Electric Potential and Capacitance

What’s a volt anyway?

Presentation 2001 Dr. Phil Dauber as modified by R. McDermott

Why Potential?

Electric potential can be visualized as Electric potential can be visualized as “height”.“height”.

This allows us to make comparisons to This allows us to make comparisons to gravity.gravity.

Potential is independent of the object in the Potential is independent of the object in the electric field.electric field.

What is Potential? By agreement from fields, we always define electrical By agreement from fields, we always define electrical

quantities in terms of positive charge.quantities in terms of positive charge.

Positive charges will always move away from other Positive charges will always move away from other positive charges and toward negative charges, much as a positive charges and toward negative charges, much as a ball always rolls downhill.ball always rolls downhill.

By analogy, we consider a region of positive charge to be By analogy, we consider a region of positive charge to be at a high potential (hill), and a region of negative charge at a high potential (hill), and a region of negative charge to be at a low potential (valley).to be at a low potential (valley).

Field lines, then, show the most direct “downhill” path.Field lines, then, show the most direct “downhill” path.

Equipotential Lines Maps dealing with hills and valleys are called contour or Maps dealing with hills and valleys are called contour or

topographical maps, and consist of closed lines that connect topographical maps, and consist of closed lines that connect points that have the same height or altitude.points that have the same height or altitude.

We draw similar “maps” for dealing with charges using We draw similar “maps” for dealing with charges using Equipotential lines, which connect points that have the same Equipotential lines, which connect points that have the same electrical height (potential).electrical height (potential).

An equipotential line will always be perpendicular to the An equipotential line will always be perpendicular to the electric field, since the field always points “downhill”.electric field, since the field always points “downhill”.

Stationary charges will not move on their own along an Stationary charges will not move on their own along an equipotential line, because that will not lead them equipotential line, because that will not lead them “downhill”.“downhill”.

Hills and Valleys, Oh My!

As a ball rolls downhill, the gravitational field does work on As a ball rolls downhill, the gravitational field does work on the ball due to the unbalanced force (weight) acting on the the ball due to the unbalanced force (weight) acting on the ball, causing it to gain speed (and kinetic energy). We say ball, causing it to gain speed (and kinetic energy). We say that potential energy stored in the field between the ball and that potential energy stored in the field between the ball and Earth is converted into kinetic energy. Earth is converted into kinetic energy.

Similarly, as a positive charge moves from high potential to Similarly, as a positive charge moves from high potential to low the electric field does work on the charge due to the low the electric field does work on the charge due to the unbalanced force acting on it, causing it to gain speed (and unbalanced force acting on it, causing it to gain speed (and kinetic energy). We say that potential energy stored by the kinetic energy). We say that potential energy stored by the field and charge is converted into kinetic energy.field and charge is converted into kinetic energy.

Hills and Valleys, Oh My!

Electric and Gravitational Fields

Property: massProperty: mass Property: chargeProperty: charge

Gravity: Electricity:

1 Sign: positive1 Sign: positive

Dependency: 1/rDependency: 1/r22 Dependency: 1/rDependency: 1/r22

2 Signs: pos, neg2 Signs: pos, neg

k = 8.99x10k = 8.99x1099 n-m n-m22/C/C22 G = 6.67x10G = 6.67x10-11-11 n-m n-m22/kg/kg22

Electric and Gravitational Fields

Gravity: Electricity:

F = (kqF = (kq11/r/r22)) qq22

F = EF = Efieldfieldqq22

F= (GmF= (GmEarthEarth/r/rEarthEarth22)) mm22

F = gF = gfieldfieldmm22

PE = mgPE = mghh PE = PE = qEqEx (x (x = d)x = d)

PE is in JoulesPE is in Joules PE is in JoulesPE is in Joules

Electric and Gravitational Fields

Gravity: Electricity: We want to treat all objects We want to treat all objects the same way, so..the same way, so.. ““Potential”, V = Eqd/q = EdPotential”, V = Eqd/q = Ed

V = (PE)/chargeV = (PE)/charge Units = Joules/Coul.Units = Joules/Coul.

L= (PE)/massL= (PE)/mass

Units = Joules/kgUnits = Joules/kg

We want to treat all We want to treat all objects the same way, so..objects the same way, so..

““Liftage”, L = mgLiftage”, L = mgh/m = gh/m = ghh

Electric and Gravitational Fields

Gravity: Electricity:

E = F/qE = F/q Units = N/cUnits = N/c E is the electric field E is the electric field strengthstrength

g = F/mg = F/m

Units = N/kgUnits = N/kg g is the grav. field g is the grav. field strengthstrength

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

field or potential. Consider uniform field:field or 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 coulombAlternate units for E: volts per coulomb

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 = qV Thus electrical work = qV

If q is in coulombs, the work is in joules; if If q is in coulombs, the work is in joules; if q is in elementary charges, the work is in q is in elementary charges, the work is in electron-volts (eV).electron-volts (eV).

Potential difference is called Potential difference is called voltagevoltage

Electrical Potential Energy

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

Electrical PE is transferred to the charge as Electrical PE is transferred to the charge as kinetic energy.kinetic energy.

A positive charge has its greatest PE near A positive charge has its greatest PE near another positive charge or positive plate.another positive charge or positive plate.

Only Only differences differences in potential energy are in potential energy are measurablemeasurable

V = 0 Arbitrary

Usually ground is zero point of potentialUsually ground is zero point of potential Sometimes potential is zero at infinitySometimes potential is 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 pressure (depth or height)Voltage is like water pressure (depth or height)

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

potential difference of 5000 voltspotential 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 PE = 8x10-16 J

V = 4.2x107 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 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 to a point 10cm from a 5 to a point 10cm from a 5 c charge?c charge?

Work required = change in potential energyWork required = change in potential energyW = qVW = qVbaba = q{kQ/r = q{kQ/rbb – kQ/r – kQ/raa}}

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 volt.accelerated through one volt.

W = qV, if q is in Coulombs, then W is in Joules. W = qV, if q is in Coulombs, then W is in Joules. If q is in elementary charges, W is in electron If q is in elementary charges, W is in electron volts (eV).volts (eV).

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

It’s about It’s about ENERGY!ENERGY! 1 KeV = 1000 eV; 1 MeV = 101 KeV = 1000 eV; 1 MeV = 1066 eV eV 1 Gev = 101 Gev = 1099 eV eV

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?

Capacitors

Store chargeStore 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 = 1f = 1 microfarad = 1-6-6 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 Inversely proportional to distance between

plates plates

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

use e = Kuse e = K00; K is called dielectric constant; K is called dielectric constant

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

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 f = 50000 pf

Capacitors

Photos courtesy Illinois Capacitor, Inc

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

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 capacitor Derivation: the work needed to charge a capacitor

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

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

V/2V/2