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Lecture 9: Faraday’s Law Of
Electromagnetic Induction;
Displacement Current; Complex
Permittivity and Permeability
Lecture 9
1



To study
Faraday’s law of electromagnetic induction;
displacement current;
and complex permittivity and permeability.
Lecture 9
2

Integral form

 E  dl  0
 E  0
  D  qev
C
 Dds  q
ev
S
Differential form
dv
V
D E
Lecture 9
3

Integral form

 H  dl   J  d s
C
Differential form
 H  J
S
B  0
 Bds  0
S
B  H
Lecture 9
4


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.
Lecture 9
5


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.
Lecture 9
6



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.
Lecture 9
7
switch
toroidal iron
core
compass
battery
secondary
coil
primary
coil
Lecture 9
8




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.
Lecture 9
9

“The electromotive force induced around
a closed loop Γ is equal to the time rate of
decrease of the magnetic flux linking the
loop.”
d
e  
dt
S
C
Lecture 9
10
   Bds
• S is any surface
bounded by Γ
S
e   E  d l
C
d
E

d
l


B

d
s
C

dt S
integral form
of Faraday’s
law
Lecture 9
11
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
Lecture 9
12

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
Lecture 9
13


Faraday’s law states that a changing
magnetic field induces an electric field.
The induced electric field is nonconservative.
Lecture 9
14



“The sense of the emf induced by the
time-varying 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.
Lecture 9
15

“The electromotive force induced
around a closed loop Γis equal to the
time rate of decrease of the magnetic
flux linking the loop.”
d
e  
dt

For a coil of N tightly wound turns
d
e   N
dt
16
Lecture 9
   Bds
S
S
Γ
• S is any surface
bounded by Γ
e   E  d l
C
Lecture 9
17

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
Lecture 9
18

The induced emf around a circuit can be
separated into two terms:
 (1) due to the time-rate of change of the
B-field (transformer emf)
 (2) due to the motion of the circuit
(motional emf)
Lecture 9
19
d
e    B  d s
dt S
B
 
ds 
t
S


v

B

d
l

C
transformer emf
motional emf
Lecture 9
20

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
Lecture 9
21

2
B
v

+
1
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
22


The electrostatic force on an electron due to
the induced electrostatic field is given by
The migration of electrons stops
 e E when
(equilibrium F
is eestablished)
F e  F m
 E  v  B
Lecture 9
23

A motional (or “flux cutting”) emf is produced
given by
1
e   v  B   d l
2
Lecture 9
24

Electrostatics:
  E  0  E  
scalar electric potential
Lecture 9
25

Electrodynamics:
B   A
B

 E  
    A
t
t
A
A

 E 
 
0  E
t 
t

Lecture 9
26

Electrodynamics:
A
E   
t
vector
magnetic
potential
• both of these
potentials are now
functions of time.
scalar
electric
potential
Lecture 9
27

The differential form of Ampere’s law in the
static case is
 H  J

The continuity equation is
qev
J 
0
t
Lecture 9
28

In the time-varying case, Ampere’s law in the
above form is inconsistent with the continuity
equation
  J      H   0
Lecture 9
29

To resolve this inconsistency, Maxwell
modified Ampere’s law to read
D
 H  J c 
t
conduction
current density
displacement
current density
Lecture 9
30

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
Lecture 9
31

Ampere’s law can be written as
 H  Jc  J d
where
D
Jd 
 displaceme nt current density (A/m 2 )
t
Lecture 9
32



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.
Lecture 9
33

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
Lecture 9
34

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 
 aˆ z
sin  t
t
d
Lecture 9
35

The displacement current is given by
id   J d  d s   J d A  
S
A
d
dv
  CV0 sin t  C
 ic
dt
V0 sin t
conduction
current in
wire
Lecture 9
36


Consider a conducting medium
characterized by conductivity s and
permittivity .
The conduction current density is given
by
J s E
c

The displacement current density is
given by
E
Jd 
t
Lecture 9
37


Assume that the electric field is a sinusoidal
function of time:
Then,
E  E0 cos t
J c  sE0 cos t
J d  E0 sin t
Lecture 9
38


We have
Therefore
Jc
max
 sE0
Jd
max
 E0
J c max
Jd
max
s


Lecture 9
39



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.
Lecture 9
40
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



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.
Lecture 9
42

The complex dielectric constant can be
written as
 c     j 

Substituting the complex dielectric
constant into the differential frequencydomain form of Ampere’s law, we have
  H  s E  j  E    E
Lecture 9
43

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
2
Lecture 9
44


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
   s eff
• The value of seff is
often determined by
measurements.
Lecture 9
45

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
Lecture 9
46
18
20

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 

Lecture 9
47


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 
Lecture 9
48
  E   j  s m  H  K i
  H   j  s e  E  J i
E 
qev
H 
qmv


Lecture 9
49
complex
permeability
  E   j H  K i
  H  j E  J i
complex
permittivity
Lecture 9
50