Electrostatics
Contents
Electrostatics
21. Point charges, Coulomb Law: electric force
22. Continuous charge distributions, Gauss Law: electric
field
23. Electric potential
24. Electrostatic energy, capacitance, dielectrics
25. Electric current, DC circuits
Ch 21 to 25
Magnetostatics
Ch 26 and 27
Electromagnetism ⇒ Light
Ch 28 to 30
1
2
Charge
Coulomb Law
Electromagnetism is 1 or the 4 fundamental forces
Charles Augustin de Coulomb
1736 - 1806
Electro[statical] attraction:
• Chemical binding
• Van der Waals force (a bit more complicated)
• Casimir force (a bit more complicated)
Electric charge
Electric force
Electric field
Examples
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Coulomb.html
Separation or charge:
• accumulator/battery
• photosynthesis
• photovoltaic cell
• polarization of nerves
3
4
Experiment
+/- charge
ebonite
glass
+
+
Experiment
−+
new force:
Felectrical >> Fgravitation
positive: + & negative: + + & - -: repulsive
+ - & - +: attractive
quantisatie: qelektron
charge conservation: Σ q = constant
ελεχτρον = amber (barnsteen)
5
6
1
Demo 1
Experiment
NB book is wrong
7
8
Force ⇒ Coulomb Law
Palais de la decouverte - Paris
1777: C. the Coulomb
Q
q F
q
G
O r
G
1 qQ ˆ
r
Fq =
4πε 0 r 2
10
11
Demo
Force ⇒ Coulomb Law
G
1 qQ
Fq =
rˆ
4π ε 0 r2
1
4πε 0
Q
q
r
= 8.99 ⋅109 Nm 2 /C 2
– unit or charge:
Units:
– Length [l]: meter m
– Time [t]: second s
– Mass [m]: kilogram kg
– charge [q]: Coulomb C
12
q elektron ≈ −1.602177 ∗10 −19 C
– permittivity:
ε 0 ≈ 8.85 ∗10−12 F/m
13
2
Quantisation of Electric charge
FElectric ↔ FGravitation
G
FE =
me m p
G
FG = G 2
r
electron
m=9.1∗10-31 kg
q=-1.6∗10-19 C
e2
4π ε 0 r 2
1
≈ 1.0 ∗ 10−47 N
10-10 m
−8
(
proton
m=1.7∗10-27 kg
q=+1.6∗10-19 C
≈ 2.3 ∗ 10 N
3
G = 6.673∗10−11 m 2
kg s
)
Conservation of charge
Example: anti-proton (p−≡p) discovery (1955):
p p
reaction:
p
+
+
+
+
+
allowed: p p →p p p p
forbidden: p+p+→p+pp p
p
14
15
Millikan Experiment
Millikan Experiment
Millikan oil-drop experiment,
First direct and compelling measurement of the electric charge of a single electron. It was performed
originally in 1909 by the American physicist Robert Millikan, who devised a straightforward method of
measuring the minute electric charge that is present on many of the droplets in an oil mist. The force
on any electric charge in an electric field is equal to the product of the charge and the electric field.
Millikan was able to measure both the amount of electric force and magnitude of electric field on the
tiny charge of an isolated oil droplet and from the data determine the magnitude of the charge itself.
Millikan's original experiment of any modified version, such as the following, is called the oil-drop
experiment. The apparatus associated with Millikan's oil-drop experiment is shown in the Figure. A
closed chamber with transparent sides is fitted with two parallel metal plates, which acquire a positive
of negative charge when an electric current is applied. At the start of the experiment, an atomizer
sprays a fine mist of oil droplets into the upper portion of the chamber. Under the influence of gravity
and air resistance, some of the oil droplets fall through a small hole cut in the top metal plate. When
the space between the metal plates is ionized by radiation (e.g., X rays), electrons from the air attach
themselves to the falling oil droplets, causing them to acquire a negative charge. A light source, set at
right angles to a viewing microscope, illuminates the oil droplets and makes them appear as bright stars
while they fall. The mass of a single charged droplet can be calculated by observing how fast it falls.
By adjusting the potential difference, or voltage, between the metal plates, the speed of the droplet's
motion can be increased or decreased; when the amount of upward electric force equals the known
downward gravitational force, the charged droplet remains stationary. The amount of voltage needed
to suspend a droplet is used along with its mass to determine the every place electric charge on the
droplet. Through repeated application of this method, the values of the electric charge on individual oil
drops are always whole-number multiples of a lowest value--that value being the elementary electric
charge itself (about 1.602 x 10-19 coulomb). From the time of Millikan's original experiment, this
method offered convincing proof that electric charge exists in basic natural units. All subsequent
distinct methods of measuring the basic unit of electric charge point to its having the same
fundamental value.
16
17
Quantisation or Electric charge
For the
interested!
The elementary particles
Step 1: determine the mass of the drop: no voltage
F d  6Rv  mg  F g
m
m
6Rv
g
4
R 3 
3
9v
2g

R2
q
(terminal velocity)
m
4
3

9v
2g
3/2

q
mg
E

4
3

9v
2g
-e
2
+ e
3
1
− e
3
Step 2: determine the charge or the drop: voltage applied
F E  qE  mg  F g
mg
q E
0
(floating)
νe
e I
u u u
d d d
q
0
-e
2
+ e
3
1
− e
3
νμ
μ II
c c c
s s s
q
0
-e
2
+ e
3
1
− e
3
ντ
τ III
t t t
b b b
remark: every quark occurs in three “colors”:
3/2
Everything on this sheet is not for the examination
g
E
18
red
yellow
blue
Everything on this sheet is not for the examination
19
3
Electrostatics: superposition
q 
F
1 qQ1
4 0 r 21
r̂ 1 
1 Q1
40 r 21
q
1 qQ2
40 r 22
r̂ 1 
r̂ 2 . . .
1 Q2
40 r22
Discrete:
qi
ri
r̂ 2 . . . .

: qE
superposition
_Q1
q
F
q

E
Charge distribution⇒ E-field
i
Q
_ 2
r1
 is the electric field
E
(in the origin)
q is at the origin
P  ∑E
 i,P  ∑
E
Q
_ 3
i
P
qi
1
40 r 2i,P
r̂ i,P
in principle all charges in nature are distrubuted
discretly: in elementairy particles
q
+
Q4
_
Fq
20
21
E-field outside the origin
Three charges: Q1, Q2 and Q3
Q3
q
r
1 
E
0 
1 Q1
40 r 2
1
 1 r 
E
r1 −r
1
4 0 |r1 −r| 3
 r 
E
1
4 0
r̂ 1 
Q1
1
40 |r 1 | 3
r 1
1 −r

r
Q1 
–
r1
–
r2
–
http://www.colorado.edu/physics/2000/waves_particles/wavpart2.html
r2 −r
|r2 −r| 3
Q2 
3 −r

r
|3
3 −r
|r
Maxwell equations in vacuum are
linear in the fields E and B
Superposition at macroscopic level is
an experimental fact!
Thousands of telephone calls at the
same time through one fibre
Non-linear effects in
•
•
Q2
Q1
| 3
1 −r
|r
–
–
r3
Q1
Superposition principle
Magnetic materials
Crystals subjected to
intense laser radiation
Non-lineair effects for field
strengths > 1021 V/m (QED)
input
system
output
A
B
C
D
A+C
B+D
Q3
22
Concept of a field
23
Virtual demo
–Petri dish filled with oil
–small wires in it
–turn on the electric field
field – a region under the
influence or a physical quantity…
Petrus Peregrinus de
Maricourt - 1269
Epistola de Magnete
French knight
• magnetic poles
• compass
• there is a field associated
with a force
24
26
4
Electric field
Two charges: electric field
–
–
force is along the line
connecting the charges.
right hand figure shows
the situation for an
infinite non-conducting
plane.
the force is
perpendicular because of
symmetry
–
–
Rotation symmetry around a
line connecting the two
charges. The line lies in the
plane of the drawing.
Can you see the attraction and
repulsion?
The second figure shows an
electric dipole.
–
–
27
28
Ex. E-field of a dipole
Discussion question 1
charges +q and -q at distance 2d:
Find the field on the line ϑ=0o
Which pattern of field lines is that of two
equal positive charges?
d
r
ϑ
2d
-q
+q
G
G
G
dipole moment : p = q2d = qL
B
A
GP
r
r,   0 
E
≃
C
q 4d
40 r 3
q
40
:
−
1
r−d 2
2|p|
40 r 3

1
rd 2
ϑ=90o
|p|
-
20 r 3
ϑ=0o
+
E
NB dipole moment: - to +
electric field: + to 29
Taylor expansion of dipolar field
Ex. E-field of dipole
charges +q and -q at distance 2d:
-q
1
 r,   0  q
E
−
4 0
r−d 2
r  d
1
Taylor: fx  1x
≃ 1−x
1
rd  2
≃
1
r 2 2dr

 r,   0 
E
≃
q
40
1
r2

1
r2
q
4 0
2d
r3
≃
1
1 2dr
−
1
r2
−
1 −
−
2d
r3
2d
r

+q
r,   90 ∘  
E
1
r2
−
2d
r3
1
rd  2

q 4d
40 r 3
r 2 +d 2
vertical component: 0
for the  -charge
?
1
rd 2
1
r−d 2
1
r2
2d
r
ϑ
q
1
40 r 2 d 2
d
-q
d +q
 r,   90 ∘  cos
horizontal component: E
q
G
G
1
d
G
 4
dipole moment : p = q2d = qL
2
2
2
2
0
r d
r d
r,   90 ∘ 
E

31
P
α
field in P at line ϑ=90 o
P
field on connecting line: X-axis
30
total
q
1
40 r 2 d 2

q
1
40 r 2 d 2
2d
r d
2
2
≃
−
d
r 2 d2
q 2d
4 0 r 3

−d
r 2 d 2
|
|p
4 0 r 3
33
5
Mathematical dipole
q ,d →0; p =constant
Applications
34
35
Cathode Ray Tube
Photo copier
2
3
4
1
5
36
Electrostatic cleaning of smoke
37
Electrostatic cleaning of smoke
38
39
6
Electrostatic painting
Inkjet printer
– Bubble formation
• Thermal
• Piezoelectrical
– Print head, including Ink
filled cartridge moves
horizontally across paper
surface
– Each of 4 cartridges
(CYMK) has 50 ink-filled
firing chambers
– Droplets: 40 μm diameter
– Speed: 40 m/s
-
+
40
41
Inkjet printer – calculate E
Inkjet printer – calculate E
Vertical displacement: Δy  12 at 2
travel time: t  Δx/v x
2Δy
2Δy
Acceleration: a  2 

Δx/v x  2
t
E-field needed: 4 3
E

F
q

 r
3
q

ma
q

Δx  2
3
2010−6 m
1000 kg/m3
8
3
2Δyv 2x
210 −9 C
2Δyv 2x
Δx  2
3 2
8 r v x Δy
3 qΔx  2
40 m/s 2 310 −3 m
0.01 m2
 1610 N / C  1610 V/ m
42
F-q
-q
O
H
G G
E =0
E
 − qE

 F
 q  F
 −q  qE
force: F
0
1
1
 q − L
F
 −q
Moment: 
  2
LF
2
− 1
 L
  qE

 12 
L  qE
L  −qE
2

E
 p

E

 qL

pE
44
H
H
O
H
H O
H
H
O
G
G
p ≡ qL
F+q
H
H
E H
O
H
G
G
E ≠ 0
H
Molecule with intrinsic dipole moment p
with E=0: orientation of p random
with E≠0: orientation p // E
+q
H
H O
H
O H
G G
E =0
+q
Polarisation polar molecuul
G
p
H
O H
O
H
Molecule with intrinsic dipole moment p
with E=0: orientation of p random
with Electrical field ?
H
H O
H
Polarisation polar molecule
H
H O
G
p
43
F-q
G
p
O
O
H
H
H
O
G
G
p ≡ qL
-q
F+q
E
H
H
O
H
45
7
Work by rotation
Potential energy of polar molecule
F
work done by moment
R

x
x  R cos
dx  −R sin d

  p  E
Work
  ds  Fdx  −FR sin d  − F
R
 d  −d
dW  F
dW  −d  −pE sin d
dU  −dW  pE sin d
Potential energy

U  −pE cos  −
pE
dW  −d



U  −
pE
46
Dipoles in dielectrics
47
Polarization of a neutral atom
G G
E=0
elektron cloud
uniform sphere
(R)
see later (Ch 24) about the capacitor:
in dielectrics, dipoles become oriented
R
G G
E≠0
-Q
+Q
G G
G
α ≡ "polarizability" p ≡ Qd ∝ E en
spherical
symmetric
⇒ dipole moment
FE
E
+Q
-Q
d
Fe
G
p
G ≡α
E
ar
Nucle
e
charg
48
Element Z
α/ε0
------------------------------Helium 2
3x10-30 m3
Neon
10
5x10-30 m3
Argon 18 20x10-30 m3
Water vapour 500x10-30 m503
What did we learn?
charge + or -
q elektron ≈ − 1.6 ∗ 10
force and E-field
(Coulomb)
−19
C
and ∑q =constant
G
G
Q rˆ
qQ rˆ
F=
and E =
2
2
ε
4π r
4πε r
0
0
field from a discrete charge distribution
Dipole moment
G
G
G G G
G G
p = qL and τ = p × E and U = −p ⋅ E
start with memorizing equations now!
51
8