EMLAB

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Chapter 2. Coulomb’s law

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RRF ˆ4

ˆ2

0

21

2

21

r

qq

r

qqk

• This law is discovered by Coulomb experimentally.

• In the free space, the force between two point charges is proportional to the charges of them, and is inversely proportional to the square of the distance between those charges.

k is the proportionality constant., whose value is determined exper-imentally to be

0

229

4

1

]CNm[109

k

+q1

+q2

O

2r

1r

12 rrR

If q1, q2 have the same po-

larity, the force is repulsive.

Coulomb’s law only states that the force between two charge is related to the distance between them and their charges. It does not tells us how the interaction occurs.

Coulomb’s law

(Permittivity of vacuum)

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3

+q1 +q2

0ˆ)( 2

21

RF xk

xd

qqk

d

x

+q1 +q2

Coulomb’s torsion balance

EMLAB

4Electric field

EF 2q+q2

Media that transfer electric forces. Charges that fall apart interact via electric field.

rE ˆ2r

Qk

1. The electric field is generated by the charge Q and spread into the universe.

2. The speed of electric field transmission is the same as the speed of light.

3. The force on q2 is proportional to the electric field near the charge q2.

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Electric field due to known charge distributions

n

n

nQQQa

rra

rra

rrrE 2

0

22

20

212

10

1

444)(

•E-field by point charges

•Continuous distributions

d

V '2

04

ˆ)()(

rr

RrrE

adS

s

'2

04

ˆ)()(

rr

RrrE

rrRrr

RrrE l

,

4

ˆ)()(

'2

0

rdC

•Volume charge

•Surface charge

•Line charge

Quantities with prime ( ′ ) symbols are associated with sources. Un-primed quantities are related with observation positions.

EMLAB

6Charges in a material

1. If an electric field problem contains a physical media, it is difficult to predict electric field in the space due to the charges contained on it.

2. If the positions of the charges are unknown, Coulomb’s law cannot be applied.

molecule

Molecules in a solid are aligned in the direction of the external electric field.

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+

-

Electron energy level

- -

- -

- -

1 atom

Electrons in an isolated atom

Tightly bound electron

Energy levels and the radii of the electron orbit are quantized and have discrete values. For each energy level, two electrons are accommodated at most.

-

-

More freely moving electron

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+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

Atoms in a solid are arranged in a lattice structure. The electrons are attracted by the nuclei. The amount of attractions differs for various material.

Electrons in a solid

Freely moving elec-tron

Tightly bound elec-tron

-

Electron energy level

To accommodate lots of electrons, the discrete energy levels are broad-ened.

extE

External E-field

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-

Energy level of insula-tor atoms

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

+-

-

Energy level of con-ductor atom

+-

+-

+-

+-

+-

+-

+-

+-

Insulator and conductorInsulator atoms Conductor atoms

Occupied energy level

Empty energy level

External E-field External E-field

EMLAB

10Movement of electrons in a conductor

EMLAB

11Charges on a conductor

1. In equilibrium, there is no charge in the interior of a conductor due to repulsive forces between like charges.

2. The charges are bound on the surface of a conduc-tor.

3. The electric field in the interior of a conductor is zero.

4. The electric field emerges on the positive charges and sinks on negative charges.

5. On the surface, tangential component of electric field becomes zero. If non-zero component exist, it induces electric current flow which generates heats on it.

0inE

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E

-q1+q1

Conductor

1. Tangential component of an external E-field causes a positive charge (+q) to move in the di-rection of the field. A negative charge (-q) moves in the opposite direction.

2. The movement of the surface charge compensates the tangential electric field of the external field on the surface, thus there is no tangential electric field on the surface of a conductor.

3. The uncompensated field component is a normal electric field whose value is proportional to the surface charge density.

4. With zero tangential electric field, the conductor surface can be assumed to be equi-potential.

0

Sn

t

ˆ

0

nE

E

-q1

Conductor

EE in

Electric field on a conductor due to external field

normal component

tangential component

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Example : infinite line charge

•The line charges are on the z-axis and extend to infinity.

0

2/

0002/32

0

2/3220

2/3220

2/3220

2/3220

'3

0

2ˆ

cos2

ˆ)/(1

/

2ˆ

4ˆ

4ˆ

4

ˆˆ

)'(4

'ˆ)'(ˆ

''4

)()(

L

LL

LL

LL

C

L

dx

dx

x

xdx

x

dx

x

dxx

zz

dzzz

dr

ρ

ρρ

zρ

zρzρ

rr

Rr'rE

dsecdx

tanx 2

zrzρrrrR ˆ'z',ˆzˆ,'

L

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'ˆ'',ˆz,' ρrzrrrR

0

S

10

S

1 20

S

0 2/320

S

0 2/3220

S

2

0 0 2/3220

S

2

0 0 2/3220

S

'S3

0

S

2ˆ

t

1

2ˆ

t

dt

2ˆ

z

'1

z

'd

z

'

2ˆ

'z

'd'z

2ˆ

'd'd''z4

)sinycosx('ˆz

'd'd''z4

'ˆ'ˆz

'da'4

)()(

zzz

zz

z

ρz

rr

Rr'rE

z

x

y

tdt2z

'd

z

'2t

z

'1

tz

'1

22

2

R

s

Example : infinite surface charge

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z

x

y

Example : Volume charge density

rrzrrrR ˆ'r',ˆz,'

3

a4

z4z3

aˆ

3

zˆ

'dr'r4z4

ˆ

'dr'r4z4

ˆ

'dr'rz'r

1

z'r

1)z'r(z'rz'r

z4ˆ

'drdy'ry

'rz1

z4ˆ

'drdt'rt'zr2z'r

t'rz

2ˆ

'd'dr'sin'r'cos'zr2z'r

'cos'rz

2ˆ

'd'd'dr'sin'r'cos'zr2z'r4

)'cosˆ'sin'sinˆ'cos'sinˆ('rˆz

'd'd'dr'sin'r)'cosˆ'sin'sinˆ'cos'sinˆ('rˆz4

)'cosˆ'sin'sinˆ'cos'sinˆ('rˆz

'd'd'dr'sin'r'ˆ'rˆz4

'ˆ'rˆz

'da'4

)()(

3

20

20

30

a

0

22

0

z

0

22

0

a

0

222

0

a

0

z'r

z'r 2

22

20

1

1

a

0

22/322

0

0

a

0

22/322

0

2

0 0

a

0

22/322

0

2

0 0

a

0

23

0

2

0 0

a

0

23

0

'V3

0

z

z

z

z

z

z

z

z

zyxz

zyxz

zyxz

rz

rz

rr

Rr'rE

R

ydydt'zryt'zr2z'r

yt'zr2z'r222

22

(z<a)

(z>a)

a

EMLAB

16Induction charging

Charging a metallic object by induction (that is, the two objects never touch each other).

(a) A neutral metallic sphere, with equal numbers of positive and nega-tive charges.(b) The charge on the neutral sphere is redistributed when a charged rubber rod is placed near the sphere. (c) When the sphere is grounded, some of its electrons leave through the ground wire. (d) When the ground connection is removed, the sphere has excess posi-tive charge that is non-uniformly distributed. (e) When the rod is removed, the excess positive charge becomes uni-formly distributed over the surface of the sphere.

EMLAB

17Induction charging example : inkjet printer

Charging a conducting liquid droplet by induction. As the droplet breaks off (d), it retains the charge induced on it by the opposing electrode.

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