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Electromagnetism revision
Electric fields
Chapter 16
Charge is quantized (量子化).
Everything is charged in multiples of e,
the elementary charge (except quarks!).
Charge is conserved (守恒的).
The total charge of the universe never
changes.
Electric field of one point charge
q

E
1
q
ˆ
r

2
40 r
Coulomb’s Law

E
1
q
ˆ
r

2
40 r
+
q is positive:
the direction of the field is away from the charge.

E
1
q
ˆ
r

2
40 r
-
q is negative:
the direction of the field is towards the charge.
Superposition Principle
电场强度叠加原理
+
-
q2


F4  q4 Enet



 q4 E1  E2  E3


E1
q1

E3

Enet
q3
+
Location of q4

E2

Superposition Principle
电场强度叠加原理
In general:




Enet  E1  E2  E3  
or:


Enet   Ei
i
The net field at a location in space is the
vector sum of the fields contributed by
all charged particles at other locations.
Dipole
电偶极子
+
Hydrogen chloride
-q
+q
d
O
H
+
H
Water
We can define a vector called the
dipole moment (电矩).
-q

p
+q
d
Magnitude:
NOTE: This p has
nothing to do with
momentum!

p  qd
Direction: from the negative (-) charge
to the positive (+) charge.


1 p
E  
40 r 3
Using superposition….

p


1 2p
Eaxis 
40 r 3
Electric forces on dipoles

F

F

p
+

E
In a uniform field,
the net force on a
dipole is zero:
 
F  F  0
Electric forces on dipoles

F

F

p
+

E
But the net torque
around the dipole’s
COM is not zero:
 
  p E

Electric forces on dipoles

F

F

p

E
This torque will
rotate the dipole
until it is parallel
with the field.
Electric forces on dipoles

F

F

p

E
The dipole-field system
has a potential energy:
 
U dipole   p  E
Electric field of distributed charges
y
y
Q

r

r
y
Q

E
x

E
Uniformly charged rod
Uniformly charged ring
Eaxis
E
2Q L 
40
r
1
Q 1

 2
40  z 
Electric field of a uniformly charged plate
Q
R
r
r
z

E
Very large plate (R >> z):

Q A
E
2 0
The field almost
doesn’t change with
distance, near the
plate.
Electric potential (电势)
Chapter 18
Electric potential (电势)
• The potential difference
between two points:
 
Vab   E  ds
b
a
• The potential energy
difference for a charge q,
moved between two
points:
• The potential near a
point charge, with
respect to infinity:
U electric  qV
1
q1
V (r ) 
40 r
Electric field is the negative
gradient (梯度) of the potential
V
V
V
Ex  
, Ey  
, Ez  
x
y
z
V
Ex
Ex
x
The potential is like
the height of the hill.
The field is like the
slope of the hill.
Just remember:
- positive charges
go down the hill
- negative charges
go up!
Potential along the axis of a ring
dQ
R
Potential obeys the
superposition principle, just
like the field.
x2  R2
x
Potential due to
one small piece:
dV 
1
dQ
40
x2  R2
Potential along the axis of a ring
dQ
Potential obeys the
superposition principle, just
like the field.
x2  R2
R
x
Integrate:
V   dV 
1
1
40
x R
2
2
 dQ
Potential along the axis of a ring
dQ
Potential obeys the
superposition principle, just
like the field.
x2  R2
R
x
Integrate:
V
1
Q
40
x2  R2
Field along the axis of a ring
The strength of the field is
the negative of the potential
gradient:

E
V
Ex  
x
 
1


40 x  x 2  R 2
Q




Field along the axis of a ring
The strength of the field is
the negative of the potential
gradient:

E
Ex 
1
Qx
40 x  R
2

2 3/ 2
Field along the axis of a ring
We already calculated this field
the hard way.
It is often easier to first calculate
the potential, then use its
gradient to get the field.

E
Ex 
1
Qx
40 x  R
2

2 3/ 2
Potential in a
conductor
B
At equilibrium,
the field inside
the conductor
must be zero.
A
+
VAB  0
Potential in a
conductor
So the potential
inside a conductor
(and at the surface)
must be constant.
+
V  constant
Capacitors and dielectrics
Chapter 19
Two uniformly charged plates:
A capacitor (电容器)
R
d
-
+
+
+
+
+
+
+
Einside

Q A

0
There is also a small
field outside the plates:
Efringe

Q A  d 

2 0
 
R
Definition of capacitance
Q
C
V
+
+
+
+
+
+
+
-
Capacitance C measures:
- how much charge Q we must
put on the plates
- to achieve a certain potential
difference ΔV.
For parallel-plate capacitor:
A = area of plates
0 A
C
d
Units: the Farad (F)
d = separation of plates

Fby you
+
+
+
+
+
+
+
Energy density in an electric
field

Eother plate

Fby plate
-
1
2
Energy / v olume   0 E
2
d
It is true for any electric field, not
just in a capacitor!
Electric fields contain energy.
(Also momentum!)
I insert a dielectric between the plates.
What happens?
+Q +
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
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+
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+
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+
+
+
+
+
+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
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+
-
+
-
+
-
+
-
+
+
-
+
-
+
-
+
-
+
 Epolar
-
+

Eplates
- -Q
-

Etotal 
Polarization of the dielectric reduces the
net electric field between the plates.

Eplates

The dielectric constant ε depends on the material.
+Q +
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
+
+
+
+
+
+
-
+
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+
-
+
-
+
-
+
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-
+
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+
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+
-
+
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+
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+
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+
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+
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+
-
+
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+
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+
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+
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+
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+
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+
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-
+
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-
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+
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+
-
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+
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+
-
+
-
+
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+
-
+
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+
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+
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+
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+
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+
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-
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+
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-
+
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+
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+
-
+
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+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+

Etotal
- -Q
-
It also reduces the potential difference
between the plates.
V 
Vvacuum

The dielectric constant ε depends on the material.
+Q +
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
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+
-
+
-
+
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+
+
+
+
+
+
+
-
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+
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+
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+
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+
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+
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-
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+
-
+
-
+
-
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-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+
-
+

Etotal
- -Q
-
Magnetic fields from currents
Chapter 21
Magnetic field of a single moving charge:
The Biot-Savart Law

B
r̂
+


v
“Permeability of
free space”

  0 qv  rˆ
B
2
4 r
Units: tesla (T)
0
tesla  m
7
 110
exactly
4
coulomb / s
For a very long wire (L >> r):
 0 2i
B
4 R
Far from a current loop (z >> R)
 0 iA
B
4 z 3

Btotal
Current loops are magnetic dipoles.
Baxis
  iA
The magnetic
dipole moment
0 

4 z 3
Magnetic field inside a solenoid 螺线管
.
Binside  0 ni
where n is the number of
loops per unit length.
Magnetic forces
Chapter 20
Moving electric charges make
magnetic fields…
and magnetic fields make forces on
moving electric charges.

 
Fmagnetic  qv  B

 
F  ( e)v  B
-

v
 
vB
The force is always perpendicular to the direction of
motion: it cannot change the particle’s speed.
Velocity selector:

B into page

FB
+
FE

E
qE  qvB
E
v
B
Magnetic force on a
current-carrying wire
i
L
 

F  i  dl  B

B
Magnetic force on a
current-carrying wire
i

F into page

L

B
  
F  iL  B
(uniform field)
Torque on a current loop

B

B
b
i
a

B


i out
Magnetic dipole moment
a
  ibB   sin 
2


  B


i in
a
  ibB   sin 
2

B


i out
i in

  B


A magnetic dipole will align
with the magnetic field.
Gauss’ Law
Chapter 17

r

E
L
  q
E

d
A


0
Uniformly charged spherical shell (outside)

E
dA
q
r
Uniformly charged spherical shell (inside)
q

E
r
Excess charge in a conductor is always on the
surface.
  q
 E  dA 
Q

E 0
0
0
0
Gaussian
surface
Net charge is
zero inside a
conductor.
Gauss’ Law for magnetic fields
 
B

d
A

0

Ampere’s Law
Chapter 21
 
B

d
s


i
0
enc

Magnetic field inside a cylindrical
current-carrying conductor

B
“Amperian loop”
i
r
R
 


B

d
s

B
2

r

r
ienclosed  i 2
R
2
Magnetic field inside a cylindrical
current-carrying conductor

B
“Amperian loop”
i
r
R
  0i 
B
r

2
 2R 
(direction from
right-hand rule!)
Faraday’s Law
Chapter 23

ENC
d B
EMF  
dt

ENC
i increasing
Which direction does the electric field curl?

dB
Right thumb along 
dt

ENC

Fingers curl in direction of ENC

dB

dt
i increasing
i2
ENC can induce
current in a wire
around the changing
magnetic field.
d B
EMF  
dt
EMF
i2 
resistance
i1
Example
Ammeter 电表
What current will the ammeter
measure?
Wire
R  0.5 
The magnetic field in the solenoid
increases from 0.1 T to 0.7 T in
0.2 seconds.

B
 B  BAsolenoid
Solenoid
Asolenoid  3 cm 2
d B
B
EMF 
 Asolenoid
dt
t
EMF
i
 1.8 10 3 A
R
Motional EMF (动生电动势)
vt

B
L
i
i
d B
EMF 
dt
v
d B
EMF 
dt
This equation works, no matter if
• the magnetic field is changing (Faraday’s Law),
• the area of the circuit is changing (motional EMF),
• or both!
Maxwell’s Equations
Chapter 21, 24
 
 B  ds  0
 
 B  ds  0i
Something missing here…
Ampere’s Law is incomplete.
d E
i  0
dt
How Maxwell fixed Ampere’s Law:
 
 B  ds  0ienclosed
d E
i  0
dt
How Maxwell fixed Ampere’s Law:
 
d E 

 B  ds  0  ienclosed   0 dt 
Maxwell’s Equations
  qinside
 E  dA 
Gauss’ Law for
electric fields
 
 B  dA  0
Gauss’ Law for
magnetic fields
0
 
d  
 E  ds   dt  B  dA
Faraday’s Law
 
d  

 B  ds  0  ienclosed   0 dt  E  dA 
Ampere-Maxwell Law
Maxwell’s equations predict
travelling waves.
v
v
1
0 0
Maxwell’s equations predict
light.
v
v  3.0  10 m/s
8