object’s position in a force field.an object’s position in a force field.

Work = FWork = F··d = d = ΔΔPE = PE = ΔΔUU

When work is done to move an object When work is done to move an object into a force field, Potential Energy into a force field, Potential Energy IncreasesIncreases

Gravitational P.E. UGravitational P.E. Ugg

To bring a mass in from To bring a mass in from infinity to near the earth, infinity to near the earth, we would encounter the we would encounter the Earth’s gravitational field.Earth’s gravitational field.

Work = FWork = F·d = ·d = ΔΔUUgg

ΔΔUUg g = U= Uff – U – Uii = U = Urr – U – Uinfinf

ΔΔUUgg= (GM= (GMeem/rm/r22)·r)·r

UUgg = -GMm/r = -GMm/r

Uniform FieldsUniform Fields

Near Earth’s surface, gravitational field is Near Earth’s surface, gravitational field is uniform. g = 9.8 m/suniform. g = 9.8 m/s22..

We calculate Work = We calculate Work = ΔΔU = mghU = mgh

This is a simplification of UThis is a simplification of Ugg = -GMm/r = -GMm/r

Work is Work is essentiallyessentially F F·d·d

Electric Potential EnergyElectric Potential Energy To bring a charge, q, in To bring a charge, q, in

from infinity to near +Q, from infinity to near +Q, we would encounter the we would encounter the charges electric field.charges electric field.

Work = FWork = F·d = ·d = ΔΔUUee

ΔΔUUe e = U= Uff – U – Uii = U = Urr – U – Uinfinf

ΔΔUUee= (kQq/r= (kQq/r22)·r)·r

UUee = kQq/r = kQq/r

Uniform Electric FieldUniform Electric Field Parallel Plates with Parallel Plates with

equal charge provide equal charge provide a uniform electric a uniform electric field.field.

Recall E = FRecall E = Fee/q/q

FFee = Eq = Eq

Potential Energy – Potential Energy – Uniform Field (plate Uniform Field (plate separation d)separation d)

ΔΔU = UU = Uff – U – Uii = U = UBB – U – UAA

ΔΔU = Work = F·dU = Work = F·d

ΔΔU = (Eq)d = EqdU = (Eq)d = Eqd

E is electric field E is electric field between plates, d is between plates, d is plate separation.plate separation.

Uniform Gravitational Uniform Gravitational and Electric Field and Electric Field ComparisonComparison

ΔΔUUgg = mgh for = mgh for constant g fieldconstant g field

ΔΔUUee = qEd for = qEd for constant E fieldconstant E field

U is measured in U is measured in JoulesJoules

Non-Uniform Fields / PENon-Uniform Fields / PE

FFgg = GMm/r = GMm/r22 U Ugg = -GMm/r = -GMm/r

FFee = kQq/r = kQq/r22 U Uee = kQq/r = kQq/r

Electric Potential, VElectric Potential, V

Electric Potential is defined as potential Electric Potential is defined as potential energy per unit charge.energy per unit charge.

Think about adding charge, one charge at Think about adding charge, one charge at a time, to a conducting sphere.a time, to a conducting sphere.

Electric Potential, VElectric Potential, V

ΔΔV = V = ΔΔUUee/q measured in Joules/Coulomb/q measured in Joules/Coulomb

Joule/Coulomb = Volt!Joule/Coulomb = Volt!

Uniform Electric Field: Uniform Electric Field: ΔΔV = qEd/q = EdV = qEd/q = Ed

ΔΔV = Ed gives the voltage difference between two V = Ed gives the voltage difference between two parallel platesparallel plates

Non-Uniform E Field: Non-Uniform E Field: ΔΔV = (kQq/r)/q = kQ/rV = (kQq/r)/q = kQ/r

Summarize New FormulasSummarize New Formulas FFee = kq = kq11qq22/r/r22

UUee = F = F· (distance)· (distance)= kq= kq11qq22/r electric potential energy between two /r electric potential energy between two

charges (non uniform fields)charges (non uniform fields) UUee = qEd = qEd electric potential energy for parallel plates with electric potential energy for parallel plates with

electric field Eelectric field E V = UV = Uee/q/q

= kq/r = kq/r electric potential due to charge q (non uniform electric potential due to charge q (non uniform field)field)

V = EdV = Ed electric potential for parallel plates with electricelectric potential for parallel plates with electric field Efield E

Electric Potential Energy Electric Potential Energy for Several Chargesfor Several Charges

To find the potential To find the potential energy of a system energy of a system of charges, add the of charges, add the potential energy potential energy between each pair of between each pair of charges.charges.

UUee = U = U1212 + U + U1313 + U + U2323

ExamplesExamples

Read Examples 16.1, 16.2, 16.3, 16.4Read Examples 16.1, 16.2, 16.3, 16.4

Copy these into your notes if you feel that Copy these into your notes if you feel that is helpful!is helpful!

Try # 11 – 14, 17, 18, 20 – 22, 24, 25, 28 Try # 11 – 14, 17, 18, 20 – 22, 24, 25, 28 page 562 { prepare to turn this page 562 { prepare to turn this assignment in}assignment in}

Equi-potential SurfacesEqui-potential Surfaces1515

Surfaces where potential energy is Surfaces where potential energy is constant are known as constant are known as equi-potential equi-potential surfaces.surfaces.

For the electric field, we are concerned For the electric field, we are concerned with electric potential, V, in addition to with electric potential, V, in addition to electric potential energy, U. Equi-electric potential energy, U. Equi-potential surfaces are surfaces with the potential surfaces are surfaces with the same potential energy same potential energy andand the same the same electric potential.electric potential.

Conservative FieldsConservative Fields A conservative force is a force A conservative force is a force

in which work done does NOT in which work done does NOT depend on the path taken. depend on the path taken.

Gravitational Force is a Gravitational Force is a conservative force.conservative force.

Electric Force is a conservative Electric Force is a conservative forceforce

Friction is non-conservative.Friction is non-conservative. In moving an object along an In moving an object along an

equi-potential surface, no work equi-potential surface, no work is done.is done.

Equi-potential SurfacesEqui-potential Surfaces

Uniform Electric FieldUniform Electric Field

Recall E = F/qRecall E = F/q

F = EqF = Eq

ΔΔU = Fd = EqdU = Fd = Eqd

ΔΔV = U/qV = U/q

For a uniform field, For a uniform field,

ΔΔV = Ed = EV = Ed = EΔΔxx

UnitsUnits

E [Joule/Coulomb] or [Volt/meter] E [Joule/Coulomb] or [Volt/meter]

These are equivalent since These are equivalent since

E = F/q = E = F/q = ΔΔV/dV/d

Example:Example:

Normally the Earth is electrically charged. This Normally the Earth is electrically charged. This creates a constant electric field pointing down creates a constant electric field pointing down near the Earth’s surface of 150 V/m.near the Earth’s surface of 150 V/m.

A) What are the shapes of the equipotential A) What are the shapes of the equipotential surfaces?surfaces?

B) How far apart are two equi-potential B) How far apart are two equi-potential surfaces with 1000V difference between them?surfaces with 1000V difference between them?

Electron Volt measure for Electron Volt measure for Energy…Energy…

An Electron-Volt is a common unit for An Electron-Volt is a common unit for energy… energy…

It is the amount of Kinetic energy It is the amount of Kinetic energy acquired by an electron if it is accelerated acquired by an electron if it is accelerated through a voltage of 1 Volt.through a voltage of 1 Volt.

1eV = 1eV = ΔKE = -ΔU = qV = 1.6 X 10 ΔKE = -ΔU = qV = 1.6 X 10 –19–19 J J

CapacitanceCapacitance A capacitor is a device that stores charge.A capacitor is a device that stores charge. A good capacitor has the ability to store charge A good capacitor has the ability to store charge

without appreciably increasing the electric without appreciably increasing the electric potential.potential.

Work is done by a battery to move charge from Work is done by a battery to move charge from one parallel plate to another. Separation of one parallel plate to another. Separation of charge creates an electric field.charge creates an electric field.

Capacitance is defined as the amount of stored Capacitance is defined as the amount of stored charge per unit of potential difference.charge per unit of potential difference.

CapacitanceCapacitance Capacitance = Charge stored / VoltageCapacitance = Charge stored / Voltage

C = Q/V or Q = CC = Q/V or Q = CVV

[Coulombs/Volt] = [Farad][Coulombs/Volt] = [Farad]

1 Farad is a huge amount of capacitance. 1 Farad is a huge amount of capacitance. It is most commonly measured in It is most commonly measured in microfarads = 10microfarads = 10-6-6 F F

CapacitanceCapacitance Capacitance of a parallel plate arrangement depends Capacitance of a parallel plate arrangement depends

on the area of and the distance between the plates, on the area of and the distance between the plates, as well as the material between them. (dielectrics as well as the material between them. (dielectrics increase capacitance)increase capacitance)

If the material between the plates is air, then If the material between the plates is air, then

C = C = ξξ00A/d A/d

Where ξWhere ξ0 0 is the permittivity of free spaceis the permittivity of free space

And ξAnd ξ0 0 = 8.85 X 10 = 8.85 X 10 –12–12 C C22/Nm/Nm22

ExampleExample

What would be the plate area of an air filled What would be the plate area of an air filled 1.0F parallel plate capacitor if the plate 1.0F parallel plate capacitor if the plate separation were 1.0 mm?separation were 1.0 mm?

Energy!Energy! Potential increases as Potential increases as

charge is added.charge is added.

Slope = Slope = ΔV/ ΔQ = 1/CΔV/ ΔQ = 1/C

Potential energy stored in a Potential energy stored in a capacitor = Work Done = capacitor = Work Done = QVQVavav = Q(V/2) = Q(V/2)

UUcc = ½ QV = ½ CV = ½ QV = ½ CV22 = Q = Q22/2C/2C

ExampleExample

During a heart attack the heart beats During a heart attack the heart beats erratically. One way to get it back to erratically. One way to get it back to normal is to shock it with electrical normal is to shock it with electrical energy. About 300J of energy is required energy. About 300J of energy is required to produce the desired effect. A to produce the desired effect. A defibrillator stores this energy in a defibrillator stores this energy in a capacitor charged by 5000V. What capacitor charged by 5000V. What capacitance is required? What is the capacitance is required? What is the charge on the plates?charge on the plates?

Homework!Homework!

Read sections 16.4 and 16.5 pages 552 Read sections 16.4 and 16.5 pages 552 – 559.– 559.

Do # 57, 59, 60, 61, 64, 65 – 67, 69- 72 Do # 57, 59, 60, 61, 64, 65 – 67, 69- 72

Dielectric MaterialsDielectric Materials

Dielectric materials Dielectric materials placed between parallel placed between parallel plates have several plates have several purposes: purposes: Keep plates from coming Keep plates from coming

in contactin contact Allow for flexible platesAllow for flexible plates Increase capacitance of Increase capacitance of

the capacitorthe capacitor Dielectric constant, Dielectric constant, κ>1κ>1 κ = C/Cκ = C/C00

Two dielectric situations:Two dielectric situations:

Either the voltage difference is removed Either the voltage difference is removed once the plates are charged, once the plates are charged, thenthen the the dielectric material inserted between dielectric material inserted between platesplates

Or the dielectric material is inserted Or the dielectric material is inserted between the plates while the voltage is between the plates while the voltage is maintained.maintained.

These are different situations, but both These are different situations, but both result in increased capacitance.result in increased capacitance.

Remove voltage then insert dielectricRemove voltage then insert dielectric

Voltage applied, VVoltage applied, V00, , separates charge, Qseparates charge, Q00 and and sets up electric field Esets up electric field E00

Dielectric inserted and Dielectric inserted and becomes polarized. Electric becomes polarized. Electric field does work to polarize field does work to polarize dielectric molecules, which dielectric molecules, which set up a smaller electric field set up a smaller electric field in the opposite direction.in the opposite direction.

Net electric field is reduced. Net electric field is reduced. Therefore voltage is reduced.Therefore voltage is reduced.

Remove voltage then insert dielectricRemove voltage then insert dielectric

Charge, Q, doesn’t change once Charge, Q, doesn’t change once dielectric is inserted.dielectric is inserted.

Induced electric field in dielectric Induced electric field in dielectric reduces original electric field to E reduces original electric field to E and original voltage difference to V.and original voltage difference to V.

κ = Eκ = E00 / E = V / E = V00 /V /V

Then C = Q/V = QThen C = Q/V = Q00/(V/(V00/ κ)/ κ)

= κ(Q= κ(Q00/V/V00) = κC) = κC00

Uc = QUc = Q22/2C = Q/2C = Q22/2(κC/2(κC00) = U) = U00/ κ/ κ

Remove voltage then insert dielectricRemove voltage then insert dielectric

Capacitance increases by factor of Capacitance increases by factor of κκ

Since C = Q/V and voltage Since C = Q/V and voltage decreases, decreases, this makes sense!this makes sense!

Stored energy decreases by factor of κStored energy decreases by factor of κ

Some stored energy goes into Some stored energy goes into polarizing the molecules in the polarizing the molecules in the dielectric, so this makes sense!dielectric, so this makes sense!

When capacitor is maintained at When capacitor is maintained at constant voltage, the battery constant voltage, the battery continues to supply charge to continues to supply charge to compensate for the induced compensate for the induced electric field. electric field.

C = Q/V = C = Q/V = κQκQ00/V/V0 0 = κC= κC00

Charge and capacitance Charge and capacitance increase by a factor of increase by a factor of κ.κ.

Uc = ½ CVUc = ½ CV22 = κC = κC00VV0022

Stored energy increased by κStored energy increased by κ

Insert dielectric and maintain constant voltage

Maintain constant voltage across Maintain constant voltage across the capacitor with dielectricthe capacitor with dielectric

Charge (and therefore capacitance) Charge (and therefore capacitance) increase by a factor of increase by a factor of κκ

Stored energy ( in the capacitor) Stored energy ( in the capacitor) increases by a factor of κ at the expense increases by a factor of κ at the expense of the battery, which does more work.of the battery, which does more work.

Capacitor JewelryCapacitor Jewelry

CapacitanceCapacitance

With dielectric, C = With dielectric, C = κCκC00 where C where C00 is is

capacitance without dielectric.capacitance without dielectric.

C = κ(εC = κ(ε00)A/d)A/d

ExampleExample

Consider a capacitor with dielectric underneath a Consider a capacitor with dielectric underneath a computer key. The capacitor is connected to 12V computer key. The capacitor is connected to 12V and has a normal (uncompressed) plate and has a normal (uncompressed) plate separation of 3.0 mm and plate area of 0.75 cmseparation of 3.0 mm and plate area of 0.75 cm22. . a) What is the required dielectric constant if the a) What is the required dielectric constant if the capacitance is 1.10 pF?capacitance is 1.10 pF?b) How much charge is stored on the plates b) How much charge is stored on the plates under normal conditions?under normal conditions?c) How much charge flows onto the plates if they c) How much charge flows onto the plates if they are compressed to a separation of 2.0 mm?are compressed to a separation of 2.0 mm?

Circuitry – Capacitors in Circuitry – Capacitors in Series and ParallelSeries and Parallel

Capacitors in series are Capacitors in series are connected one after connected one after another.another.

The voltage from the The voltage from the battery is shared battery is shared between the series between the series capacitors.capacitors.

Want to find ‘equivalent Want to find ‘equivalent capacitance’.capacitance’.

Series and Parallel Series and Parallel CapacitorsCapacitors

Capacitors in parallel are Capacitors in parallel are connected in branches connected in branches parallel to one another.parallel to one another.

Each capacitor in parallel Each capacitor in parallel receives the same receives the same voltage from the battery.voltage from the battery.

Want to find the Want to find the ‘equivalent capacitance’.‘equivalent capacitance’.

SeriesSeries To find equivalent capacitance, To find equivalent capacitance,

consider what is constant. For consider what is constant. For series capacitors, the charge on series capacitors, the charge on each capacitor must be constant. each capacitor must be constant. (why?)(why?)

Voltages across each capacitor Voltages across each capacitor add to the total voltage supplied.add to the total voltage supplied.

VV11 + V + V22 + V + V33 = V = Vtottot

Q/CQ/C11 + Q/C + Q/C22 + Q/C + Q/C33 =Q/C =Q/Ceqeq

1/C1/Ceqeq = 1/C = 1/C11 + 1/C + 1/C22 + 1/C + 1/C33

ParallelParallel In parallel, each capacitor gets In parallel, each capacitor gets

the same voltage. With different the same voltage. With different capacitances, the charge on capacitances, the charge on each is different.each is different.

Charge adds to total charge Charge adds to total charge separated by batteryseparated by battery

QQtottot = Q = Q1 1 + Q+ Q22 + Q + Q33

CCeffeffV = CV = C11V + CV + C22V + CV + C33VV

CCeffeff = C = C11 + C + C22 + C + C33

ExampleExample

Given two capacitors, one with a Given two capacitors, one with a capacitance of 2.5 capacitance of 2.5 μF and the other of μF and the other of 5.0 μF, what are the charge on each and 5.0 μF, what are the charge on each and the total charge stored if they are the total charge stored if they are connected to a 12 V battery inconnected to a 12 V battery in

a) seriesa) series

b) parallelb) parallel

ExampleExample

Three capacitors are connected as Three capacitors are connected as shown on page 560. Find the voltage shown on page 560. Find the voltage across each capacitor.across each capacitor.