Chapter 15 Electric Forces and Electric Fields. A Bit of History Ancient Greeks Ancient Greeks...

Chapter 15Chapter 15

Electric Forces andElectric Forces and

Electric FieldsElectric Fields

A Bit of HistoryA Bit of History

Ancient GreeksAncient Greeks Observed electric and magnetic Observed electric and magnetic

phenomena as early as 700 BCphenomena as early as 700 BC Found that amber, when rubbed, became Found that amber, when rubbed, became

electrified and attracted pieces of straw electrified and attracted pieces of straw or feathersor feathers

Magnetic forces were discovered by Magnetic forces were discovered by observing magnetite attracting ironobserving magnetite attracting iron

A Bit More HistoryA Bit More History

William GilbertWilliam Gilbert 16001600 Found that electrification was not Found that electrification was not

limited to amberlimited to amber Charles CoulombCharles Coulomb

17851785 Confirmed the inverse square Confirmed the inverse square

relationship of electrical forcesrelationship of electrical forces

History FinalHistory Final Hans OerstedHans Oersted

18201820 Compass needle deflects when placed near an electrical Compass needle deflects when placed near an electrical

currentcurrent

Michael FaradayMichael Faraday A wire moved near a magnet, an electric current is A wire moved near a magnet, an electric current is

observed in the wireobserved in the wire

James Clerk MaxwellJames Clerk Maxwell 1865-18731865-1873 Formulated the laws of electromagnetismFormulated the laws of electromagnetism

HertzHertz Verified Maxwell’s equationsVerified Maxwell’s equations

Properties of Electric Properties of Electric ChargesCharges

Two types of charges existTwo types of charges exist They are called positive and negativeThey are called positive and negative Named by Benjamin FranklinNamed by Benjamin Franklin

Like charges repel and unlike charges Like charges repel and unlike charges attract one anotherattract one another

Nature’s basic carrier of positive charge is Nature’s basic carrier of positive charge is the protonthe proton Protons do not move from one material to Protons do not move from one material to

another because they are held firmly in the another because they are held firmly in the nucleusnucleus

More Properties of ChargeMore Properties of Charge

Nature’s basic carrier of negative Nature’s basic carrier of negative charge is the electroncharge is the electron Gaining or losing electrons is how an Gaining or losing electrons is how an

object becomes chargedobject becomes charged Electric charge is always conservedElectric charge is always conserved

Charge is not created, only exchangedCharge is not created, only exchanged Objects become charged because Objects become charged because

negative charge is transferred from negative charge is transferred from one object to anotherone object to another

Properties of Charge, finalProperties of Charge, final

Charge is quantizedCharge is quantized All charge is a multiple of a fundamental All charge is a multiple of a fundamental

unit of charge, symbolized by eunit of charge, symbolized by e Quarks are the exceptionQuarks are the exception

Electrons have a charge of –eElectrons have a charge of –e Protons have a charge of +eProtons have a charge of +e The SI unit of charge is the Coulomb (C)The SI unit of charge is the Coulomb (C)

e = 1.6 x 10e = 1.6 x 10-19-19 C C

ConductorsConductors

Conductors are materials in which Conductors are materials in which the electric charges move freelythe electric charges move freely Copper, aluminum and silver are good Copper, aluminum and silver are good

conductorsconductors When a conductor is charged in a When a conductor is charged in a

small region, the charge readily small region, the charge readily distributes itself over the entire distributes itself over the entire surface of the materialsurface of the material

InsulatorsInsulators

Insulators are materials in which Insulators are materials in which electric charges do not move freelyelectric charges do not move freely Glass and rubber are examples of Glass and rubber are examples of

insulatorsinsulators When insulators are charged by When insulators are charged by

rubbing, only the rubbed area rubbing, only the rubbed area becomes chargedbecomes charged

There is no tendency for the charge to There is no tendency for the charge to move into other regions of the materialmove into other regions of the material

SemiconductorsSemiconductors

The characteristics of The characteristics of semiconductors are between those semiconductors are between those of insulators and conductorsof insulators and conductors

Silicon and germanium are Silicon and germanium are examples of semiconductorsexamples of semiconductors

Charging by ConductionCharging by Conduction A charged object (the rod) A charged object (the rod)

is placed in contact with is placed in contact with another object (the sphere)another object (the sphere)

Some electrons on the rod Some electrons on the rod can move to the spherecan move to the sphere

When the rod is removed, When the rod is removed, the sphere is left with a the sphere is left with a chargecharge

The object being charged is The object being charged is always left with a charge always left with a charge having the same sign as having the same sign as the object doing the the object doing the chargingcharging

Charging by InductionCharging by Induction When an object is When an object is

connected to a conducting connected to a conducting wire or pipe buried in the wire or pipe buried in the earth, it is said to be earth, it is said to be groundedgrounded

A negatively charged A negatively charged rubber rod is brought near rubber rod is brought near an uncharged spherean uncharged sphere

The charges in the sphere The charges in the sphere are redistributedare redistributed Some of the electrons in Some of the electrons in

the sphere are repelled the sphere are repelled from the electrons in the from the electrons in the rodrod

Charging by Induction, Charging by Induction, contcont

The region of the sphere nearest The region of the sphere nearest the negatively charged rod has an the negatively charged rod has an excess of positive charge because excess of positive charge because of the migration of electrons away of the migration of electrons away from this locationfrom this location

A grounded conducting wire is A grounded conducting wire is connected to the sphereconnected to the sphere Allows some of the electrons to move Allows some of the electrons to move

from the sphere to the groundfrom the sphere to the ground

Charging by Induction, Charging by Induction, finalfinal

The wire to ground is removed, the The wire to ground is removed, the sphere is left with an excess of sphere is left with an excess of induced positive chargeinduced positive charge

The positive charge on the sphere The positive charge on the sphere is evenly distributed due to the is evenly distributed due to the repulsion between the positive repulsion between the positive chargescharges

Charging by induction requires no Charging by induction requires no contact with the object inducing contact with the object inducing the chargethe charge

PolarizationPolarization

In most neutral atoms or molecules, the In most neutral atoms or molecules, the center of positive charge coincides with center of positive charge coincides with the center of negative chargethe center of negative charge

In the presence of a charged object, these In the presence of a charged object, these centers may separate slightlycenters may separate slightly This results in more positive charge on one This results in more positive charge on one

side of the molecule than on the other sideside of the molecule than on the other side This realignment of charge on the surface This realignment of charge on the surface

of an insulator is known as of an insulator is known as polarizationpolarization

Examples of Examples of PolarizationPolarization

The charged object The charged object (on the left) (on the left) induces charge on induces charge on the surface of the the surface of the insulatorinsulator

A charged comb A charged comb attracts bits of attracts bits of paper due to paper due to polarization of the polarization of the paperpaper

QUICK QUIZ 15.1If a suspended object A is attracted to object B, which is charged, we can conclude that (a) object A is uncharged, (b) object A is charged, (c) object B is positively charged, or (d) object A may be either charged or uncharged.

QUICK QUIZ 15.1 ANSWER(d). Object A could possess a net

charge whose sign is opposite that of the excess charge on B. If object A is neutral, B would also attract it by creating an induced charge on the surface of A. This situation is illustrated in Figure 15.5 of the textbook.

Coulomb’s LawCoulomb’s Law

Coulomb shows that an electrical force Coulomb shows that an electrical force has the following properties:has the following properties: It is inversely proportional to the square of It is inversely proportional to the square of

the separation between the two particles the separation between the two particles and is along the line joining themand is along the line joining them

It is proportional to the product of the It is proportional to the product of the magnitudes of the charges qmagnitudes of the charges q11 and q and q22 on the on the two particlestwo particles

It is attractive if the charges are of opposite It is attractive if the charges are of opposite signs and repulsive if the charges have the signs and repulsive if the charges have the same signssame signs

Coulomb’s Law, cont.Coulomb’s Law, cont.

Mathematically,Mathematically,

kkee is called the is called the Coulomb ConstantCoulomb Constant kkee = 8.99 x 10 = 8.99 x 1099 N m N m22/C/C22

Typical charges can be in the µC rangeTypical charges can be in the µC range Remember, Coulombs must be used in the Remember, Coulombs must be used in the

equationequation Remember that force is a Remember that force is a vector vector quantityquantity

2

21e r

qqkF

Vector Nature of Electric Vector Nature of Electric ForcesForces

Two point charges Two point charges are separated by a are separated by a distance distance rr

The like charges The like charges produce a repulsive produce a repulsive force between themforce between them

The force on qThe force on q11 is is equal in magnitude equal in magnitude and opposite in and opposite in direction to the force direction to the force on qon q22

Vector Nature of Forces, Vector Nature of Forces, cont.cont.

Two point charges Two point charges are separated by a are separated by a distance distance rr

The unlike charges The unlike charges produce a attractive produce a attractive force between themforce between them

The force on qThe force on q11 is is equal in magnitude equal in magnitude and opposite in and opposite in direction to the force direction to the force on qon q22

Electrical Forces are Field Electrical Forces are Field ForcesForces

This is the second example of a field forceThis is the second example of a field force Gravity was the firstGravity was the first

Remember, with a field force, the force is Remember, with a field force, the force is exerted by one object on another object exerted by one object on another object even though there is no physical contact even though there is no physical contact between thembetween them

There are some important differences There are some important differences between electrical and gravitational forcesbetween electrical and gravitational forces

Electrical Force Compared Electrical Force Compared to Gravitational Forceto Gravitational Force

Both are inverse square lawsBoth are inverse square laws The mathematical form of both The mathematical form of both

laws is the samelaws is the same Electrical forces can be either Electrical forces can be either

attractive or repulsiveattractive or repulsive Gravitational forces are always Gravitational forces are always

attractiveattractive

QUICK QUIZ 15.2Object A has a charge of +2 µC, and object B has a charge of +6 µC. Which statement is true:(a) FAB = –3FBA, (b) FAB = –FBA, or (c) 3FAB = –FBA

QUICK QUIZ 15.2 ANSWER

(b). By Newton’s third law, the two objects will exert forces having equal magnitudes but opposite directions on each other.

The Superposition The Superposition PrinciplePrinciple

The resultant force on any one The resultant force on any one charge equals the vector sum of charge equals the vector sum of the forces exerted by the other the forces exerted by the other individual charges that are individual charges that are present.present. Remember to add the forces Remember to add the forces

vectoriallyvectorially

Superposition Principle Superposition Principle ExampleExample

The force exerted The force exerted by qby q11 on q on q3 3 is is FF1313

The force exerted The force exerted by qby q22 on q on q33 is is FF2323

The The total forcetotal force exerted on qexerted on q33 is is the vector sum of the vector sum of FF1313 and and FF2323

Electrical FieldElectrical Field

Maxwell developed an approach to Maxwell developed an approach to discussing fieldsdiscussing fields

An An electric fieldelectric field is said to exist in is said to exist in the region of space around a the region of space around a charged objectcharged object When another charged object enters When another charged object enters

this electric field, the field exerts a this electric field, the field exerts a forceforce on the second charged object on the second charged object

Electric Field, cont.Electric Field, cont.

A charged particle, A charged particle, with charge Q, with charge Q, produces an produces an electric field in the electric field in the region of space region of space around itaround it

A small A small test test chargecharge, q, qoo, placed , placed in the field, will in the field, will experience a forceexperience a force

Electric FieldElectric Field

Mathematically,Mathematically,

Use this for the magnitude of the fieldUse this for the magnitude of the field The electric field is a vector quantityThe electric field is a vector quantity The direction of the field is defined to The direction of the field is defined to

be the direction of the electric force be the direction of the electric force that would be exerted on a small that would be exerted on a small positive test charge placed at that positive test charge placed at that pointpoint

2e

o r

Qk

q

FE

Direction of Electric FieldDirection of Electric Field

The electric field The electric field produced by a produced by a negative charge negative charge is directed toward is directed toward the chargethe charge A positive test A positive test

charge would be charge would be attracted to the attracted to the negative source negative source chargecharge

Direction of Electric Field, Direction of Electric Field, contcont

The electric field The electric field produced by a produced by a positive charge is positive charge is directed away directed away from the chargefrom the charge A positive test A positive test

charge would be charge would be repelled from the repelled from the positive source positive source chargecharge

More About a Test Charge More About a Test Charge and The Electric Fieldand The Electric Field

The test charge is required to be a small The test charge is required to be a small chargecharge It can cause no rearrangement of the It can cause no rearrangement of the

charges on the source chargecharges on the source charge The electric field exists whether or not The electric field exists whether or not

there is a test charge presentthere is a test charge present The Superposition Principle can be The Superposition Principle can be

applied to the electric field if a group of applied to the electric field if a group of charges is presentcharges is present

Problem Solving StrategyProblem Solving Strategy UnitsUnits

When using kWhen using kee, charges must be in Coulombs, , charges must be in Coulombs, distances in meters and force in Newtonsdistances in meters and force in Newtons

If values are given in other units, they must be If values are given in other units, they must be convertedconverted

Applying Coulomb’s Law to point chargesApplying Coulomb’s Law to point charges Use the superposition principle for more than Use the superposition principle for more than

two chargestwo charges Use Coulomb’s Law to find the individual forcesUse Coulomb’s Law to find the individual forces Directions of forces are found by noting that Directions of forces are found by noting that

like charges repel and unlike charges attractlike charges repel and unlike charges attract

Problem Solving Problem Solving Strategies, contStrategies, cont

Calculating Electric Fields of point Calculating Electric Fields of point chargescharges The Superposition Principle can be The Superposition Principle can be

applied if more than one charge is applied if more than one charge is presentpresent

Use the equation to find the electric Use the equation to find the electric field due to the individual chargesfield due to the individual charges

The direction is given by the direction The direction is given by the direction of the force on a positive test chargeof the force on a positive test charge

QUICK QUIZ 15.3A test charge of +3 µC is at a point P where the electric field due to other charges is directed to the right and has a magnitude of 4 106 N/C. If the test charge is replaced with a –3 µC charge, the electric field at P (a) has the same magnitude but changes direction, (b) increases in magnitude and changes direction, (c) remains the same, or (d) decreases in magnitude and changes direction.

QUICK QUIZ 15.3 ANSWER(c). The electric field at point P is

due to charges other than the test charge. Thus, it is unchanged when the test charge is altered. However, the direction of the force this field exerts on the test change is reversed when the sign of the test charge is changed.

QUICK QUIZ 15.4A Styrofoam ball covered with a conducting paint has a mass of 5.0 10-3 kg and has a charge of 4.0 C. What electric field directed upward will produce an electric force on the ball that will balance the weight of the ball? (a) 8.2 102 N/C (b) 1.2 104 N/C (c) 2.0 10-2 N/C (d) 5.1 106 N/C


(b). The magnitude of the upward electrical force must equal the weight of the ball. That is: qE = mg, so

C 100.4

)m/s 80.9)(kg 100.5(6

23

q

mgE

= 1.2 × 104 N/C

QUICK QUIZ 15.5A circular ring of radius b has a total charge q uniformly distributed around it. The magnitude of the electric field at the center of the ring is (a) 0 (b) keq/b2 (c) keq2/b2 (d) keq2/b (e) none of these.

QUICK QUIZ 15.5 ANSWER(a). If a test charge is at the center

of the ring, the force exerted on the test charge by charge on any small segment of the ring will be balanced by the force exerted by charge on the diametrically opposite segment of the ring. The net force on the test charge, and hence the electric field at this location, must then be zero.

A "free" electron and "free" proton are placed in an identical electric field. Which of the following statements are true? (a) Each particle experiences the same electric force and the same acceleration. (b) The electric force on the proton is greater in magnitude than the force on the electron but in the opposite direction. (c) The electric force on the proton is equal in magnitude to the force on the electron, but in the opposite direction. (d) The magnitude of the acceleration of the electron is greater than that of the proton. (e) Both particles experience the same acceleration.

QUICK QUIZ 15.6


(c) and (d). The electron and the proton have equal magnitude charges of opposite signs. The forces exerted on these particles by the electric field have equal magnitude and opposite directions. The electron experiences an acceleration of greater magnitude than does the proton because the electron’s mass is much smaller than that of the proton.

Electric Field LinesElectric Field Lines

A convenient aid for visualizing A convenient aid for visualizing electric field patterns is to draw electric field patterns is to draw lines pointing in the direction of lines pointing in the direction of the field vector at any pointthe field vector at any point

These are called These are called electric field lineselectric field lines and were introduced by Michael and were introduced by Michael FaradayFaraday

Electric Field Lines, cont.Electric Field Lines, cont.

The field lines are related to the The field lines are related to the field byfield by The electric field vector, E, is tangent The electric field vector, E, is tangent

to the electric field lines at each pointto the electric field lines at each point The number of lines per unit area The number of lines per unit area

through a surface perpendicular to through a surface perpendicular to the lines is proportional to the the lines is proportional to the strength of the electric field in a given strength of the electric field in a given regionregion

Electric Field Line PatternsElectric Field Line Patterns

Point chargePoint charge The lines radiate The lines radiate

equally in all equally in all directionsdirections

For a positive For a positive source charge, source charge, the lines will the lines will radiate outwardradiate outward


For a negative For a negative source charge, source charge, the lines will point the lines will point inwardinward


An electric An electric dipoledipole consists of two consists of two equal and opposite equal and opposite chargescharges

The high density of The high density of lines between the lines between the charges indicates charges indicates the strong electric the strong electric field in this regionfield in this region

Electric Field Line PatternsElectric Field Line Patterns Two equal but like point Two equal but like point

charges charges At a great distance from At a great distance from

the charges, the field the charges, the field would be approximately would be approximately that of a single charge of that of a single charge of 2q2q

The bulging out of the field The bulging out of the field lines between the charges lines between the charges indicates the repulsion indicates the repulsion between the chargesbetween the charges

The low field lines between The low field lines between the charges indicates a the charges indicates a weak field in this regionweak field in this region

Electric Field PatternsElectric Field Patterns

Unequal and Unequal and unlike chargesunlike charges

Note that two Note that two lines leave the lines leave the +2q charge for +2q charge for each line that each line that terminates on -terminates on -qq

Rules for Drawing Electric Rules for Drawing Electric Field LinesField Lines

The lines for a group of charges must The lines for a group of charges must begin on positive charges and end on begin on positive charges and end on negative chargesnegative charges In the case of an excess of charge, some lines In the case of an excess of charge, some lines

will begin or end infinitely far awaywill begin or end infinitely far away The number of lines drawn leaving a The number of lines drawn leaving a

positive charge or ending on a negative positive charge or ending on a negative charge is proportional to the magnitude of charge is proportional to the magnitude of the chargethe charge

No two field lines can cross each otherNo two field lines can cross each other

QUICK QUIZ 15.7

Rank the magnitudes of the electric field at points A, B, and C in the figure below, largest magnitude first.


A, B, and C. The field is greatest at point A because this is where the field lines are closest together. The absence of lines at point C indicates that the electric field there is zero.

Conductors in Electrostatic Conductors in Electrostatic EquilibriumEquilibrium

When no net motion of charge occurs within a When no net motion of charge occurs within a conductor, the conductor is said to be in conductor, the conductor is said to be in electrostatic equilibriumelectrostatic equilibrium

An isolated conductor has the following properties:An isolated conductor has the following properties: The electric field is zero everywhere inside the The electric field is zero everywhere inside the

conducting materialconducting material Any excess charge on an isolated conductor resides Any excess charge on an isolated conductor resides

entirely on its surfaceentirely on its surface The electric field just outside a charged conductor is The electric field just outside a charged conductor is

perpendicular to the conductor’s surfaceperpendicular to the conductor’s surface On an irregularly shaped conductor, the charge On an irregularly shaped conductor, the charge

accumulates at locations where the radius of curvature of accumulates at locations where the radius of curvature of the surface is smallest (that is, at sharp points)the surface is smallest (that is, at sharp points)

Property 1Property 1

The electric field is zero everywhere The electric field is zero everywhere inside the conducting materialinside the conducting material Consider if this were Consider if this were notnot true true

if there were an electric field inside the if there were an electric field inside the conductor, the free charge there would conductor, the free charge there would move and there would be a flow of chargemove and there would be a flow of charge

If there were a movement of charge, the If there were a movement of charge, the conductor would not be in equilibriumconductor would not be in equilibrium


Any excess charge on an isolated Any excess charge on an isolated conductor resides entirely on its conductor resides entirely on its surfacesurface A direct result of the 1/rA direct result of the 1/r22 repulsion repulsion

between like charges in Coulomb’s Lawbetween like charges in Coulomb’s Law If some excess of charge could be placed If some excess of charge could be placed

inside the conductor, the repulsive forces inside the conductor, the repulsive forces would push them as far apart as possible, would push them as far apart as possible, causing them to migrate to the surfacecausing them to migrate to the surface

Property 3Property 3 The electric field just The electric field just

outside a charged outside a charged conductor is conductor is perpendicular to the perpendicular to the conductor’s surfaceconductor’s surface Consider what would Consider what would

happen it this was not happen it this was not truetrue

The component along The component along the surface would the surface would cause the charge to cause the charge to movemove

It would not be in It would not be in equilibriumequilibrium


On an irregularly On an irregularly shaped conductor, shaped conductor, the charge the charge accumulates at accumulates at locations where locations where the radius of the radius of curvature of the curvature of the surface is smallest surface is smallest (that is, at sharp (that is, at sharp points)points)

Property 4, cont.Property 4, cont.

Any excess charge moves to its surfaceAny excess charge moves to its surface The charges move apart until an equilibrium is achievedThe charges move apart until an equilibrium is achieved The amount of charge per unit area is greater at the flat endThe amount of charge per unit area is greater at the flat end The forces from the charges at the sharp end produce a The forces from the charges at the sharp end produce a

larger resultant force away from the surfacelarger resultant force away from the surface Why a lightning rod worksWhy a lightning rod works

Experiments to Verify Experiments to Verify Properties of ChargesProperties of Charges

Faraday’s Ice-Pail ExperimentFaraday’s Ice-Pail Experiment Concluded a charged object suspended inside a Concluded a charged object suspended inside a

metal container causes a rearrangement of metal container causes a rearrangement of charge on the container in such a manner that charge on the container in such a manner that the sign of the charge on the inside surface of the sign of the charge on the inside surface of the container is opposite the sign of the charge the container is opposite the sign of the charge on the suspended objecton the suspended object

Millikan Oil-Drop ExperimentMillikan Oil-Drop Experiment Measured the elementary charge, eMeasured the elementary charge, e Found every charge had an integral multiples of Found every charge had an integral multiples of

ee q = n eq = n e

Van de GraaffVan de GraaffGeneratorGenerator

An electrostatic An electrostatic generator designed generator designed and built by Robert J. and built by Robert J. Van de Graaff in 1929Van de Graaff in 1929

Charge is transferred Charge is transferred to the dome by to the dome by means of a rotating means of a rotating beltbelt

Eventually an Eventually an electrostatic electrostatic discharge takes placedischarge takes place

Electric FluxElectric Flux

Field lines Field lines penetrating an penetrating an area A area A perpendicular to perpendicular to the fieldthe field

The product of EA The product of EA is the flux, is the flux, ΦΦ

In general:In general: ΦΦEE = E A sin θ = E A sin θ

Electric Flux, cont.Electric Flux, cont.

ΦΦEE = E A sin θ = E A sin θ The perpendicular to the area A is at The perpendicular to the area A is at

an angle an angle θ to the fieldθ to the field When the area is constructed such When the area is constructed such

that a closed surface is formed, use that a closed surface is formed, use the convention that flux lines passing the convention that flux lines passing into the interior of the volume are into the interior of the volume are negative and those passing out of the negative and those passing out of the interior of the volume are positiveinterior of the volume are positive

Gauss’ LawGauss’ Law Gauss’ Law states that the electric flux Gauss’ Law states that the electric flux

through any closed surface is equal to the through any closed surface is equal to the net charge Q inside the surface divided by net charge Q inside the surface divided by εεoo

εεoo is the is the permittivity of free spacepermittivity of free space and equals and equals 8.85 x 108.85 x 10-12-12 C C22/Nm/Nm22

The area in The area in Φ is an imaginary surface, a Gaussian Φ is an imaginary surface, a Gaussian surface, it does not have to coincide with the surface, it does not have to coincide with the surface of a physical objectsurface of a physical object

oE

Q

QUICK QUIZ 15.8

For a surface through which the net flux is zero, the following four statements could be true. Which of the statements must be true? (a) There are no charges inside the surface. (b) The net charge inside the surface is zero. (c) The electric field is zero everywhere on the surface. (d) The number of electric field lines entering the surface equals the number leaving the surface.


Statements (b) and (d) are true and follow from Gauss’s law. Statement (a) is not necessarily true because Gauss’s law says that the net flux through any closed surface equals the net charge inside the surface divided by ε0. For example, a positive and a negative charge could be inside the surface. Statement (c) is not necessarily true. Although the net flux through the surface is zero, the electric field in that region may not be zero.

Electric Field of a Charged Electric Field of a Charged Thin Spherical ShellThin Spherical Shell

The calculation of the field outside the shell is identical The calculation of the field outside the shell is identical to that of a point chargeto that of a point charge

The electric field inside the shell is zeroThe electric field inside the shell is zero

2eo

2 r

Qk

r4

QE

Electric Field of a Nonconducting Electric Field of a Nonconducting Plane Sheet of ChargePlane Sheet of Charge

Use a cylindrical Use a cylindrical Gaussian surfaceGaussian surface

The flux through the The flux through the ends is EA, there is no ends is EA, there is no field through the field through the curved part of the curved part of the surfacesurface

The total charge is Q The total charge is Q = = σAσA

Note, the field is Note, the field is uniformuniform

o2E

Chapter 15 Electric Forces and Electric Fields. A Bit of History Ancient Greeks Ancient Greeks...

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