notes 13 3317

clappergappawpawUrban and Civil

Nov 16, 2013 (4 years and 1 month ago)

118 views

Prof. D. R. Wilton

Adapted from notes by Prof. Stuart A. Long

Notes 13

Maxwell’s Equations

ECE 3317

Electromagnetic Fields


Four vector quantities

E



electric field strength [Volt/meter] = [kg
-
m/sec
3
]

D


electric flux density [Coul/meter
2
] = [Amp
-
sec/m
2
]

H


magnetic field strength [Amp/meter] = [Amp/m]

B


magnetic flux density [Weber/meter
2
] or [Tesla] = [kg/Amp
-
sec
2
]

each are functions of space and time

e.g.

E
(
x
,
y
,
z
,
t
)

J



electric current density [Amp/meter
2
]


ρ
v


electric charge density [Coul/meter
3
] = [Amp
-
sec/m
3
]



Sources generating
electromagnetic fields

MKS units

length


meter [m]

mass


kilogram [kg]

time


second [sec]

Some common prefixes and the power of ten each represent are listed below

femto
-

f

-

10
-
15

pico
-

p

-

10
-
12

nano
-

n

-

10
-
9

micro
-

μ

-

10
-
6

milli
-

m

-

10
-
3



mega
-

M

-

10
6

giga
-

G

-

10
9

tera
-

T

-

10
12

peta
-

P

-

10
15



centi
-

c
-

10
-
2

deci
-

d
-

10
-
1

deka
-

da

-

10
1

hecto
-

h
-

10
2

kilo
-

k

-

10
3


0
v
B
E
t
D
H J
t
B
D
ρ

 


  

 
 
Maxwell’s Equations

(time
-
varying, differential form)

Maxwell’s Equations

James Clerk Maxwell (1831

1879)


James Clerk Maxwell

was a Scottish mathematician and
theoretical physicist. His most significant achievement was the
development of the classical electromagnetic theory, synthesizing
all previous unrelated observations, experiments and equations
of electricity, magnetism and even optics into a consistent theory.
His set of equations

Maxwell's equations

demonstrated that
electricity, magnetism and even light are all manifestations of the
same phenomenon: the electromagnetic field. From that moment
on, all other classical laws or equations of these disciplines
became simplified cases of Maxwell's equations. Maxwell's work
in electromagnetism has been called the "
second great
unification in physics
", after the first one carried out by Isaac
Newton.


Maxwell demonstrated that electric and magnetic fields travel
through space in the form of waves, and at the constant speed of
light. Finally, in 1864 Maxwell wrote
A Dynamical Theory of the
Electromagnetic Field

where he first proposed that light was in
fact undulations in the same medium that is the cause of electric
and magnetic phenomena. His work in producing a unified model
of electromagnetism is considered to be one of the greatest
advances in physics.

(Wikipedia)

Maxwell’s Equations (cont.)

ˆ
ˆ ˆ
ˆ
0
ˆ
C S
C S S
S
v
S V
d
E dr B ndS
dt
d
H dr J ndS D ndS
dt
B ndS
D ndS
ρ dV
  
    
 
 
 
  

 
Faraday’s law

Ampere’s law

Magnetic Gauss law

Electric Gauss law

(Time
-
varying, integral form)

The above are the most fundamental form of Maxwell’s equations since
all differential forms, all boundary forms, and all frequency domain forms
derive from them!

Maxwell’s Equations (cont.)

0
v
B
E
t
D
H J
t
B
D
ρ

 


  

 
 
Faraday’s law

Ampere’s law

Magnetic Gauss law

Electric Gauss law

(Time
-
varying, differential form)





0

D
H J
t
D
H J
t
J D
t

  


 
    
 

 

   


Law of Conservation of Electric
Charge (Continuity Equation)

v
J
t


  

Flow of electric
current out of volume
(per unit volume)

Rate of decrease of electric
charge (per unit volume)

[2.20]
Continuity Equation (cont.)

v
J
t


  

Apply the divergence theorem:

Integrate both sides over an arbitrary volume
V
:

v
V V
J dV dV
t


  

 
ˆ
v
S V
J ndS dV
t


  

 
V

S

ˆ
n
Continuity Equation (cont.)

Physical interpretation:

V

S

ˆ
n
ˆ
v
S V
J ndS dV
t


  

 
v
out v
V V
i dV dV
t t


 
   
 
 
encl
out
Q
i
t

 

(This assumes that the
surface is stationary.)

encl
in
Q
i
t



or

Maxwell’s Equations

Decouples



0

0 0


v
v
E
B D
E H J B D
t t
E D H J B


 
         
 
       
Time-Dependent
Time-Independent (Static

s)

and is a function of and is a functio
n of

v
H E H J



E B
H J D
B 0
D
v
j
j



  
  
 
 

Maxwell’s Equations

Time
-
harmonic (phasor) domain

j
t




Constitutive Relations

Characteristics of media relate

D

to
E

and
H

to
B

0
0
0
0
( = permittivity )
(

= permeability)

D E
B µ H
µ





-12
0
-7
0
[F/m]
8.8541878 10
= 4 10
H/m] ( )

[
µ




exact
[2.24]
[2.25]
[p. 35]
0 0
1
c


c = 2.99792458


10
8

[m/s]

(exact value that is defined)

Free Space

Constitutive Relations (cont.)

Free space, in the
phasor domain
:

0
0
0
0
( = permittivity )
(
D
=
= E
B
pe
=
rmeability

)
H
µ
µ



This follows from the fact that



V
aV t a

(where
a

is a real number)

Constitutive Relations (cont.)

In a
material medium
:

( = permittivity )
( = permeability
D = E
B=
)
H
µ
µ



0
0
=
r
r
µ µ
 



r

= relative permittivity


r

= relative permittivity


μ
or
ε


Independent of


Dependent on




space


homogenous


inhomogeneous



frequency non
-
dispersive dispersive



time


stationary


non
-
stationary



field strength linear



non
-
linear



direction of isotropic


anisotropic


E

or
H

Terminology

Isotropic Materials

ε

(or
μ
) is a scalar quantity,


which means that
E

||
D

(and
H

||
B

)

x x
B = H

y y
B = H

y
H
x
H
x

y

x x
D = E

y y
D = E

y
E
x
E
x

y

ε

(or
μ
) is a tensor (can be written as a matrix)

This results in
E

and
D

being


NOT

parallel to each other; they are
in different directions.

0 0
0 0
0 0
x x x
y y y
x x x
z z z
y y y
z z z
D E
D E
D E
D E
D E
D E






     
     

 





   
     
   





Anisotropic Materials

D
E