1

Electric Forces and Fields: Coulomb’s Law

Chapter 20

Reading: Chapter 20

2

Electric Charges

There are two kinds of electric charges

- Called positive and negative Negative charges are the type possessed by

electrons Positive charges are the type possessed by protons

Charges of the same sign repel one another and

charges with opposite signs attract one anotherElectric charge is always conserved in isolated system

Neutral – equal number of positive and negative charges

Positively charged

3

Electric Charges: Conductors and Isolators

Electrical conductors are materials in which some of the electrons are free electrons

These electrons can move relatively freely through the material Examples of good conductors include copper, aluminum and silver

Electrical insulators are materials in which all of the electrons are bound to atoms

These electrons can not move relatively freely through the material Examples of good insulators include glass, rubber and wood

Semiconductors are somewhere between insulators and conductors

Electric Charges

Electric Charges

Electric Charges

Electric Charges

Electric Charges

Electric Charges

Electric Charges

Electric Charges

Electric Charges

Electric Charges

Electric Charges

Electric Charges

16

Coulomb’s Law

• Mathematically, the force between two electric charges:

• The SI unit of charge is the coulomb (C)

• ke is called the Coulomb constant

– ke = 8.9875 x 109 N.m2/C2 = 1/(4πeo)

– eo is the permittivity of free space

– eo = 8.8542 x 10-12 C2 / N.m2

Electric charge: electron e = -1.6 x 10-19 C proton e = 1.6 x 10-19 C

1 212 2e

q qF k

r

17

Coulomb’s Law

1 212 21 2

| || |e

q qF F k

r

21 12F F

opposite directions, the same magnitude

1 2

1 2 0q q

1 2

1 2 0q q

The force is attractive if the charges are of opposite signThe force is repulsive if the charges are of like sign

21

21F

12F

21F

12F

21F

12F

Magnitude: 1 212 21 2

| || |e

q qF F k

r

Direction depends on the sign of the product

1 2q q 1 2 0q q

18

Coulomb’s Law: Superposition Principle

• The force exerted by q1 on q3 is F13

• The force exerted by q2 on q3 is F23

• The resultant force exerted on q3 is the vector

sum of F13 and F23

19

Coulomb’s Law

1 221 2e

q qF k

r

1 2 13F

23F

33 21 31F F F

3F1 1q μC 2 2q μC

3 3q μC

5m 6m

Magnitude:

6 69 43 1

13 2 2

| || | 10 3 108.9875 10 2.2 10

11e

q qF k N N

r

6 69 43 2

23 2 2

| || | 2 10 3 108.9875 10 15 10

6e

q qF k N N

r

13F

23F

4 4 43 23 13 (15 10 2.2 10 ) 12.8 10F F F N N

Resultant force:

20

Coulomb’s Law

1 221 212

ˆe

q qF k r

r

1 2 32F

12F

32 12 32F F F

2F

1 1q μC 2 2q μC

3 3q μC

5m 6m

Magnitude:

6 69 41 2

12 2 2

| || | 10 2 108.9875 10 7.2 10

5e

q qF k N N

r

6 69 43 2

32 2 2

| || | 2 10 3 108.9875 10 15 10

6e

q qF k N N

r

32F12F

4 4 42 32 12 (15 10 7.2 10 ) 7.8 10F F F N N

Resultant force:

21

Coulomb’s Law

1 221 212

ˆe

q qF k r

r

1 221F

31F

31 21 31F F F

1F

1 1q μC 2 2q μC

3 3q μC

5m 6m

Magnitude:

6 69 41 2

21 2 2

| || | 10 2 108.9875 10 7.2 10

5e

q qF k N N

r

21F31F

4 4 41 21 31 (7.2 10 2.2 10 ) 5 10F F F N N

6 69 43 1

31 2 2

| || | 10 3 108.9875 10 2.2 10

11e

q qF k N N

r

Resultant force:

22

Coulomb’s Law

1 221 2e

q qF k

r

1

2

31F

21F

3

1 21 31F F F

31F

21F

1F

1 1q μC 2 2q μC

3 3q μC

5m

7m

6m

Magnitude:

6 69 33 1

31 2 2

| || | 10 3 108.9875 10 1.1 10

5e

q qF k N N

r

6 69 41 2

21 2 2

| || | 10 2 108.9875 10 5 10

6e

q qF k N N

r

1

213F

23F

3

5m

7m

6m

1

232F

12F

3

5m

7m

6m

13F

23F 3F

12F

1F32F

Resultant force:

23

Conservation of Charge

1 1q μC2 2q μC

Electric charge is always conserved in isolated system

Two identical sphere

They are connected by conducting wire. What is the electric charge of each sphere?

The same charge q. Then the conservation of charge means that :

1 2 1 20.5

2 2

q qq μC μC

1 22q q q

For three spheres: 1 1q μC 2 2q μC 3 3q μC

1 2 33q q q q

1 2 3 1 2 31

3 3

q q qq μC μC

24

Electric Field

Chapter 20

25

Electric Field

An electric field is said to exist in the region of space around a

charged object This charged object is the source charge

When another charged object, the test charge, enters this electric

field, an electric force acts on it.The electric field is defined as the electric force on the test charge

per unit charge

If you know the electric field you can find the force

If q is positive, F and E are in the same directionIf q is negative, F and E are in opposite directions

0

FE

q

F qE

26

Electric Field

The direction of E is that of the force on a positive test charge The SI units of E are N/C

0

FE

q

02e

qqF k

rCoulomb’s Law:

20

e

F qE k

q r

Then

27

Electric Field

02e

qqF k

r

q is positive, F is directed away

from q The direction of E is also away

from the positive source charge q is negative, F is directed toward q E is also toward the negative

source charge

20

e

F qE k

q r

28

Electric Field: Superposition Principle

• At any point P, the total electric field due to a group of

source charges equals the vector sum of electric

fields of all the charges

ii

E E

29

Electric Field

1 2 1E

2E

Electric Field:

1 2E E E

1 1q μC 2 2q μC

5m 6m

Magnitude:

69 21

1 2 2

| | 108.9875 10 / 0.7 10 /

11e

qE k N C N C

r

69 22

2 2 2

| | 2 108.9875 10 / 5 10 /

6e

qE k N C N C

r

2 2 22 1 (5 10 0.7 10 ) / 4.3 10 /E E E N C N C

2e

qE k

r

2E

1E

E

30

Electric Field

1 2

1E

2E

Electric field

1 1q μC 2 2q μC5m

7m

6m

Magnitude:

69 21

1 2 2

| | 108.9875 10 / 0.37 10 /

5e

qE k N C N C

r

69 22

2 2 2

| | 2 108.9875 10 / 5 10 /

6e

qE k N C N C

r

1 2E E E

2E

1E

E

2e

qE k

r

31

Electric Field

1 2

1 10q μC 2 10q μC

5m

6m

5m

2e

qE k

r

E

Direction of electric field?

z

32

Electric Field

1 2

1E

2E

Electric field

1 10q μC 2 10q μC

5m

6m

5m

Magnitude:

69 32

2 1 2 2

| | 10 108.9875 10 / 3.6 10 /

5e

qE E k N C N C

r

1 2E E E 2e

qE k

r

E 2E

12 cosφE E

2 25 3 4cosφ

5 51

8

5E E

33

Example

1 2

EF

Electric field

1 10q μC 2 10q μC1l m

0ET mg F

1 22E e

q qF k

r

21l m

T

mg

g

1

2cosT mg 2sin ET F

1 21 2tan E

e

F q qk

mg r mg

r

1 22 2tan E

e

F q qk

mg r mg

1 2m m m

2 1

34

0Q

σ>0planeE

planeE

e0

σ=2πk σ=

2εplaneE

Q=σA

0Q

σ<0planeE

planeE

e0

|σ|2πk |σ|=

2εplaneE

Q=σA

35

Find electric field

σ>0

E

E

0

σ

2εE

Important Example

σ>0

-σ<0

σ<0

E

E

0

σ

2εE

36

Find electric field

σ>0E

E

Important Example

σ>0

σ<0

σ<0E

0

σ

2εE

0

σ

2εE

E

E

E

E E E

0E E E

0E E E

0

σ

εE E E

37

σ>0

Important Example

0σ

0σ

σ<0

E

0E

0 0

σ

ε ε

QE

A

0E

38

Electric field due to a thin uniformly charged spherical shell

• When r < a, electric field is ZERO: E = 0

1r

1A

2A

2r

1 1q A 2 2q A

the same solid angle

21 1A r 2

2 2A r

1E

2E

21 1 1

1 2 2 21 1 1

e e e e

q A rE k k k k

r r r

22 2 2

2 2 2 22 2 2

e e e e

q A rE k k k k

r r r

1 2E E

1 2 0E E

2

1E

rOnly because in Coulomb law

39

Motion of Charged Particle

40

Motion of Charged Particle

• When a charged particle is placed in an electric field, it experiences an electrical force

• If this is the only force on the particle, it must be the net force

• The net force will cause the particle to accelerate according to Newton’s second law

F qE

F ma

- Newton’s second law

- Coulomb’s law

qa E

m

41

Motion of Charged Particle

What is the final velocity?

F qE

F ma

- Newton’s second law

- Coulomb’s law

| |y

qa E

m

fv

0

ltv

- travel time

Motion in x – with constant acceleration 0v | |

y

qa E

m

Motion in x – with constant velocity

After time t the velocity in y direction becomes

| |y y

qv a t Et

m

fv

0xv v

yvthen

2

20f

qv v Et

m

42

Conductors in Electric Field

Chapter 20

43

Electric Charges: Conductors and Isolators

Electrical conductors are materials in which some of the electrons are free electrons

These electrons can move relatively freely through the material Examples of good conductors include copper, aluminum and silver

Electrical insulators are materials in which all of the electrons are bound to atoms

These electrons can not move relatively freely through the material Examples of good insulators include glass, rubber and wood

Semiconductors are somewhere between insulators and conductors

44

Electrostatic Equilibrium

Definition:

when there is no net motion of charges

within a conductor, the conductor is

said to be in electrostatic equilibrium

Because the electrons can move freely through the

material no motion means that there are no electric forces no electric forces means that the electric field

inside the conductor is 0

If electric field inside the conductor is not 0, then

there is an electric force and, from the second

Newton’s law, there is a motion of free electrons.

0E

F qE

F qE

45

Conductor in Electrostatic Equilibrium

• The electric field is zero everywhere inside the conductor

• Before the external field is applied, free electrons are distributed throughout the conductor

• When the external field is applied, the electrons redistribute until the magnitude of the internal field equals the magnitude of the external field

• There is a net field of zero inside the conductor

46

Conductor in Electrostatic Equilibrium

0E

• If an isolated conductor carries a charge, the charge resides on its surface