Download Electric field

1
Chapter 2. Coulomb’s law
EMLAB
2
Coulomb’s law
•
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
experimentally to be
q1q 2 ˆ
q1q 2 ˆ
Fk 2 R
R
2
r
4  0r
If q1, q2 have the same
polarity, the force is
repulsive.
k  9  109 [ Nm2C 2 ]

+q2
1
40
(Permittivity of vacuum)
+q1
R  r2  r1
r2
r1
O
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.
EMLAB
Coulomb’s torsion balance
3
d
+q1
+q1
+q2
x
+q2
q 1q 2

ˆ
R  0
F   k

k

x
2
(
d


x
)


EMLAB
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Electric field
Media that transfer electric forces. Charges that fall apart interact via electric field.
Q
E  k 2 rˆ
r
+q2
1.
F  q2E
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.
EMLAB
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Electric field due to known charge distributions
•E-field by point charges
E(r ) 
Q1
40 r  r1
2
a1 
Q2
40 r  r2
2
a2   
Qn
40 r  rn
2
an
•Continuous distributions
•Volume charge
ˆ
 (r) R
E(r)  
d 
2
V ' 40 r  r
•Surface charge
 s (r) Rˆ
E(r )  
da
2
S ' 40 r  r
•Line charge
 l (r) Rˆ
E(r )  
dr, R  r  r
2
C ' 40 r  r
Quantities with prime ( ′ ) symbols are
associated with sources. Un-primed
quantities are related with observation
positions.
EMLAB
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Charges 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.
EMLAB
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Electrons in an isolated atom
Electron
energy
level
1 atom
-
+
-
-
-
-
-
-
-
Tightly bound
electron
-
More freely moving 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.
EMLAB
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Electrons in a solid
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.
Freely moving
electron
+
Eext
External E-field
+
+
Tightly bound
electron
+
+
-
+
-
-
-
+
+
+
-
+
-
-
-
+
+
+
-
+
-
-
-
+
+
-
Electron
energy
level
-
-
-
To accommodate lots of electrons,
the discrete energy levels are
broadened.
EMLAB
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Insulator and conductor
Insulator atoms
+
+
Conductor atoms
+
-
+
-
+
-
+
-
+
-
+
-
External E-field
+
-
+
-
+
-
+
-
-
-
External E-field
+
+
-
-
+
+
-
-
+
+
-
-
+
+
-
-
+
+
-
-
+
+
-
-
Empty energy level
-
Energy level of
insulator atoms
Occupied energy level
-
Energy level of
conductor atom
EMLAB
Movement of electrons in a conductor
10
EMLAB
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Charges 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
conductor.
Ein  0
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.
EMLAB
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Electric field on a conductor due to external field
tangential component
Et  0
E
E n  nˆ
normal component
+q1
-q1
S
0
-q1
Ein  E
Conductor
Conductor
1. Tangential component of an external E-field causes a positive charge (+q) to move in the
direction 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.
EMLAB
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Example : infinite line charge
•The line charges are on the
z-axis and extend to infinity.
R  r  r' , r   ρˆ  z zˆ , r'  z' zˆ
E(r ) 
 L ρˆ  ( z  z ' )zˆ dz '


 4 

L
 L (r' ) R
C ' 4 r  r' 3 dr '
0
0
 
 ρˆ L
40
2
 ( z  z' )2

 

dx
2
 x2

3/ 2

3/ 2


 L ρˆ  x zˆ dx
 4 


 zˆ L
40
0

 

2
 x2
xdx
2
 x2

3/ 2

3/ 2
L 
dx / 
L  /2
 ρˆ
 ρˆ
cos  d
3/ 2


2
20  0 1  ( x /  ) 
20  0
 ρˆ
L
20 

x
dx
 tan  
 sec 2 d


EMLAB
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Example : infinite surface charge
R  r  r' , r  z zˆ , r'  ' ρˆ '
x̂
S (r' ) R
E(r )  
S'
s
40 r  r '

2

2
0
R
ẑ
ŷ
0
 zˆ

da '
S z zˆ  ' ρˆ '



2 3/ 2
' d' d'
40 z  '
  z z
ˆ  ' ( x̂ cos   ŷ sin )
S
0

3

40 z  '
0
S
2 0

 zˆ S
2 0
2
z' d'

 z
0


1
2
2
 '2

3/ 2

2 3/ 2
 zˆ
S
2 0

' d' d'
' d'
z z

0
  '  2 
1    
  z  
3/ 2

S  1 
S
dt
ˆ
ˆ

z


z
t2
2 0  t 1
2 0
 ' 
 1    t
 z
2
' d'
 ' 
1    t2  2
 2tdt
z z
 z
2
EMLAB
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Example : Volume charge density
E(r ) 
ẑ
 4
0 r  r'
V'

2

2

2
0
R
0

0
ŷ
a
x̂
R  r  r' , r  z zˆ , r'  r' rˆ
(r' ) R
3
da '
z zˆ  r ' rˆ '
r '2 sin ' dr ' d' d'
3
40 z zˆ  r ' rˆ '


a
0
0


a
z zˆ  r ' (xˆ sin ' cos ' yˆ sin ' sin ' zˆ cos ' )
0
0
40 z zˆ  r ' (xˆ sin ' cos ' yˆ sin ' sin ' zˆ cos ' )

a
0
0

 zˆ

2 0
 zˆ

2 0
z zˆ  r ' (xˆ sin ' cos ' yˆ sin ' sin ' zˆ cos ' )


a
0
0
z  r ' cos '
  r' z
1

40 r '  z  2zr ' cos '
2
2

3/ 2
3/ 2
3
r '2 sin ' dr ' d' d'
r '2 sin ' dr ' d' d'
r '2 sin ' dr ' d'
 2zr ' cos '
z  r' t 
r '2 dtdr '
3
/
2
r '2  z 2  2zr ' t
2
a

1 0
2

 z 2  r '2 
0 r ' z 1  y 2  r' dydr' 
 1
a 

1  

2
2
 zˆ
r
'

z

r
'

z

(
r
'

z
)


 r ' dr '

4 0 z 2 0 
 r ' z r 'z  
 zˆ
(z<a)
(z>a)

4 0 z 2


ˆ
z
 4 z 2
0


zˆ
 4 0 z 2
r ' z
a

z

a
0
0
z

 r '2  z 2  2zr ' t  y
zˆ
4r '2 dr ' 
3 0

r '2  z 2  2zr ' t  y 2  zr ' dt  ydy

a 3 
 4a 3 
4r '2 dr ' zˆ



 3 0 z 2  40 z 2 3 
EMLAB
16
Induction 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
negative 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
positive charge that is non-uniformly distributed.
(e) When the rod is removed, the excess positive charge becomes
uniformly distributed over the surface of the sphere.
EMLAB
Induction charging example : inkjet printer
17
Charging a conducting liquid droplet by induction. As the droplet breaks off
(d), it retains the charge induced on it by the opposing electrode.
EMLAB
18
EMLAB