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Chapter 3 Electromagnetic Theory, Photons
and Light
August 31, September 2 Electromagnetic waves
3.1 Basic laws of electromagnetic theory
Lights are electromagnetic waves.
Electric fields are generated by electric charges or time-varying magnetic fields.
Magnetic fields are generated by electric currents or time-varying electric fields.
Maxwell’s wave equation is derived from the following four laws (Maxwell’s equations).
3.1.1 Faraday’s induction law
Electromotive force (old term, actually a voltage):
d M
d
emf   E  dl  
   B  dS
C
dt
dt A
B
dl
E
dS
B
 dS
A t
 E  dl  
C
A time-varying magnetic field produces an electric field.
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3.1.2 Gauss’s law - electric
Flux of electric field:  E   E  dS
A
q
For a point charge:  E 
, e0 is the permittivity of free space.
e0
Generally,
 E  dS 
A
1
e0

V
dV
For general material, the permittivity e  K Ee 0 , where KE is the relative permittivity
(dielectric constant).
3.1.3 Gauss’s law- magnetic
There is no isolated magnetic monopoles:
 M   B  dS  0
A
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3.1.4 Ampere’s circuital law
For electric currents:
J
 B  dl  m  J  dS
C
0
dl
B
A
m0 is the permeability of free space.
dS
For general materials, the permeability m  K M m0 ,
where KM is the relative permeability.
Moving charges are not the only source for a
magnetic field. Example: in a charging capacitor,
there is no current across area A2 (bounded by C).
E
Q
E i
e

eA
t A
Ampere’s law:
 JD  e
E
t
E 

B

d
l

m
J

e

  dS
C
A 
t 
A time-varying electric field produces a magnetic field.
i
B
E
C
A1
A2
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3.1.5 Maxwell’s equations
Gaussian’s divergence theorem:
 F  dS     FdV
A
V
Stokes’s theorem:
 F  dl    F  dS
C
A
 ˆ  ˆ  ˆ
i
j k
x
y
z



  F  Fx  Fy  Fz
x
y
z

iˆ

F 
x
Fx
ˆj

y
Fy
kˆ

z
Fz
Maxwell’s equations in differential form:
(integrals in finite regions  derivatives at individual points)
B
 
B
 dS


E


 
C
A t
t
 
E 

CB  dl  m A  J  e t   dS    B  m  J  e Et 


1
 

E

d
S


dV


E

A
 
e V
e
 
B  0
 
AB  dS  0
 E  dl  
In free space,
e  e 0 , m  m0 ,   0, J  0.
B



E



t

  B  m e E
0 0

t

E  0


B  0
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3.2 Electromagnetic waves
  ()  ()  2
Applying to free space Maxwell’s equations, we have the 3D wave equations:
 2
 E  e 0 m 0

 2 B  e 0 m 0

 2E
t 2
 2B
t 2
c
1
e 0 m0
 2.9979 108 m/s
A great interllect ral triumph
3.2.1 Transverse waves
For a plane EM wave propagating in vacuum in the x direction: E  E( x, t )
E
In free space the plane EM waves are transverse.
  E  0  x  0  Ex  0
x
For a linearly polarized wave, choose E  ˆjE y ( x, t ).
iˆ
B

E  

t
x
0
ˆj

y
E y ( x, t )
kˆ

z
0
 Bx


0
 t

 B

Bx ˆ B y ˆ Bz
y
ˆ
 i
j
k

0
  B  kˆBz ( x, t )
t
t
t
 t


E

B
y
 z 

 t
x 
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Harmonic waves:
E y ( x, t )  E0 y ei ( kx t e )
E
Bz
 y 
t
x
E
E
Bz ( x, t )    y dt  0 y ei ( kx t e )
x
c
x
E y ( x, t )  cBz ( x, t ) (in vacuum)
E y ( x, t )  vBz ( x, t ) 
c
Bz ( x, t ) (in a medium)
n
Characteristics of the electromagnetic fields of a harmonic wave:
1)E and B are in phase, and are interdependent.
2)E and B are mutually perpendicular.
3)E × B points to the wave propagation direction.
6
Read: Ch3: 1-2
Homework: Ch3: 1,3,4,8,12
Due: September 9
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September 7 Energy and momentum
3.3 Energy and momentum
3.3.1 Poynting vector
E-field and B-field store energy:
Energy density (energy per unit volume) of any E- and B-field in free space:
1

2
u

e
E
E
0

2

1 2
B
uB 

2 m0
(The first equation can be obtained from charging a capacitor:
E=q/e0A, dW=El·dq).
2
2
For light, applying E=cB, we have uE  uB , u  uE  uB  e 0 E  B / m0
The energy stream of light is shared equally between its E-field and B-field.
u  Act
The energy transport per unit time across per unit area: S 
 uc  c 2e 0 EB
At
Assuming energy flows along the light propagation direction,
Poynting vector: S  c 2e 0 E  B  E  B is the power across a unit area whose
m0
normal is parallel to S.
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For a harmonic, linearly polarized plane wave:
E  E 0 cos(k  r  ωt )
B  B 0 cos(k  r  ωt )
S  c 2e 0 E 0  B 0 cos 2 (k  r  ωt )
3.3.2 Irradiance
Irradiance (intensity): The average energy transport across a unit area in a unit time.
Time averaging:
I S
I
T
cos 2 t
T

1
( when T   )
2
 c 2e 0 | E0  B 0 | cos 2 (k  r  ωt )
1
ce 0 E02
2
In a medium I  ve E 2
T

1 2
c e 0 | E0  B 0 |
2
Note I  E0 .
2
T
The inverse square law: The irradiance from a point source is proportional to 1/r2.
Total power I·4pr2 = constant, I  E0 2 E01/r.
Example 3.2,3.3
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3.3.3 Photons
The electromagnetic wave theory explains many things (propagation, interaction with
matter, etc.). However, it cannot explain the emission and absorption of light by atoms
(black body radiation, photoelectric effect, etc.).
Planck’s assumption: Each oscillator could absorb and emit energy of hn, where n is
the oscillatory frequency.
Einstein’s assumption: Light is a stream of photons, each photon has an energy of
h  6.626 10 34 J  s
e  hn  hc / .
3.3.3 Radiation pressure and momentum
Let P be the radiation pressure.
Work done = PAct = uAct  P=u (radiation pressure = energy density)
1
1 2
P  uE  uB  e 0 E 2 
B ,
2
2m0
For light
P(t )
T

S (t )
c

T
or
P(t ) 
S (t )
c
I
c
10
Momentum density of radiation ( pV):
PA 
p pV ctA S
S

 A  pV  2
t
t
c
c
Momentum of a photon (p):
S 
c   p  u  p  hn  h
V
c

S
c
pV  2 
c 
u
Vector momentum: p  k
The energy and momentum of photons are confirmed by Compton scattering.
11
Read: Ch3: 3
Homework: Ch3: 13,14,22,24,27,35
Due: September 16
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September 9 Radiation
3.4 Radiation
3.4.1 Linearly accelerating charges
Field lines of a moving charge
ct2
ct2
c(t2-t1)
0
t1
c(t2-t1)
t2
Constant speed
0
t1
t2
Analogy: A train emits smoke columns at
speed c from 8 chimneys over 360º. What
do the trajectories of the smoke look like
when the train is:
1) still,
2) moving at a constant speed,
3) moving at a constant acceleration.
With acceleration
Assuming the E-field information propagates at speed c.
Gauss’s law suggests that the field lines are curved when the charge is accelerated.
The transverse component of the electric field will propagate outward.
 A non-uniformly moving charge produces electromagnetic waves.
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Examples:
1. Synchrotron radiation. Electromagnetic radiation
emitted by relativistic charged particles curving in
magnetic or electric fields. Energy is mostly radiated
perpendicular to the acceleration direction.
2. Electric dipole radiation.
p  p0 cos t  qd0 cos t
+

Far from the dipole (radiation zone):

p0k 2 sin  cos( kr  t )
E 
4pe0
r

B  E / c

2
1
p0  4 sin 2 
2
Irradiance: I ( )  ce 0 E0 
2
32p 2c 3e 0 r 2
Important features:
1) Inverse square law.
2) Angular distribution (toroidal).
3) Frequency dependence.
4) Directions of E, B, and S.
S
B
E
-
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3.4.4 The emission of light from atoms
Bohr’s model of H atom:
E∞
(Excited states)
a0
E1
Pump
Relaxation (E = hn)
E0 (Ground state)
15
Read: Ch3: 4
Homework: Ch3: 46
Due: September 16
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September 12 Dispersion
3.5 Light in bulk matter
Phase speed in a dielectric (non-conducting material): v  1 / em .
Index of refraction (refractive index): n 
c
e m


 K E K M.
v
e 0 m0
KE and KM are the relative permittivity and relative permeability.
For nonmagnetic materials K M  1, n  K E  e e 0 .
Dispersion: The phenomenon that the index of refraction is wavelength dependent.
3.5.1 Dispersion
 n( )  e ( ) e 0 . How do we get e ()?
Lorentz model of determining n (): The behavior of a dielectric medium in an external
field can be represented by the averaged contributions of a large number of molecules.
Electric polarization: The electric dipole moment per unit volume induced by an
external electric field.
For most materials P  (e  e 0 )E.
Examples: Orientational polarization, electronic polarization, ionic polarization.
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Atom = electron cloud + nucleus. How is an atom polarized ?
Restoring force: F  k E x  m02 x
E
Natural (resonant) frequency: 0  k E me
+
Forced oscillator: FE  qe E (t )  qe E0 exp( it )
dx
Damping force:   me
(does negative work)
dt
dx
d 2x
 me 2
Newton’s second law of motion: qe E0 exp( it )  me x   me
dt
dt
q E /m
q /m
E (t )
Solution: x(t )  x0 exp( it )  2 e 02 e exp( it )  2 e 2 e
0    i
0    i
2
0
Nqe2 / me
Electric polarization (= dipole moment density): P(t )  Nqe x(t )  2
E (t )
2
0    i
2
Nq / m
P(t )
 e  e0 
 e0  2 e 2 e
E (t )
0    i
2


Nq
e
1
e
Dispersion equation: n ( )   1 
 2

2
e0
e 0 me  0    i 
2
n 2 1  N
Frequency dependent x0 ( )  frequency dependent n (): n()  e ()  P  x0 ()
18
Quantum theory: 0 is the transition frequency.
Nqe2
For a material with several transition frequencies: n ( )  1 
e 0 me
Oscillator strength:  f j  1
2

j
fj
2
0j
  2  i j 
j
Normal dispersion: n increases with frequency.
Anomalous dispersion: n decreases with frequency.
1.5
n  n'in"
2p
 2p

exp( ikx)  exp   i
n' x 
n' ' x 




1.0
Re (n)
0.5
n'  Phase velocity
n"  Absorption (or amplification)
0.0
0.5
1.5
0
2.0
Im (n)
0.5
19
Sellmeier equation: An empirical relationship between refractive index n and wavelength
 for a particular transparent medium:
Nqe2
Refer to n ( )  1 
e 0 me
2

j
fj
2
0j
  2  i j 
.
• Sellmeier equations work fine when the wavelength range of interests is far from the
absorption of the material.
2
• Beauty of Sellmeier equations: n( ),
dn d n
,
,  are obtained analytically.
d d2
• Sellmeier equations are tremendously helpful in designing various optical elements.
Examples: 1) Control the polarization of lasers. 2) Control the phase and pulse
duration of ultra-short laser pulses. 3) Phase-match in nonlinear optical processes.
Example: BK7 glass
Coefficient
Value
B1
1.03961212
B2
2.31792344×10
B3
1.01046945
C1
6.00069867×10 μm
C2
2.00179144×10 μm
C3
1.03560653×10 μm
−1
−3
2
−2
2
2
2
20
Read: Ch3: 5-7
Homework: Ch3: 54,55,57,62,66 (Optional)
Due: September 23
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