Chapter 25

Electric Potential

Introduction

• We’ve used conservation of Energy and the idea of potential energy associated with conservative forces (spring/gravity) in our study of mechanics.

• The electrostatic force is also conservative (note similarities to gravity)

• We can use Electric Potential Energy to study phenomena and also to define a new scalar quantity, Electric Potential.

25.1 Potential Difference and Electric Potential

• When a test charge qo is placed into an E-field, E, created by a charge distribution, we know the electric force on qo is found by

(conservative)• If some external agent moves the charge within

the electric field, the work done by the electric field on the charge is equal to the negative work done by the agent.– Gravitational Analog

EF oE q

25.1

• When talking about Electric and Magnetic fields often ds will be used to represent infinitesimal displacement vectors tangent to a path through space.

• Adding up the work done along the path (either curved or straight) is called the “path integral” or “line integral”

25.1

• For an infinitely small displacement, ds, of a charge, qo, the work (W = Fd) done by the electric field is

• As this amount of work is done by the field, the potential energy of the charge-field system is changed by a small amount

sEsF dqd o

sE dqdU o

25.1

• If we look at a finite path from points A to B, the change in potential energy of the system is

• This integration is performed along the path that qo follows as it moves from A to B.

• Because qoE is conservative, this line integration does not depend on the path taken from A to B.

AB UUU

B

Ao dqU sE

25.1

• If we divide qo, we can define a term that measures “Potential Energy per unit test charge” which now solely depends on the source charge distribution.

• This quantity U/qo is called the electric potential (or simply “potential”) V.

oq

UV

25.1

• Since U is a scalar, V is also a scalar.• If we move a test charge between points A

and B, the system experiences a change in potential energy.

• The Potential Difference between A and B is found by

AB VVV

B

Ao

dq

UV sE

25.1

• Just as we saw with (grav) potential energy, only the differences are meaningful.

• We will take the value of electric potential to be zero at some convenient point in the field.

• Electric Potential is a scalar characteristic of an electric field, independent of any charges that may be placed in the field.

25.1

• If an external agent moves a test charge from A to B, without changing the kinetic energy, the work done is simply equal to the change in potential energy.

and therefore

UW

VqW

25.1

• Units- Electric Potential is a measure of potential energy per unit charge, the SI unit is a Joule/Coulomb, defined as a volt (V).

• Or, 1 J of work must be done to move a 1-C charge through a potential difference of 1 volt.

C

J 1V 1

25.1

• Since potential difference also has units of E-field times distance, E-field can also be expressed in volts per meter.

• E-field can now be interpreted as a measure of the rate of change of electric potential, with position.

m

V 1

C

N 1

25.1

• A common energy unit for Atomic and Nuclear physics is the electron volt (eV)– Defined as the energy gained/lost by a system

when an electron/proton moves through a potential difference of 1 V.

• Since 1 V = 1 J/C, and e = 1.60 x 10-19 J

VC 10 x 1.60 eV 1 -19

J 10 x 1.60 eV 1 -19

25.1

ExampleThe electron beam of a typical CRT television reaches a speed of 3.0 x 107 m/s.a. What is the kinetic energy (in eV) of a single electron?b. What potential difference is required to accelerate this electron from rest?

25.1

Quick Quizzes p 765

25.2 Potential Differences in a Uniform Electric Field

• While the equations for Electric PotentialEnergy and Potential Difference hold in any field, they can be simplified if the field is uniform. • First consider a uniform E field in the negative y direction.

25.2

• We can calculate the potential difference between points A and B, separated by a distance |s| = d, where displacement vector s is parallel to the field lines.

B

AdV sE

B

A

o dsEV 0cos

B

AEdsV

25.2

• Since E is constant we can remove it giving

• The negative indicates that the potential at B is lower than potential at A.

• Electric field lines point in the direction of decreasing electric potential.

B

AdsEV

B

AdsEV

EdV

25.2

• Now if we move a test charge qo from A to B, we can calculate the change in potential energy of the charge-field system

• A system of a positive charge and electric field loses potential energy when the charge moves in the direction of the field.

EdqVqU oo

25.2

• We can imagine what would happen if we release a positive test charge from rest in a field.– The net force would be qoE– The charge would accelerate.• A gain of kinetic energy• Loss of Potential Energy

25.2

• If the test charge qo is negative, the opposite is true. The system gains potential energy if the charge moves in the direction of the field. – A negative charge would accelerate in a direction

opposite to the field, gaining K, losing U.

25.2

• A more general case is if the charge moves a displacement vector s, that is not parallel to the field lines.

25.2

• Again with a uniform E field, it can be removed from the integral.

• So the Potential Energy of the charge-field system is.

B

AdV sE

B

AdV sE

sEV

sE oo qVqU

25.2

• Now, all points in a plane that is perpendicular to the uniform field have the same Electric Potential.

• We can see this is true from the cosine component within the dot product.• The Potential Difference VB-VA is equal to VC-VA

25.2

• Equipotential Surface- any surface having a continuous distribution of points having the same electric potential

• For a uniform E-Field, equipotential surfaces are a family of parallel planes that are all perpendicular to the field.

• Quick Quizzes p 766• Examples 25.1, 25.2

25.3 Electric Potential and Potential Energy Due to Point Charges

• We can determine the electric potential around a single point charge.

• Consider points A and B neara source charge q.• As it moves through ds, its radial distance changes by dr, where dscosθ = dr

25.3

• We can then determine the change in electric potential from points A to B.

B

AAB dVV sE

B

A eAB dr

qkVV sr̂

2 drdsd cos)1(ˆ sr

drr

qkVVB

A

r

reAB 2

1

ABe

r

r

eAB rr

qkr

qkVV

B

A

11

25.3

• We see that this result is independent of the path from A to B, and therefore the Electric field of a fixed point charge is conservative.

• Electric Potential at any distance from a charge is

• V = 0 at ∞

r

qkV e

25.3

• As a scalar quantity the electric potential around multiple charges is simply the sum of electric potentials.

• Example DipoleThe steep slope indicatesA strong E field betweenthe charges.

i i

ie r

qkV

25.3

• Potential energy is

• And therefore, between two charges

VqU o

12

21

r

qqkU e

25.3

• And with several charges (Example of 3)

• Using q’s +/- takes into account whether postive or negative work must be done to keep the charges in place.

23

32

13

31

12

21

r

qq

r

qq

r

qqkU e

25.3

• Like V, U = 0 at ∞

• Quick Quizzes p 770• Example 25.3

25.4 Obtaining E-Field from Electric Potential

• Equipotential Surfaces are must always be perpendicular to the electric field lines passing through them.

• Uniform Field

25.4

• Point Charge

25.4

• Electric Dipole

25.5 Electric Potential from Continuous Charge Distributions

• Adding up each little potential of each piece of charge.

• So the total potential will equal

r

dqkdV e

r

dqkV e

25.5

• Examples 25.5-25.8

25.6 Electric Potential due to Charged Conductors

• Charged Conductors– Charge resides on the surface.– E-field just outside the surface is perpendicular

and equal to σ/εo

– E-field inside is zero.

25.6

• Consider two points on a charged conductor.• Since E is always perp. to the surface,any small displacement along the surface ds will be perp to E. (E.ds = 0)• Therefore ΔV = 0 • The surface of a charged conductor is equipotential

25.6

• Also, since the E-Field inside the conductor is zero, the rate of change of voltage dV is zero, so V must be constant.

• The potential inside is equal to the potential at the surface.

25.6

• The surface density is uniform on a conducting sphere.

• An irregular conductor will have greater charge density (and also E) at convex points with small radii of curvature.

• Sharp points on the conductor will have the highest charge density.

25.6

• Example 25.9 p 780

25.6

• Corona Discharge- High Voltage conductors can cause ionization in the air molecules– Separated electrons are accelerated away from

parent molecules, causing additional ionizations– Eventually the electrons/molecules recombine,

giving off a dim glow (excited state -> ground state)

25.6

• The Corona Discharge effect tends to occur in at sharp points and edges of conductors.– Useful for identifying fraying wire strands, broken

insulators etc. – Still difficult because the majority of the radiation

is in the UV band, washed out by sunlight.

25.7-25.8

• Read p. 781-784• Millikan Oil-Drop Experiement– Determined the value of e.

• Other Applications of Electrostatics