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Optical Equations 1
Maxwell’s equations – A historical review for a
better understanding (continued)
1
Acceleration (AC Current)
• If the charge stream is accelerated – the magnetic flux
(swirl) becomes dynamic; what makes it more important
is that an electric field swirl will be generated by the
dynamic magnetic flux, which leads to the Faraday’s law.
• Why would a dynamic magnetic flux generate the
electric field swirl?
• Implication - the system is attempting to stay still: once a
change is found in magnetic flux, an electric field is
excited in such a way that it tries to cancel out the
magnetic flux change by generating a new magnetic flux
against the original one; i.e.,
 
 

B  ds   E  dl  0

s
l
t


B
 E  
t
2
Extended Summary beyond Static Fields
- A Wrap up of Experimental Observations
• Stay still charge distribution generates divergence driven,
swirl free electric field (which can be sensed by any
charged object, hence we have the name “electric”).
• Charge in static motion generates not only the above
mentioned electric field, but also swirl driven, divergence
free magnetic field (which differs from the electric field as
it can only be sensed by the charged moving object,
hence we have the name “magnetic”).
- So far, the fields are static (with spatial dependence
only, no temporal dependence) and non-coupled
(between the electric and magnetic ones).
3
Extended Summary beyond Static Fields
- A Wrap up of Experimental Observations
• Accelerated charge generates dynamic (time-varying)
magnetic field, which induces the swirl to the electric
field.
• Consequently, the electric field will be driven by both
divergence and curl sources; the former comes from the
stay still or constantly moving charges, whereas the
latter is induced by the time-varying of the magnetic field
which comes from the charge acceleration. Also, the
electric and magnetic fields becomes coupled, but still in
one way (from magnetic to electric only).
4
Extended Summary beyond Static Fields
- A Wrap up of Experimental Observations
• We can express these conclusions mathematically to
obtain the governing equations for any electromagnetic
effect in vacuum


B
 E  
t


  B  0 J

  E   /0

B  0
Faraday’s law
Ampere’s law
Gauss’ law (E)
Gauss’ law (M)
5
Maxwell’s Equations
- Power of Logic and Math
• Maxwell found inconsistency in the 2nd equation if the
charge accelerates


    B  0  J

0


 0

t
• He then mended the 2nd equation by

 
    B   0 (  J 
)
t


0

0



E
  B   0 J   0 0
t
with Gauss’ law applied to
the last term on the RHS
6
Maxwell’s Equations
- Power of Logic and Math
• Implication of the added term – time-varying electric field,
similar to the current, also generates magnetic field.
• Hence we name the time change rate of the electric field
the displacement current (more accurately, the timederivative of the displacement vector), and the
conventional current the conduction current.
7
Maxwell’s Equations
- Power of Logic and Math
• As a consequence
– 1. charge acceleration generates dynamic magnetic field in its
neighborhood (Ampere’s law);
– 2. dynamic magnetic field induces dynamic electric field
(Faraday’s law);
– 3. dynamic electric field in its neighborhood generates dynamic
magnetic field (Maxwell’s displacement current equivalence +
Ampere’s law), such sequence repeats endlessly in a area which
is not necessarily limited to the location of the source where the
charge accelerates
• This process describes the electromagnetic wave
generation and propagation.
8
Maxwell’s Equations
- The Ultimate Form in Vacuum


B
 E  
t



E
  B   0 J   0 0
t

  E   / 0

B 0
9
Maxwell’s Equations
- The Ultimate Form in Media


B
 E  
t


 D
 H  J 
t

D  

B  0


D  E


B  H


J  E
• There are 16 scalar variables, but 17 equations. One
equation is redundant.
• Normally, we don’t need the last one (the magneto Gauss’
law ), as it is embedded in the 1st equation.
• The carrier continuity equation is embedded in the 2nd
equation.
10
Optical Equations 2
A simple phenomenological (Lorentz’s) model to
understand wave-material interaction
11
Material Lorentz’s Model
• The material is viewed as a group of spring bonded
flexible electrons on fixed ion centers
dx
d 2x
eE (t )  kx  
 m0 2
The motion of a single electron:
dt
dt
~

k ~ eE ( )
d 2 x  dx k
eE (t )
2~
~


x

j

x

x


x

2
m
m
m0
dt
m0 dt m0
m0
0
0
~
x
~
eE ( )
m0 ( 2 

k
j  )
m0
m0
~
eE ( )

m0 ( 02   2  j )
k

e2 N
2
 

p 
m0
m0
m0V
2
0
~
~
 p2 E ( )
N ~
e 2 NE ( )
~
The (dipole) polarization: P ( )  ex 

V
m0V (02   2  j ) 02   2  j
The displacement:
~
~
~
D( )   0 E ( )  P ( )  ( 0 
 p2
~
)
E
( )
02   2  j
12
Material Lorentz’s Model
• Dielectric constant (permittivity) for insulators and
semiconductors
 p2
   0 (1  2
)
2
 0    j 
 p2 (02   2 )
 p2 
  0 [1  2
j 2
]
(0   2 ) 2   2 2
(0   2 ) 2   2 2
1
0
Normal

For frequency far away from  0
the real part decays more slowly
than the imaginary part – that’s
why we often take a real dielectric
constant with the lossy part ignored.
Normal
Abnormal
13
Material Lorentz’s Model
• Dielectric constant (permittivity) for metals –
The Drude Model 0  0
   0 (1 
  0 [1 
If the loss is negligible,   0
 p2
(  j)
 p2
 
2
2
j
)
 p2
we find  ~  2  0  0

 p2 
(   )
2
2
]
The refractive index becomes
imaginary. Therefore, inside metals,
there is no EM wave can possibly be
traveling – only exponentially decayed
(i.e., evanescent) wave is allowed.
1
0  0
Normal
Abnormal
14