Review – The Wave Equation
2
y
M2
1
T
y  x, t 
M1
T
δx
x
x+δx
x
2
2 y T 2 y

y
2

c
2
2
t
 x
x 2
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1
Review – Boundary Condition
y A  yB
y A y B

x
x
y1 t  x vA  y3 t  x vB 
apply continuity conditions
for separate components
y2 t  x v A 
y
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A
combine forward and
reflected waves to give total
fields for each region
B
x
hence derive fractional
transmission and reflection
2
Review – Fourier Transform
We desire a measure of the frequencies present in a wave. This will
lead to a definition of the term, the "spectrum.“
Plane waves have only
one frequency, w.
This light wave has many
frequencies – wave group.
The frequency increases in
time (from red to blue).
It will be nice if our measure also tells us when each frequency occurs.
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Review – Wave Group
When two waves of different frequency interfere, they produce "beats"
Individual
Waves
Sum
Envelope
Irradiance:
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Spring 2016
Ch6. Electromagnetic Waves
1. A Brief History of Electromagnetism
2. The Wave Equation for Light
3. The Electromagnetic Spectrum
4. Vector Field Representation and Operators
5. The Maxwell's Equations
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Light is an Electromagnetic Wave
Electric (E) and magnetic (B) fields are in phase.
The electric field, the magnetic field, and the propagation
direction are all perpendicular.
2

E
2
 E   2  0
t
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2

B
2
 B   2  0
dt
The 1D wave equation for EM waves
 2E
 2E
  2  0
2
x
t
where E is the
light electric field
w
 is the permittivity
 is the permeability of the
medium
k
 v 
E(x,t) = A cos[( w t- k x + q ]
A = Amplitude
q = Absolute phase (or initial phase)
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1

A Brief History of Electromagnetism
17th-century
Kepler,
Huygens
….
18th-century 19th-century
Newton…
Total internal reflection,
Telescope, geometrical optics, the
wave theory, prism dispersion, the
particle theory of light
Fresnel,
Young…
20th-century
Maxwell
Michelson…
Interference, diffraction,
expressions for reflected
and transmitted waves,
unified electricity and
magnetism
Einstein
Louis de Broglie
…
Light is
(1) “a phenomenon of
empty space”
(2) both a wave and
a particle
“Light is, in short, the most
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refined form of matter.”
More Definitions
Spatial quantities:
Temporal quantities:
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The Electromagnetic Spectrum
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The Electromagnetic Spectrum
gamma-ray
microwave
2
10
1
106
10
visible
radio
infrared
0
10
105
-1
4
10
10
3
10
UV
2
10
wavelength (nm)
The transition wavelengths are a bit arbitrary…
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X-ray
1
10
0
10
-1
10
The Electromagnetic Waves
GPS: ~1.5 GHz
Microwave antenna
The solar corona
THz gap
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Photonics
IR for night vision
Electronics
Why Study Electromagnetic Waves?
One reason: fiber optics have replaced or will soon replace most wires.
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Infrared Lie-Detection
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Vector Fields
EM waves are a 3D vector
field.
A 3D vector field f (r )
assigns a 3D vector (i.e., an
arrow having both direction
and length) to each point in
3D space.
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Vector Derivatives
The “Del” operator:
 


  
,
,


x

y

z


The “Gradient” of a scalar function f :
 f
f
f 
f  
,
,


x

y

z


The gradient points in the direction of steepest ascent.
The “Divergence” of a vector function:
f x f y f z
 f 


x y z
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Vector Derivatives
The “Laplacian” of a scalar function :
2 f
  f
 f
f
f 
  
,
,


x

y

z



2 f
2 f
2 f


2
2
x
y
z 2
The “Laplacian” of a vector function is the same,
but for each component of :
2
2
2
2
2
2
2
2
2


f

f

f

f

f

f

f

f

fz
y
y
y
2
x
x
x
z
z
 f   2 
 2 ,


, 2 
 2
2
2
2
2
2
 x
y
z
x
y
z
x
y
z

The Laplacian tells us the curvature of a function.
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


The Wave Equations in Vacuum
 E
 E   2  0
t
2E 2E 2E
2E

 2   2  0
2
2
x
y
z
t
2
2

i (wt  k r q )
E (r , t )  E 0e
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This is really just
three independent
wave equations,
one each for the
x-, y-, and zcomponents of E.
The Equations of EM Waves
Are Maxwell’s Equations
E  r /
B  0
B
 E  
t
E
  B  
t
where E is the electric field, B is the magnetic field,
r is the charge density,  is the permittivity, and  is
the permeability of the medium.
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Divergence of Vector Fields
Divergence is a number indicating the ‘outwardness’ of the vector field
 

1
Div F  lim
F  dS
V 0 V 
The divergence of a vector field is done through differentiation operator
The change in a vector field in the direction of pointing
 
 
1
  F  lim
F  dS
V  0 V 
  


  F  Fx  Fy  Fz
x
y
z
Tip: you can view divergence as a ‘source density’
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Curl of Vector Fields
The curl of a field defines the ‘amount of rotation’ in the field

 F
x


 f 
x
fx
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y

y
fy
z

z
fz
Curl of Vector Fields
The "Curl" of a vector function f :
 f
 f z f y f x f z f y f x 
 

,

,



y
dz

z
dx

x
dy


The curl can be treated as a matrix determinant :
x



 f 
 x

 f x
y

y
fy
z 


z 

f z 
Functions that tend to "curl around" have large curls
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Quiz
f ( x, y, z )  (  y, x, 0)
f (1, 0, 0)  (0,1, 0)
f (0,1, 0)  (1, 0, 0)
f (1, 0, 0)  (0, 1, 0)
f (0, 1, 0)  (1, 0, 0)
f y f x 
 f z f y
f x f z
 f  

,

,



y

z

z

x

x

y


  0  0, 0  0, 1  (1) 
Calculate
, curl
2 of this function
 0 , 0 the
So this function has a curl of 2z
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(I) Gauss’ Law
In CGS unit, integrating perpendicular component of E-field over
closed surface  gives total charge within surface.

 
 E (r )  dS  qtot / 
 
E  r /
Consider rtot(r) = total charge density

qtot   r tot (r ) dV
V

 
 
 E (r )  dS     E dV
V
Simple check from a “known” formula
Coulomb force field:
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E
q
4r
2
r
(II) Gauss’ Law for B field
Following Gauss’ Law
  

 Br  ds  qm  0     Bdv
s
v
 
 B  0
There are no magnetic charges or “monopoles”
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(III) Faraday Law
The interplay between B and E
You may know steady current generates magnetic field (or flux B)
I
B
B
I
But also: increasing B generates counter-current because of a ‘rotating’ E field
Or in formula form :
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 
  
  
C E  dl   t S B  dS     E  dS
S

 
B
 E  
t
(IV) Ampere Law
For an entirely free current (quite rare, but possible - see sketch) the
magnetic flux is related to the current through



B

d
l


I




B

d
s
total


c
s
Note last part is after applying Stoke’s theorem.
Defining the total current density Jtot (current per unit area):
I free


  J free  ds
s
We have


  B  J tot
Is this enough?
No. Similar to the way a material influences E due to induced polarization,
materials can also affect B due to induced magnetization current
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(IV) Ampere Law – Continued
The time varying part is missing for the E field
E
dt
This leads to the final result


D
  B  J free  
dt
Displacement current


D  E

E
  B  
dt
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Maxwell’s Equations in Vacuum
E  r /
B  0
B
 E  
t
E
  B  
t
where E is the electric field, B is the magnetic field,
r is the charge density,  is the permittivity, and  is
the permeability of the medium.
2

E
2
 E   2  0
t
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2

B
2
 B   2  0
dt