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General Physics (PHY 2140)

Lecture IILecture II

Electrostatics

Coulomb’s lawElectric field

Chapter 15

http://www.physics.wayne.edu/~apetrov/PHY2140/

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Lightning ReviewLightning Review

Last lecture:

1. Properties of electric chargetwo types: positive and negativealways conserved and quantized

2. Insulators and conductorscharges move freely in conductors; opposite is true for insulatorsconductors can be charged by conduction and induction; insulators can be polarized

Review Problem: OperatingOperating--room personnel must wear special conducting room personnel must wear special conducting shoes while working around oxygen. Why? What might shoes while working around oxygen. Why? What might happen if personnel wore ordinary rubber shoes happen if personnel wore ordinary rubber shoes (sneakers)?(sneakers)?

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15.3 Coulomb’s Law 15.3 Coulomb’s Law -- ObservationObservationCharles Coulomb discovered in 1785 the fundamental law of Charles Coulomb discovered in 1785 the fundamental law of electrical force between two stationary charged particles.electrical force between two stationary charged particles.An electric force has the following properties:An electric force has the following properties:

Inversely proportionalInversely proportional to the to the square of the separationsquare of the separation, , rr, between the , between the particles, and is along a line joining them.particles, and is along a line joining them.Proportional to the product of the magnitudes of the charges Proportional to the product of the magnitudes of the charges |q|q11|| and and |q|q22|| on the two particles. on the two particles. AttractiveAttractive if the charges are of if the charges are of opposite signopposite sign and and repulsiverepulsive if the charges if the charges have have the same signthe same sign..

q1 q2

r

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15.3 Coulomb’s Law 15.3 Coulomb’s Law –– Mathematical Mathematical FormulationFormulation

kkee known as the Coulomb constant.known as the Coulomb constant.Value of Value of kkee depends on the choice of units.depends on the choice of units.SI unitsSI units

Force: the Newton (N)Force: the Newton (N)Distance: the meter (m).Distance: the meter (m).Charge: the coulomb ( C).Charge: the coulomb ( C).Current: the ampere (A =1 C/s).Current: the ampere (A =1 C/s).

EExperimentally measurement: xperimentally measurement: kkee = 8.9875= 8.9875××101099 NmNm22/C/C22..Reasonable approximate value: Reasonable approximate value: kkee = 8.99= 8.99××101099 NmNm22/C/C22..

1 22e

q qF k

r=

How do we know the units of ke?

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Example: Fun with unitsExample: Fun with units

Recall that units can be manipulated:

[ ] [ ] [ ][ ][ ]1 2

2e

q qF k

r=1 2

2e

q qF k

r=

[ ] [ ] [ ][ ][ ]2e

Coulomb CoulombNewton k

meter=

[ ] 22e

N mk C⋅=

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ExampleExample

1e = 1e = --1.60 1.60 ××1010--1919 ccTakes 1/e=6.6 Takes 1/e=6.6 ××10101818 protons to create a total charge of 1Cprotons to create a total charge of 1CNumber of free electrons in 1 cmNumber of free electrons in 1 cm33 copper ~ 10copper ~ 102323

Charge obtained in typical electrostatic experiments with Charge obtained in typical electrostatic experiments with rubber or glass 10rubber or glass 10--66 C = 1 C = 1 µµccA very small fraction of the total available chargeA very small fraction of the total available charge

Charge and Mass of the Electron, Charge and Mass of the Electron, Proton and Neutron.Proton and Neutron.

1.67 1.67 ××1010--272700NeutronNeutron1.67 1.67 ××1010--2727+1.60 +1.60 ××1010--1919ProtonProton9.11 9.11 ××1010--3131--1.60 1.60 ××1010--1919ElectronElectronMass (kg)Mass (kg)Charge ( C)Charge ( C)ParticleParticle

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15.3 Coulomb’s Law 15.3 Coulomb’s Law –– RemarksRemarks

The electrostatic force is often called Coulomb force.The electrostatic force is often called Coulomb force.It is a force (thus, a It is a force (thus, a vectorvector): ):

a magnitude a magnitude a direction.a direction.

Second example of action at a distance.Second example of action at a distance.

++

r

q1q2

F21

F21 +-

r

q1

q2F21F21

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MiniMini--QuizQuiz

Name the first action at a distance force you have Name the first action at a distance force you have encountered in physics so far.encountered in physics so far.

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Example: Electrical ForceExample: Electrical Force

Question:Question:The electron and proton of a hydrogen atom are separated (on theThe electron and proton of a hydrogen atom are separated (on theaverage) by a distance of about 5.3x10average) by a distance of about 5.3x10--1111 m. Find the magnitude of the m. Find the magnitude of the electric force that each particle exerts on the other.electric force that each particle exerts on the other.

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Question:Question:The electron and proton of a hydrogen atom are separated (on theThe electron and proton of a hydrogen atom are separated (on the average) by average) by

a distance of about 5.3x10a distance of about 5.3x10--1111 m. Find the magnitude of the electric force that m. Find the magnitude of the electric force that each particle exerts on the other.each particle exerts on the other.

1 22e

q qF k

r=

Observations:Observations:We are interested in finding the magnitude of the force between We are interested in finding the magnitude of the force between two two particles of known charge, and a given distance of each other.particles of known charge, and a given distance of each other.The magnitude is given by Coulomb’s law.The magnitude is given by Coulomb’s law.

qq11 ==--1.60x101.60x10--1919 CCqq22 =1.60x10=1.60x10--1919 CCr = 5.3x10r = 5.3x10--1111 mm

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Question:Question:The electron and proton of a hydrogen atom are separated (on theThe electron and proton of a hydrogen atom are separated (on the average) by average) by a distance of about 5.3x10a distance of about 5.3x10--1111 m. Find the magnitude of the electric force that m. Find the magnitude of the electric force that each particle exerts on the other.each particle exerts on the other.

Observations:Observations:We are interested in finding the magnitude of the force between We are interested in finding the magnitude of the force between two two particles of known charge, and a given distance of each other.particles of known charge, and a given distance of each other.The magnitude is given by Coulomb’s law.The magnitude is given by Coulomb’s law.qq11 ==--1.60x101.60x10--1919 CCqq22 =1.60x10=1.60x10--1919 CCr = 5.3x10r = 5.3x10--1111 mm

Solution:Solution:

Attractive force with a magnitude of 8.2x10Attractive force with a magnitude of 8.2x10--88 N.N.

( )( )

2

2

22 199 8

22 11

1.6 108.99 10 8.2 10

5.3 10Nm

e e C

CeF k N

r m

−−

−

×= = × = ×

×

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Superposition PrincipleSuperposition Principle

From observations: one finds that whenever multiple From observations: one finds that whenever multiple charges are present, the net force on a given charge is charges are present, the net force on a given charge is the the vectorvector sum of all forces exerted by other charges.sum of all forces exerted by other charges.Electric force obeys a Electric force obeys a superposition principlesuperposition principle..

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Example: Using the Superposition PrincipleExample: Using the Superposition Principle

Consider three point charges at the corners of a triangle, as shConsider three point charges at the corners of a triangle, as shown own below. Find the resultant force on qbelow. Find the resultant force on q33 if if qq11 = 6.00 x 10= 6.00 x 10--99 CCqq2 2 = = --2.00 x 102.00 x 10--99 CCqq3 3 = 5.00 x 10= 5.00 x 10--99 CC

+ x

y

- +q2

q1

3.00 m

4.00 mq3

F32

F31

37.0o

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Consider three point charges at the corners of a triangle, as shConsider three point charges at the corners of a triangle, as shown own below. Find the resultant force on qbelow. Find the resultant force on q33..

+ x

y

- +q2

q1

3.00 m

4.00 mq3

F32

F31

37.0o

Observations:Observations:The superposition principle tells us that the net force on qThe superposition principle tells us that the net force on q33 is the vector sum is the vector sum of the forces Fof the forces F3232 and Fand F3131..The magnitude of the forces FThe magnitude of the forces F3232 and Fand F3131 can calculated using Coulomb’s can calculated using Coulomb’s law.law.

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Consider three point charges at the corners of a triangle, as shConsider three point charges at the corners of a triangle, as shown own below. Find the resultant force on qbelow. Find the resultant force on q33..

5.00 m+ x

y- +q2

q1

3.00 m4.00 m

q3

F32

F31

37.0o

( )( )( )

( )( )( )

2

2

2

2

9 93 2 9 9

32 22

9 93 1 9 8

31 22

932 31

931

2 2 9

5.00 10 2.00 108.99 10 5.62 10

4.00

5.00 10 6.00 108.99 10 1.08 10

5.00

cos37.0 3.01 10

sin 37.0 6.50 10

7.16 10

Nme C

Nme C

ox

oy

x y

C Cq qk N

r m

C Cq qF k N

r m

F F F NF F N

F F F N

− −−

− −−

−

−

−

× ×= = × = ×

× ×= = × = ×

= − = ×

= = ×

= + = ×

65.2oθ =

FSolution:Solution:

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15.4 Electric Field 15.4 Electric Field -- DiscoveryDiscovery

Electric forces act through space even in the absence of Electric forces act through space even in the absence of physical contact.physical contact.Suggests the notion of Suggests the notion of electrical fieldelectrical field (first introduced (first introduced by Michael Faraday (1791by Michael Faraday (1791--1867).1867).An electric field is said to exist in a region of space An electric field is said to exist in a region of space surrounding a charged object.surrounding a charged object.If another charged object enters a region where an If another charged object enters a region where an electrical field is present, it will be subject to an electricalelectrical field is present, it will be subject to an electricalforce.force.

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15.4 Electric Field 15.4 Electric Field –– Quantitative DefinitionQuantitative Definition

A field : generally changes with position (location)A field : generally changes with position (location)A vector quantity : magnitude and direction.A vector quantity : magnitude and direction.Magnitude at a given locationMagnitude at a given location

Expressed as a function of the force imparted by the field on Expressed as a function of the force imparted by the field on a given test charge.a given test charge.

o

FE

q=

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15.4 Electric Field 15.4 Electric Field –– Quantitative Definition (2)Quantitative Definition (2)

Direction defined as the direction of the electrical force Direction defined as the direction of the electrical force exerted on a small positive charge placed at that exerted on a small positive charge placed at that location. location.

- -- - -

- - - -- - -

- -

E

+ ++ + +

+ + + ++ + +

+ +

E

+

+ ++ + + +

+ + ++ +

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15.4 Electric Field 15.4 Electric Field –– Electric Field of a Electric Field of a Charge “q”Charge “q”

2o

e

q qF k

r=GivenGiven

One findsOne finds2e

qE k

r=

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• If q>0, field at a given point is radially outward from q.

+q

qo

rE

• If q<0, field at a given point is radially inward from q.

-q

r

E qo

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ProblemProblem--Solving StrategySolving Strategy

Electric Forces and FieldsElectric Forces and FieldsUnits: Units:

For calculations that use the Coulomb constant, For calculations that use the Coulomb constant, kkee, charges must , charges must be in coulombs, and distances in meters.be in coulombs, and distances in meters.Conversion are required if quantities are provided in other unitConversion are required if quantities are provided in other units.s.

Applying Coulomb’s law to point charges.Applying Coulomb’s law to point charges.It is important to use the superposition principle properly.It is important to use the superposition principle properly.Determine the individual forces first.Determine the individual forces first.Determine the vector sum.Determine the vector sum.Determine the magnitude and/or the direction as needed.Determine the magnitude and/or the direction as needed.

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Example:Example:An electron moving horizontally passes between two An electron moving horizontally passes between two horizontal planes, the upper plane charged negatively, horizontal planes, the upper plane charged negatively, and the lower positively. A uniform, upwardand the lower positively. A uniform, upward--directed directed electric field exists in this region. This field exerts a force electric field exists in this region. This field exerts a force on the electron. Describe the motion of the electron in on the electron. Describe the motion of the electron in this region.this region.

-vo

- - - - - - - - - - - - - - - - - - - - - -

+ + + + + + + + + + + + + + + + + + + + + +

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-vo

- - - - - - - - - - - - - - - - - - - - - -

+ + + + + + + + + + + + + + + + + + + + + +

000

x

x

x

x o

o

EFav vx v t

=

=

==

=

Observations:Observations:Horizontally: Horizontally:

No electric field No electric field No forceNo forceNo accelerationNo accelerationConstant horizontal velocityConstant horizontal velocity

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-vo

- - - - - - - - - - - - - - - - - - - - - -

+ + + + + + + + + + + + + + + + + + + + + +

2

/

/

1 /2

y o

y o o

y o o o

y o o o

o o o

E EF q Ea q E mv q E t m

y q E t m

=

=

=

=

=

Observations:Observations:Vertically: Vertically:

Constant electric field Constant electric field Constant forceConstant forceConstant accelerationConstant accelerationVertical velocity increase Vertical velocity increase linearly with time.linearly with time.

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-

- - - - - - - - - - - - - - - - - - - - - -

+ + + + + + + + + + + + + + + + + + + + + +

Conclusions:Conclusions:The charge will follow a parabolic path downward.The charge will follow a parabolic path downward.Motion similar to motion under gravitational field only except tMotion similar to motion under gravitational field only except the he downward acceleration is now larger.downward acceleration is now larger.

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Example: Electric Field Due to Two Point ChargesExample: Electric Field Due to Two Point ChargesQuestion: Question: Charge qCharge q11=7.00 =7.00 µµCC is at the origin, and charge qis at the origin, and charge q22==--10.00 10.00 µµCC is on the x is on the x axis, 0.300 m from the origin. Find the electric field at point axis, 0.300 m from the origin. Find the electric field at point P, which P, which has coordinates (0,0.400) m.has coordinates (0,0.400) m.

x

y

0.300 mq1 q2

0.40

0 m

P

E1

E2

E

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Question: Question: Charge qCharge q11=7.00 =7.00 µµCC is at the origin, and charge qis at the origin, and charge q22==--10.00 10.00 µµCC is on the x is on the x axis, 0.300 m from the origin. Find the electric field at point axis, 0.300 m from the origin. Find the electric field at point P, which P, which has coordinates (0,0.400) m.has coordinates (0,0.400) m.

Observations:Observations:First find the field at point P due to charge qFirst find the field at point P due to charge q11 and qand q22..Field EField E11 at P due to qat P due to q11 is vertically upward.is vertically upward.Field EField E22 at due to qat due to q22 is directed towards qis directed towards q22..The net field at point P is the vector sum of EThe net field at point P is the vector sum of E11 and Eand E22..The magnitude is obtained withThe magnitude is obtained with

2e

qE k

r=

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Question: Question: Charge qCharge q11=7.00 =7.00 µµCC is at the origin, and charge qis at the origin, and charge q22==--10.00 10.00 µµCC is on the x is on the x axis, 0.300 m from the origin. Find the electric field at point axis, 0.300 m from the origin. Find the electric field at point P, which P, which has coordinates (0,0.400) m.has coordinates (0,0.400) m.

Solution:Solution:

( )( )( )( )

2

2

2

2

61 9 5

1 221

62 9 5

2 222

5325

541 2 1 25

2 2 5

7.00 108.99 10 3.93 10 /

0.400

10.00 108.99 10 3.60 10 /

0.500

2.16 10 /

sin 1.05 10 /

2.4 10 /

arctan( / ) 25.9

Nme C

Nme C

x

y

x y

oy x

CqE k N C

r m

CqE k N C

r m

E E N C

E E E E E N C

E E E N C

E E

θ

φ

−

−

×= = × = ×

×= = × = ×

= = ×

= − = − = ×

= + = ×

= =

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15.5 Electric Field Lines15.5 Electric Field Lines

A convenient way to visualize field patterns is to draw A convenient way to visualize field patterns is to draw lines in the direction of the electric field.lines in the direction of the electric field.Such lines are called Such lines are called field linesfield lines..Remarks:Remarks:1.1. Electric field vector, E, is tangent to the electric field linesElectric field vector, E, is tangent to the electric field lines at at

each point in space.each point in space.2.2. The number of lines per unit area through a surface The number of lines per unit area through a surface

perpendicular to the lines is proportional to the strength of thperpendicular to the lines is proportional to the strength of the e electric field in a given region.electric field in a given region.

E is large when the field lines are close together and small E is large when the field lines are close together and small when far apart.when far apart.

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15.5 Electric Field Lines (2)15.5 Electric Field Lines (2)

Electric field lines of single positive (a) and (b) negative Electric field lines of single positive (a) and (b) negative charges.charges.

+ q

a)

- q

b)

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15.5 Electric Field Lines (3)15.5 Electric Field Lines (3)

Rules for drawing electric field lines for any charge Rules for drawing electric field lines for any charge distribution.distribution.1.1. Lines must begin on positive charges (or at infinity) and must Lines must begin on positive charges (or at infinity) and must

terminate on negative charges or in the case of excess charge terminate on negative charges or in the case of excess charge at infinity.at infinity.

2.2. The number of lines drawn leaving a positive charge or The number of lines drawn leaving a positive charge or approaching a negative charge is proportional to the magnitude approaching a negative charge is proportional to the magnitude of the charge.of the charge.

3.3. No two field lines can cross each other.No two field lines can cross each other.

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15.5 Electric Field Lines (4)15.5 Electric Field Lines (4)

Electric field lines of a Electric field lines of a dipoledipole..

+ -

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Application: Measurement of the atmospheric electric fieldApplication: Measurement of the atmospheric electric field

The electric field near the surface of the Earth is about The electric field near the surface of the Earth is about 100 N/C downward. Under a thundercloud, the electric 100 N/C downward. Under a thundercloud, the electric field can be as large as 20000 N/C.field can be as large as 20000 N/C.How can such a (large) field be measured?How can such a (large) field be measured?

A

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AAAAAAAAAAAA

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15.6 Conductors in Electrostatic Equilibrium15.6 Conductors in Electrostatic Equilibrium

Good conductors (e.g. copper, gold) contain charges Good conductors (e.g. copper, gold) contain charges (electron) that are not bound to a particular atom, and (electron) that are not bound to a particular atom, and are free to move within the material.are free to move within the material.When no net motion of these electrons occur the When no net motion of these electrons occur the conductor is said to be in conductor is said to be in electroelectro--static equilibriumstatic equilibrium..

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15.6 Conductors in Electrostatic Equilibrium15.6 Conductors in Electrostatic Equilibrium

Properties of an isolated conductor (insulated from the Properties of an isolated conductor (insulated from the ground).ground).1.1. Electric field is zero everywhere within the conductor.Electric field is zero everywhere within the conductor.2.2. Any excess charge field on an isolated conductor resides Any excess charge field on an isolated conductor resides

entirely on its surface. entirely on its surface. 3.3. The electric field just outside a charged conductor is The electric field just outside a charged conductor is

perpendicular to the conductor’s surface. perpendicular to the conductor’s surface. 4.4. On an irregular shaped conductor, the charge tends to On an irregular shaped conductor, the charge tends to

accumulate at locations where the radius of curvature of the accumulate at locations where the radius of curvature of the surface is smallest surface is smallest –– at sharp points.at sharp points.

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1.1. Electric field is zero everywhere within the conductor.Electric field is zero everywhere within the conductor.

If this was not trueIf this was not true, the field inside would be finite., the field inside would be finite.Free charge there would move under the influence of the Free charge there would move under the influence of the field.field.A current would be induced.A current would be induced.The conductor would not be in an electrostatic state.The conductor would not be in an electrostatic state.

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2.2. Any excess charge field on an isolated conductor resides entirelAny excess charge field on an isolated conductor resides entirely y on its surface.on its surface.

This property is a direct result of the 1/rThis property is a direct result of the 1/r22 repulsion repulsion between like charges.between like charges.If an excess of charge is placed within the volume, the If an excess of charge is placed within the volume, the repulsive force pushes them as far apart as they can go.repulsive force pushes them as far apart as they can go.They thus migrate to the surface.They thus migrate to the surface.

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3.3. The electric field just outside a charged conductor is The electric field just outside a charged conductor is perpendicular to the conductor’s surface. perpendicular to the conductor’s surface.

If not true, the field would have components parallel to If not true, the field would have components parallel to the surface of the conductor.the surface of the conductor.This field component would cause free charges of the This field component would cause free charges of the conductor to move.conductor to move.A current would be created.A current would be created.There would no longer be a electroThere would no longer be a electro--static equilibrium.static equilibrium.

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4.4. On an irregular shaped conductor, the charge tends to accumulateOn an irregular shaped conductor, the charge tends to accumulate at at locations where the radius of curvature of the surface is smallelocations where the radius of curvature of the surface is smallest st –– at at sharp points.sharp points.

Consider, for instance, a conductor fairly flat at one end and rConsider, for instance, a conductor fairly flat at one end and relatively pointed at the elatively pointed at the other. other. Excess of charge move to the surface.Excess of charge move to the surface.Forces between charges on the flat surface, tend to be parallel Forces between charges on the flat surface, tend to be parallel to the surface. to the surface. Those charges move apart until repulsion from other charges creaThose charges move apart until repulsion from other charges creates an equilibrium.tes an equilibrium.At the sharp ends, the forces are predominantly directed away frAt the sharp ends, the forces are predominantly directed away from the surface.om the surface.There is less of tendency for charges located at sharp edges to There is less of tendency for charges located at sharp edges to move away from one move away from one another. another. Produces large fields (and force) near sharp edges.Produces large fields (and force) near sharp edges.

-

- --

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RemarksRemarks

Property 4 is the basis for the use of lightning rods near Property 4 is the basis for the use of lightning rods near houses and buildings. (Very important application)houses and buildings. (Very important application)

Most of any charge on the house will pass through the sharp Most of any charge on the house will pass through the sharp point of the lightning rod.point of the lightning rod.First developed by B. Franklin.First developed by B. Franklin.

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Faraday’s iceFaraday’s ice--pail experimentpail experiment

++++ ++

+

+

+

+++++

-- -

-

---

- -

-

-+

++ +

+

+

++

+

+

+

+

++ +

+

+

++

+

+

+

+

++ +

+

+

++

Demonstrates that the charge resides on the surface of a conductor.

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MiniMini--quizquizQuestion:Question:Suppose a point charge +Q is in empty space. Wearing rubber glovSuppose a point charge +Q is in empty space. Wearing rubber gloves, es, we sneak up and surround the charge with a spherical conducting we sneak up and surround the charge with a spherical conducting shell. shell. What effect does this have on the field lines of the charge?What effect does this have on the field lines of the charge?

+ q +

?

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Question:Question:Suppose a point charge +Q is in empty space. Wearing rubber glovSuppose a point charge +Q is in empty space. Wearing rubber gloves, we sneak up and surround the es, we sneak up and surround the

charge with a spherical conducting shell. What effect does this charge with a spherical conducting shell. What effect does this have on the field lines of the have on the field lines of the charge?charge?

Answer:Answer:Negative charge will build up on the inside of the shell.Negative charge will build up on the inside of the shell.Positive charge will build up on the outside of the shell.Positive charge will build up on the outside of the shell.There will be no field lines inside the conductor but the field There will be no field lines inside the conductor but the field lines will remain outside the shell.lines will remain outside the shell.

+ q+

--

-

-

--

- --

-

-

-

++

+

+

+

+

++

+

+

+

+

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MiniMini--QuizQuiz

Question:Question:Is it safe to stay inside an automobile during a lightning Is it safe to stay inside an automobile during a lightning storm? Why?storm? Why?

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Question:Question:Is it safe to stay inside an automobile during a lightning stormIs it safe to stay inside an automobile during a lightning storm? Why?? Why?Answer:Answer:Yes. It is. The metal body of the car carries the excess chargesYes. It is. The metal body of the car carries the excess charges on its on its external surface. Occupants touching the inner surface are in noexternal surface. Occupants touching the inner surface are in nodanger.danger.

SAFE