pc chapter 25

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Chapter 25

Electric Potential

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Electrical Potential Energy When a test charge is placed in an

electric field, it experiences a force F = qoE

The force is conservative

ds is an infinitesimal displacement

vector that is oriented tangent to a path

through space

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Electric Potential Energy, cont The work done by the electric field is

F.ds = qoE.ds

As this work is done by the field, the

potential energy of the charge-field

system is changed by U= -qoE.ds

For a finite displacement of the charge

from A to B,B

B A oA

U U U q d

E s

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Electric Potential Energy, final Because qoE is conservative, the line

integral does not depend on the path

taken by the charge

This is the change in potential energy of

the system

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Electric Potential The potential energy per unit charge, U/qo,

is the electric potential The potential is independent of the value ofqo The potential has a value at every point in an

electric field

The electric potential iso

UV q

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Electric Potential, cont. The potential is a scalar quantity

Since energy is a scalar

As a charged particle moves in an

electric field, it will experience a change

in potentialB

Ao

UV dq E s

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Electric Potential, final The difference in potential is the

meaningful quantity

We often take the value of the potentialto be zero at some convenient point inthe field

Electric potential is a scalarcharacteristic of an electric field,independent of any charges that maybe placed in the field

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Work and Electric Potential Assume a charge moves in an electric

field without any change in its kinetic

energy The work performed on the charge is

W= V= qV

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Units 1 V = 1 J/C

V is a volt

It takes one joule of work to move a 1-coulomb charge through a potentialdifference of 1 volt

In addition, 1 N/C = 1 V/m This indicates we can interpret the electric

field as a measure of the rate of changewith position of the electric potential

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Electron-Volts Another unit of energy that is commonly used

in atomic and nuclear physics is the electron-

volt One electron-volt is defined as the energy a

charge-field system gains or loses when a

charge of magnitude e (an electron or a

proton) is moved through a potentialdifference of 1 volt 1 eV = 1.60 x 10-19 J

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Potential Difference in a

Uniform Field The equations for electric potential can

be simplified if the electric field is

uniform:

The negative sign indicates that the

electric potential at point B is lower thanat pointA

B B

B AA A

V V V d E d Ed E s s

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Energy and the Direction of

Electric Field When the electric field

is directed downward,

point B is at a lowerpotential than pointA

When a positive test

charge moves fromA

to B, the charge-fieldsystem loses potential

energy

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More About Directions A system consisting of a positive charge and

an electric field loses electric potential energywhen the charge moves in the direction of thefield An electric field does work on a positive charge

when the charge moves in the direction of theelectric field

The charged particle gains kinetic energyequal to the potential energy lost by thecharge-field system Another example of Conservation of Energy

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Directions, cont. Ifqo is negative, then Uis positive

A system consisting of a negative

charge and an electric field gains

potential energy when the charge

moves in the direction of the field In order for a negative charge to move in

the direction of the field, an external agent

must do positive work on the charge

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Equipotentials Point B is at a lower

potential than pointA

PointsA and Care at thesame potential

The name equipotential

surface is given to any

surface consisting of a

continuous distribution of

points having the same

electric potential

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Charged Particle in a Uniform

Field, Example A positive charge is

released from rest andmoves in the direction

of the electric field The change in potential

is negative The change in potential

energy is negative The force and

acceleration are in thedirection of the field

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Potential and Point Charges A positive point

charge produces a

field directed radiallyoutward

The potential

difference between

pointsA and B willbe

1 1B A e

B A

V V k qr r

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Potential and Point Charges,

cont. The electric potential is independent of

the path between pointsA and B

It is customary to choose a reference

potential ofV= 0 at rA =

Then the potential at some point ris

e

qV k

r

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Electric Potential of a Point

Charge The electric potential

in the plane around

a single pointcharge is shown

The red line shows

the 1/rnature of the

potential

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Electric Potential of a Dipole The graph shows

the potential (y-axis)

of an electric dipole The steep slope

between the

charges represents

the strong electricfield in this region

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Potential Energy of Multiple

Charges Consider two

charged particles

The potential energyof the system is

1 2

12

e

q qU k

r

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Active Figure 25.10

(SLIDESHOW MODE ONLY)

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More About Uof Multiple

Charges If the two charges are the same sign, U

is positive and work must be done to

bring the charges together If the two charges have opposite signs,

Uis negative and work is done to keep

the charges apart

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Uwith Multiple Charges, final If there are more than

two charges, then findUfor each pair ofcharges and add them

For three charges:

The result isindependent of the

order of the charges

1 3 2 31 2

12 13 23e

q q q qq qU k

r r r

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Finding E From V Assume, to start, that E has only anx

component

Similar statements would apply to the yand zcomponents

Equipotential surfaces must always beperpendicular to the electric field linespassing through them

x dVEdx

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E and Vfor an Infinite Sheet

of Charge The equipotential lines

are the dashed blue

lines The electric field lines

are the brown lines

The equipotential lines

are everywhere

perpendicular to the

field lines

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E and Vfor a Point Charge The equipotential lines

are the dashed blue


are the brown lines


are everywhere


field lines

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E and Vfor a Dipole The equipotential lines

are the dashed blue


are the brown lines


are everywhere


field lines

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Electric Field from Potential,

General In general, the electric potential is a

function of all three dimensions

Given V(x, y, z) you can find Ex, Ey and

Ezas partial derivatives

x y z

V V V

E E E x y z

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Electric Potential for a

Continuous Charge Distribution Consider a small

charge element dq

Treat it as a pointcharge

The potential at

some point due to

this charge elementis

e

dqdV k

r

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Vfor a Continuous Charge

Distribution, cont. To find the total potential, you need to

integrate to include the contributions

from all the elements

This value forVuses the reference ofV=

0 when Pis infinitely far away from the

charge distributions

e

dqV k

r

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Vfor a Uniformly Charged

Ring Pis located on the

perpendicular central

axis of the uniformlycharged ring The ring has a radius

a and a total charge Q

2 2

ee

k QdqV k

r x a

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Disk The ring has a

radius a and surface

charge density of

1

2 2 22 eVk x a x

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Vfor a Finite Line of Charge A rod of line has a

total charge ofQ

and a linear chargedensity of

2 2

lnek Q a

Va

l l

l

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Sphere A solid sphere of

radius Rand total

charge Q Forr> R,

Forr< R,

e

QV k

r

2 2

32e

D C

k QV V R r R

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Sphere, Graph The curve forVD is

for the potentialinside the curve It is parabolic It joins smoothly with

the curve forVB

The curve forVB isfor the potentialoutside the sphere It is a hyperbola

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VDue to a Charged

Conductor Consider two points on

the surface of thecharged conductor as

shown E is always

perpendicular to thedisplacement ds

Therefore, E ds = 0 Therefore, the potential

difference betweenAand B is also zero

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VDue to a Charged

Conductor, cont. Vis constant everywhere on the surface of a

charged conductor in equilibrium V= 0 between any two points on the surface

The surface of any charged conductor inelectrostatic equilibrium is an equipotentialsurface

Because the electric field is zero inside theconductor, we conclude that the electricpotential is constant everywhere inside theconductor and equal to the value at thesurface

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E Compared to V The electric potential is

a function ofr The electric field is a

function ofr2 The effect of a charge

on the spacesurrounding it: The charge sets up a

vector electric field whichis related to the force

The charge sets up ascalar potential which isrelated to the energy

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Irregularly Shaped Objects The charge density is high

where the radius of

curvature is small And low where the radius of

curvature is large

The electric field is large

near the convex points

having small radii ofcurvature and reaches

very high values at sharp

points

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Irregularly Shaped Objects,

cont. The field lines are

still perpendicular to

the conductingsurface at all points

The equipotential

surfaces are

perpendicular to thefield lines

everywhere

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Cavity in a Conductor Assume an

irregularly shaped

cavity is inside aconductor

Assume no charges

are inside the cavity

The electric field

inside the conductor

must be zero

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Cavity in a Conductor, cont The electric field inside does not depend on

the charge distribution on the outside surfaceof the conductor

For all paths betweenA and B,

A cavity surrounded by conducting walls is afield-free region as long as no charges areinside the cavity

0B

B AA

V V d E s

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Corona Discharge If the electric field near a conductor is

sufficiently strong, electrons resulting

from random ionizations of airmolecules near the conductor

accelerate away from their parent

molecules These electrons can ionize additional

molecules near the conductor

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Corona Discharge, cont. This creates more free electrons The corona discharge is the glow that

results from the recombination of thesefree electrons with the ionized airmolecules

The ionization and corona dischargeare most likely to occur near very sharppoints

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Millikan Oil-Drop Experiment

Experimental Set-Up

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Millikan Oil-Drop Experiment Robert Millikan measured e, the

magnitude of the elementary charge on

the electron He also demonstrated the quantized

nature of this charge

Oil droplets pass through a small holeand are illuminated by a light

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Active Figure 25.27

(SLIDESHOW MODE ONLY)

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Oil-Drop Experiment, 2 With no electric field

between the plates,

the gravitationalforce and the drag

force (viscous) act

on the electron

The drop reachesterminal velocity with

FD = mg

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Oil-Drop Experiment, 3 When an electric field

is set up between theplates The upper plate has a

higher potential

The drop reaches anew terminal velocity

when the electricalforce equals the sumof the drag force andgravity

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Oil-Drop Experiment, final The drop can be raised and allowed to

fall numerous times by turning the

electric field on and off After many experiments, Millikan

determined:

q = ne where n = 1, 2, 3, e = 1.60 x 10-19 C

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Van de Graaff

Generator Charge is delivered continuously to

a high-potential electrode by meansof a moving belt of insulating

material The high-voltage electrode is a

hollow metal dome mounted on aninsulated column

Large potentials can be developedby repeated trips of the belt

Protons accelerated through suchlarge potentials receive enoughenergy to initiate nuclear reactions

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Electrostatic Precipitator An application of electrical discharge in

gases is the electrostatic precipitator

It removes particulate matter from

combustible gases The air to be cleaned enters the duct and

moves near the wire

As the electrons and negative ions created

by the discharge are accelerated toward the

outer wall by the electric field, the dirtparticles become charged

Most of the dirt particles are negatively

charged and are drawn to the walls by the

electric field

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Application Xerographic

Copiers The process of xerography is used for

making photocopies

Uses photoconductive materials A photoconductive material is a poor

conductor of electricity in the dark but

becomes a good electric conductor whenexposed to light

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Application Laser Printer The steps for producing a document on a laser

printer is similar to the steps in the xerographicprocess Steps a, c, and d are the same The major difference is the way the image forms on

the selenium-coated drum A rotating mirror inside the printer causes the beam of the

laser to sweep across the selenium-coated drum The electrical signals form the desired letter in positive

charges on the selenium-coated drum Toner is applied and the process continues as in the

xerographic process

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