Transcript Maxwells eqn

Faraday’s Law Of Electromagnetic
Induction; Displacement Current;
Complex Permittivity and
Permeability
1
Objectives

To study Faraday’s law of electromagnetic
induction; displacement current; and
complex permittivity and permeability.
2
Lecture 9
Fundamental Laws of
Electrostatics

Integral form

 E  dl  0
 E  0
  D  qev
C
 Dds  q
ev
S
Differential form
dv
V
D E
3
Lecture 9
Fundamental Laws of
Magnetostatics

Integral form

 H  dl   J  d s
C
Differential form
 H  J
B  0
S
 Bds  0
S
B  H
4
Lecture 9
Electrostatic, Magnetostatic, and
Electromagnetostatic Fields


In the static case (no time variation), the electric
field (specified by E and D) and the magnetic
field (specified by B and H) are described by
separate and independent sets of equations.
In a conducting medium, both electrostatic and
magnetostatic fields can exist, and are coupled
through the Ohm’s law (J = sE). Such a
situation is called electromagnetostatic.
5
Lecture 9
Electromagnetostatic Fields


In an electromagnetostatic field, the electric field
is completely determined by the stationary
charges present in the system, and the magnetic
field is completely determined by the current.
The magnetic field does not enter into the
calculation of the electric field, nor does the
electric field enter into the calculation of the
magnetic field.
6
Lecture 9
The Three Experimental Pillars
of Electromagnetics



Electric charges attract/repel each other as
described by Coulomb’s law.
Current-carrying wires attract/repel each other
as described by Ampere’s law of force.
Magnetic fields that change with time induce
electromotive force as described by Faraday’s
law.
7
Lecture 9
Faraday’s Experiment
switch
toroidal iron
core
compass
battery
secondary
coil
primary
coil
8
Lecture 9
Faraday’s Experiment (Cont’d)




Upon closing the switch, current begins to flow
in the primary coil.
A momentary deflection of the compass needle
indicates a brief surge of current flowing in the
secondary coil.
The compass needle quickly settles back to
zero.
Upon opening the switch, another brief
deflection of the compass needle is observed.
9
Lecture 9
Faraday’s Law of
Electromagnetic Induction

“The electromotive force induced around a
closed loop C is equal to the time rate of
decrease of the magnetic flux linking the loop.”
Vind
d

dt
S
C
10
Lecture 9
Faraday’s Law of Electromagnetic
Induction (Cont’d)
   Bds
• S is any surface
bounded by C
S
Vind   E  d l
C
d
E

d
l


B

d
s
C

dt S
11
integral form
of Faraday’s
law
Lecture 9
Faraday’s Law (Cont’d)
Stokes’s theorem
E

d
l



E

d
s


C
S
d
B
  B  d s  
ds
dt S
t
S
assuming a stationary surface S
12
Lecture 9
Faraday’s Law (Cont’d)

Since the above must hold for any S, we have
differential form
of Faraday’s law
(assuming a
stationary frame
of reference)
B
 E  
t
13
Lecture 9
Faraday’s Law (Cont’d)
Faraday’s law states that a changing
magnetic field induces an electric field.
 The induced electric field is nonconservative.

14
Lecture 9
Lenz’s Law



“The sense of the emf induced by the timevarying magnetic flux is such that any current it
produces tends to set up a magnetic field that
opposes the change in the original magnetic
field.”
Lenz’s law is a consequence of conservation of
energy.
Lenz’s law explains the minus sign in Faraday’s
law.
15
Lecture 9
Faraday’s Law

“The electromotive force induced around a
closed loop C is equal to the time rate of
decrease of the magnetic flux linking the
loop.”
Vind

d

dt
For a coil of N tightly wound turns
Vind
d
 N
dt
16
Lecture 9
Faraday’s Law (Cont’d)
   Bds
S
S
C
• S is any surface
bounded by C
Vind   E  d l
C
17
Lecture 9
Faraday’s Law (Cont’d)

Faraday’s law applies to situations where
 (1)
the B-field is a function of time
 (2) ds is a function of time
 (3) B and ds are functions of time
18
Lecture 9
Faraday’s Law (Cont’d)

The induced emf around a circuit can be
separated into two terms:
 (1)
due to the time-rate of change of the Bfield (transformer emf)
 (2) due to the motion of the circuit (motional
emf)
19
Lecture 9
Faraday’s Law (Cont’d)
Vind
d
   Bds
dt S
B
 
ds 
t
S


v

B

d
l

C
transformer emf
motional emf
20
Lecture 9
Moving Conductor in a Static
Magnetic Field

Consider a conducting bar moving with
velocity v in a magnetostatic field:
• The magnetic force on an
electron in the conducting
bar is given by
2
B
v
+
F m  ev  B
1
21
Lecture 9
Moving Conductor in a Static
Magnetic Field (Cont’d)

2
B
v
+

1
22
Electrons are pulled
toward end 2. End 2
becomes negatively
charged and end 1
becomes + charged.
An electrostatic force
of attraction is
established between the
two ends of the bar.
Lecture 9
Moving Conductor in a Static
Magnetic Field (Cont’d)
The electrostatic force on an electron
due to the induced electrostatic field is
given by
F e  e E
 The migration of electrons stops
(equilibrium is established) when

F e  F m  E  v  B
23
Lecture 9
Moving Conductor in a Static
Magnetic Field (Cont’d)

A motional (or “flux cutting”) emf is
produced given by
1
Vind   v  B   d l
2
24
Lecture 9
Electric Field in Terms of
Potential Functions

Electrostatics:
  E  0  E  
scalar electric potential
25
Lecture 9
Electric Field in Terms of
Potential Functions (Cont’d)

Electrodynamics:
B   A
B

 E  
    A
t
t
A
A

 E 
 
0  E
t 
t

26
Lecture 9
Electric Field in Terms of
Potential Functions (Cont’d)

Electrodynamics:
A
E   
t
vector
magnetic
potential
• both of these
potentials are now
functions of time.
scalar
electric
potential
27
Lecture 9
Ampere’s Law and the Continuity
Equation

The differential form of Ampere’s law in
the static case is
 H  J

The continuity equation is
qev
 J 
0
t
28
Lecture 9
Ampere’s Law and the Continuity
Equation (Cont’d)

In the time-varying case, Ampere’s law in
the above form is inconsistent with the
continuity equation
  J      H   0
29
Lecture 9
Ampere’s Law and the Continuity
Equation (Cont’d)

To resolve this inconsistency, Maxwell
modified Ampere’s law to read
D
 H  J c 
t
conduction
current density
30
displacement
current density
Lecture 9
Ampere’s Law and the Continuity
Equation (Cont’d)

The new form of Ampere’s law is
consistent with the continuity equation as
well as with the differential form of
Gauss’s law

  J c    D       H   0
t
qev
31
Lecture 9
Displacement Current

Ampere’s law can be written as
 H  J c  J d
where
D
Jd 
 displaceme nt current density (A/m 2 )
t
32
Lecture 9
Displacement Current (Cont’d)



Displacement current is the type of current
that flows between the plates of a capacitor.
Displacement current is the mechanism
which allows electromagnetic waves to
propagate in a non-conducting medium.
Displacement current is a consequence of
the three experimental pillars of
electromagnetics.
33
Lecture 9
Displacement Current in a
Capacitor

Consider a parallel-plate capacitor with plates of
area A separated by a dielectric of permittivity 
and thickness d and connected to an ac
generator:
z
A
z=d
z=0

ic
+
v(t )  V0 cos t
-
id
34
Lecture 9
Displacement Current in a
Capacitor (Cont’d)

The electric field and displacement flux
density in the capacitor is given by
V0
v(t )
E  aˆ z
 aˆ z cos  t
d
d
 V0
D   E  aˆ z
cos  t
d

• assume
fringing is
negligible
The displacement current density is
given by
 V0
D
Jd 
t
 aˆ z
35
d
sin  t
Lecture 9
Displacement Current in a
Capacitor (Cont’d)

The displacement current is given by
id   J d  d s   J d A  
S
A
d
dv
  CV0 sin t  C
 ic
dt
36
V0 sin t
conduction
current in
wire
Lecture 9
Conduction to Displacement
Current Ratio



Consider a conducting medium characterized by
conductivity s and permittivity .
The conduction current density is given by
Jc s E
The displacement current density is given by
E
Jd 
t
37
Lecture 9
Conduction to Displacement
Current Ratio (Cont’d)

Assume that the electric field is a sinusoidal
function of time:
E  E0 cos t

Then,
J c  sE0 cos t
J d  E0 sin t
38
Lecture 9
Conduction to Displacement
Current Ratio (Cont’d)


We have
Therefore
Jc
max
 sE0
Jd
max
 E0
J c max
Jd
max
s


39
Lecture 9
Conduction to Displacement
Current Ratio (Cont’d)



The value of the quantity s/ at a specified
frequency determines the properties of the
medium at that given frequency.
In a metallic conductor, the displacement
current is negligible below optical frequencies.
In free space (or other perfect dielectric), the
conduction current is zero and only
displacement current can exist.
40
Lecture 9
Conduction to Displacement
Current Ratio (Cont’d)
10
10
10
10
10
10
s

10
10
10
10
10
Humid Soil ( r = 30, s = 10-2 S/m)
6
5
4
good
conductor
3
2
1
0
-1
-2
-3
good insulator
-4
10
0
10
2
10
4
10
6
Frequency (Hz)
41
10
8
10
10
Lecture 9
Complex Permittivity



In a good insulator, the conduction current (due to
non-zero s) is usually negligible.
However, at high frequencies, the rapidly varying
electric field has to do work against molecular forces
in alternately polarizing the bound electrons.
The result is that P is not necessarily in phase with
E, and the electric susceptibility, and hence the
dielectric constant, are complex.
42
Lecture 9
Complex Permittivity (Cont’d)

The complex dielectric constant can be written
as
 c     j 

Substituting the complex dielectric constant into
the differential frequency-domain form of
Ampere’s law, we have
  H  s E  j E   E
43
Lecture 9
Complex Permittivity (Cont’d)


Thus, the imaginary part of the complex
permittivity leads to a volume current density
term that is in phase with the electric field, as
if the material had an effective conductivity
given by s eff  s   
The power dissipated per unit volume in the
medium is given by
s eff E  sE   E
2
2
44
2
Lecture 9
Complex Permittivity (Cont’d)


The term  E2 is the basis for microwave
heating of dielectric materials.
Often in dielectric materials, we do not
distinguish between s and , and lump them
together in  as
• The value of seff is
often determined by
measurements.
   s eff
45
Lecture 9
Complex Permittivity (Cont’d)
In general, both  and  depend on
frequency, exhibiting resonance
characteristics at several frequencies.

1
Imag Part of Dielectric Constant
Real Part of Dielectric Constant
2.5
2
1.5
1
0.5
0
0
2
4
6
8
10
12
14
16
18
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
20
Normalized Frequency
0
2
4
6
8
10
12
14
16
Normalized Frequency
46
Lecture 9
18
20
Complex Permittivity (Cont’d)

In tabulating the dielectric properties of
materials, it is customary to specify the real part
of the dielectric constant ( / 0) and the loss
tangent (tand) defined as
 
tan d 

47
Lecture 9
Complex Permeability


Like the electric field, the magnetic field
encounters molecular forces which require work
to overcome in magnetizing the material.
In analogy with permittivity, the permeability
can also be complex
c     j 
48
Lecture 9
Maxwell’s Equations in Differential Form for
Time-Harmonic Fields in Simple Medium
  E   j  s m  H  K i
  H   j  s e  E  J i
E 
qev
H 
qmv


49
Lecture 9
Maxwell’s Curl Equations for Time-Harmonic
Fields in Simple Medium Using Complex
Permittivity and Permeability
complex
permeability
  E   j H  K i
  H  j E  J i
complex
permittivity
50
Lecture 9