Electric Potential and Capacitance

What’s a volt anyway?

Why Potential?

Energy makes some problems easier to Energy makes some problems easier to solvesolve

Energy is a scalarEnergy is a scalar The concept of a potential isThe concept of a potential is

useful to have in our physicsuseful to have in our physics

toolboxtoolbox

Define Change in Electric Potential Energy When a positive charge released near the When a positive charge released near the

positive plate moves to the right it loses positive plate moves to the right it loses potential energy equal in magnitude to the potential energy equal in magnitude to the work done by the field on the chargework done by the field on the charge

Positive plate

Negative plate

b a

Electrical Potential Energy

The change in potential energy equals the The change in potential energy equals the negative of the work done by the fieldnegative of the work done by the field

Electrical PE changes to kineticElectrical PE changes to kinetic The positive charge has its greatest PE near The positive charge has its greatest PE near

positive platepositive plate Only Only differences differences in potential energy are in potential energy are

measurablemeasurable

Electrical Potential

Potential is potential energy per unit chargePotential is potential energy per unit charge Analogous to field which is force per unit Analogous to field which is force per unit

chargecharge Symbol of potential is V; VSymbol of potential is V; Vaa = PE = PEaa/q /q Only potential Only potential differencesdifferences are measurable; are measurable;

zero point of potential is arbitraryzero point of potential is arbitrary VVabab = V = Vaa – V – Vbb = - W = - Wbaba/q/q WWba ba is work done to move q from b to ais work done to move q from b to a

Units

Unit of electric potential is the Unit of electric potential is the voltvolt AbbreviationAbbreviation V V 1 V = 1 Joule/Coulomb = 1 J/C1 V = 1 Joule/Coulomb = 1 J/C Thus electrical work = qVThus electrical work = qV Potential difference is called Potential difference is called voltagevoltage

V = 0 Arbitrary

Usually ground is zero point of potentialUsually ground is zero point of potential Sometimes potential chosen zero at infinitySometimes potential chosen zero at infinity + terminal of 12V battery is+ terminal of 12V battery is

said to be at 12V higher said to be at 12V higher

potential than – terminalpotential than – terminal

Electric Potential and Potential Energy PE = PEPE = PEbb – PE – PEaa = qV = qVbaba

If object with charge q moves through a potential If object with charge q moves through a potential difference Vdifference Vba ba its potential energy changes by qVits potential energy changes by qVbaba

Example: What is the gain of electrical PE when 1 C Example: What is the gain of electrical PE when 1 C of charge moves between the terminals of a 12 volt of charge moves between the terminals of a 12 volt battery?battery?

Water Analogy:Water Analogy: Voltage is like water pressureVoltage is like water pressure

12 joules

Example: Electron in Computer Monitor An electron is accelerated from rest through An electron is accelerated from rest through

a potential difference of 5000 voltsa potential difference of 5000 volts Find its change in potential energyFind its change in potential energy Find its speed after accelerationFind its speed after acceleration

a b

Vba = 5000 v = Vb - Va

Example: Electron in Computer Monitor An electron is accelerated from rest through An electron is accelerated from rest through

a potential difference of 5000 voltsa potential difference of 5000 volts Find its change in potential energyFind its change in potential energy Find its speed after accelerationFind its speed after acceleration PE = qV = 1.6x 10PE = qV = 1.6x 10-19 -19 C x 5 x 10C x 5 x 103 3 V =V =

8 x 108 x 10-16-16 J = ½ m v J = ½ m v22

m = 9.11 x 10m = 9.11 x 10-31-31 kg ; v = 4.2 x 10 kg ; v = 4.2 x 1077 m/s m/s

Questions on Preceding Example

Does the energy depend on the particle’s Does the energy depend on the particle’s mass?mass?

Does the final speed depend on the mass?Does the final speed depend on the mass?

Electric Potential and Electric Field Can describe charge distribution in terms field or Can describe charge distribution in terms field or

potential. Consider uniform field:potential. Consider uniform field: F = EqF = Eq W = qVW = qVbaba

W = Fd =W = Fd = qE dqE d

Thus VThus Vba ba = Ed or E = V= Ed or E = Vbaba/d/d

Alternate units for E: volts per meter v/mAlternate units for E: volts per meter v/m

Instead of?Instead of? N/C

Equipotential Lines

An equipotential surface is perpendicular to An equipotential surface is perpendicular to the electric field at any pointthe electric field at any point

If not charges would move along the If not charges would move along the surfacesurface

This contradicts the idea of an equipotentialThis contradicts the idea of an equipotential

Describe the equipotentials…

……between large oppositely charged between large oppositely charged conducting platesconducting plates

+

-

Electric Potential Due to Point Charges V = k Q/r derived from CalculusV = k Q/r derived from Calculus Here V = 0 at r = infinity; V represents Here V = 0 at r = infinity; V represents

potential difference between r and infinitypotential difference between r and infinity

Example: Work to force two point charges together What work is needed to bring a 2What work is needed to bring a 2C charge C charge

from far away to within 10cm of a 5 from far away to within 10cm of a 5 c c charge?charge?

Work required = change in potential energyWork required = change in potential energy

W = qVW = qVbaba = q{kQ/r = q{kQ/rbb – kQ/r – kQ/raa} = qkQ/r} = qkQ/rbb0.90 J

Potential at an Arbitrary Point Near Several Point Charges Add the potentials due to each point chargeAdd the potentials due to each point charge Use the right sign for the chargeUse the right sign for the charge Relax; potential isn’t a vectorRelax; potential isn’t a vector

What is true about the mid-plane between What is true about the mid-plane between two equal point charges of opposite sign?two equal point charges of opposite sign?Potential everywhere zero

Find…

The potential due to a microcoulomb charge The potential due to a microcoulomb charge at a point one meter away at a point one meter away

The potential energy of two microcoulomb The potential energy of two microcoulomb charges one meter apartcharges one meter apart

9000 volts

9 x 10-3 J

The Electron Volt The energy acquired by a particle carrying a The energy acquired by a particle carrying a

charge equal to that of the electron when charge equal to that of the electron when accelerated through one voltaccelerated through one volt

1 eV = 1.6 x 101 eV = 1.6 x 10-19-19 joules joules

It’s about It’s about ENERGYENERGY 1 KeV = 1000 eV = 1.6 x 101 KeV = 1000 eV = 1.6 x 10-16-16 joules joules

1 MeV = 101 MeV = 1066 eV eV 1 Gev = 101 Gev = 1099 eV eV

Capacitance

Stores chargeStores charge Two conducting platesTwo conducting plates NOT touchingNOT touching May have insulatingMay have insulating

material betweenmaterial between

Q = CVQ = CV

Capacitance

Symbol CSymbol C Unit: coulombs per volt = faradUnit: coulombs per volt = farad 1 pf = 1 picofarad = 101 pf = 1 picofarad = 10-12-12 farad farad 1nf = 1 nanofarad = 101nf = 1 nanofarad = 10-9-9 f f 11f = 1 microfarad = 10f = 1 microfarad = 10-6-6 f f

C is Constant for a Given Capacitor Does not depend on Q or VDoes not depend on Q or V Proportional to areaProportional to area Inversely proportional to distance between plates Inversely proportional to distance between plates

C = C = 00 A/d A/d If dielectric like oil or paper between plates use If dielectric like oil or paper between plates use

= K= K00; K is called dielectric constant; K is called dielectric constant isiscalled permittivity. called permittivity. 0 0 is permittivity of free is permittivity of free

spacespace

Find the Capacitance

A capacitor can hold 5 A capacitor can hold 5 C of charge at a C of charge at a potential difference of 100 volts. What is potential difference of 100 volts. What is its capacitanceits capacitance

Q = CV

C = Q/V = 5 x 10C = Q/V = 5 x 10-6-6 C/ 100V = C/ 100V =

5 x 105 x 10-8-8 f = 50000 pf = 0.05 f = 50000 pf = 0.05 ff

Why is the Electric Field inside the dielectric less than outside? Less since fewer field lines

Capacitors

Photos courtesy Illinois Capacitor, Inc

Supercapacitors -

Supercapcitors use double layer electrolyte technology, usually with activated carbon and sulfuric acid. The carbon has huge surface area

Applications

In automotive ignitionsIn automotive ignitions In strobe lightsIn strobe lights In electronic flashIn electronic flash In power suppliesIn power supplies In nearly all electronicsIn nearly all electronics

Supercapacitor applications

Power motor vehicles for short burstsPower motor vehicles for short bursts Backup power for computersBackup power for computers Operate emergency doors and slides in Operate emergency doors and slides in

commercial aircraftcommercial aircraft

Lower energy density but greater power Lower energy density but greater power density than batteriesdensity than batteries

A Capacitor Stores Electric Energy A battery A battery producesproduces electric energy bit by bit electric energy bit by bit A capacitor is NOT a type of batteryA capacitor is NOT a type of battery A battery can be used to charge a capacitorA battery can be used to charge a capacitor

Energy in a Capacitor Holding Charge Q at Voltage V U = QV/2 = CVU = QV/2 = CV22/2 = Q/2 = Q22/2C/2C Derivation: the work needed to charge a Derivation: the work needed to charge a

capacitor by bringing charge onto a plate capacitor by bringing charge onto a plate when some is already there (use W =QV)when some is already there (use W =QV)

Initially V = 0Initially V = 0 Average voltage during the charging Average voltage during the charging

process is V/2process is V/2

Example

A 20 A 20 f capacitor stores 20 millijoules of f capacitor stores 20 millijoules of energy. What is the voltage across it?energy. What is the voltage across it?

U = CVU = CV22/2/2 V = (2U/C)V = (2U/C)1/2 1/2 = (2x20 x 10= (2x20 x 10-3-3J/20x10J/20x10-6-6F)F)1/2 1/2 ==

44.7 V44.7 V