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ELEC 3105 Basic EM and Power
Engineering
Faraday’s LawLenz’s Law
Displacement Current
Faraday’s Law
Introduction: So far we have
E
0 E
JH�
0 B
�
These equations are OK for static fields, i.e. those fields independent of time. When fields vary as a function of time the curl equations acquire an additional term.
0 E
gets a t
B
gets a t
D
JH�
ELEC 3105 Basic EM and Power
Engineering
Faraday’s Law
Consider the following experiment. Pull a wire loop through a region of non-uniform magnetic field .
v
Wire loop, path encloses a surface SB
Magnetic field vector points into the page
1B
2B
tvtv
area gained in time t
area lost in time t
tv
tv
21BB
Faraday’s Law
Consider the following experiment. Pull a wire loop through a region of non-uniform magnetic field .
v
B
Magnetic field vector points into the page
1B
2B
tvtv
21BB
v
Charge in wire
11BvqF
Charge in wire
22BvqF
1F
2F
Faraday’s Law
Consider the work done on +1C test charge moved around the loop; this is the “emf” electromotive force.
v1
B
2B
tvtv
21BB
v
Charge in wire
11BvqF
Charge in wire
22BvqF
1F
2F
21* FFqemf
t
tvBtvBemf
21
Faraday’s Law𝑊𝑞=𝑉Early definition of potential
ExamineExpression
vBvBemf 21
Consider the work done on +1C test charge moved around the loop; this is the “emf” electromotive force.
v1
B
2B
tvtv
v1
F
2F
t
tvBtvBemf
21Now: tvB
1Flux change at right side of loop
tvB2
Flux change at left side of loop
t
Fluxemf
S
adB flux
Note that B and da are in opposite directionsdt
demf
Faraday’s Law
1B
2B
t
Fluxemf
dt
demf
This is a general result. Even if we hold the loop stationary and change B, the emf is still given by the negative rate of change of the flux.
Move magnet
N S
Faraday’s Law
1B
2B
Further generalization is possible. Consider moving loop in time varying magnetic field.
Move magnetN
S
Move loop
S
adt
BdBvemf
S
adt
B
dBv MOTIONAL emf
TRANSFORMER emf
The induced emf always opposes the change in
flux
Faraday’s Law
ELEC 3105 Basic EM and Power
Engineering
Lenz’s Law
magnetB
loopB
N S
MOVE LOOP
dBvemf
motional emf
The induced emf always opposes the
change in flux
v
• Move loop towards magnet• B increases in loop• Flux increases in loop• Current induced through emf• Current produces magnetic field in loop: 2nd postulate• This magnetic field in opposite in direction to magnetic field of magnet
I
Faraday’s Law / Lenz’s Law
magnetB
loopB
N S
MOVE LOOP
dBvemf motional emf
The induced emf always opposes
the change in flux
v
• Move loop away from magnet• B decreases in loop• Flux decreases in loop• Current induced• Current produces magnetic field in loop• This magnetic field in same direction to magnetic field of magnet
I
Faraday’s Law / Lenz’s Law
magnetB
loopB
N S
MOVE MAGNET
The induced emf always opposes the
change in flux
v
• Move magnet towards loop• B increases in loop• Flux increases in loop• Current induced• Current produces magnetic field in loop• This magnetic field in opposite direction to magnetic field of magnet
I
S
adt
Bemf
transformer emf
Faraday’s Law / Lenz’s Law
magnetB
loopB
N S
MOVE MAGNET
The induced emf always opposes the
change in flux
v
• Move magnet away from loop• B decreases in loop• Flux decreases in loop• Current induced• Current produces magnetic field in loop• This magnetic field is in same direction to magnetic field of magnet
I
S
adt
Bemf
transformer emf
Faraday’s Law / Lenz’s Law
Suppose loop is stationary so we have only transformer emf.
MOVE MAGNET
N S
S
adt
Bemf
I
dEemf Equivalent battery to drive current around loop
S
adt
BdE
S
adt
BadE
t
BE
Faraday’s Law
t
BE
Faraday’s Law in derivative form: Valid for all points in space
The induced emf always opposes the change in
flux
S
adBt
emf
Note on: Faraday’s Law / Lenz’s Law
dt
demf
WE KNOW THAT �� (𝑡 )∝𝐼 (𝑡 )
constant
�� (𝑡 )=�� ∗ 𝐼 (𝑡 )
t
tIadKemf
S
Inductance: relates the induced emf to the time rate of change of the current 𝑣 (𝑡 )=−𝐿
𝜕 𝐼 (𝑡 )𝜕𝑡
S
adBt
emf
Note on: Faraday’s Law / Lenz’s Law
tBAemf sin
Rotating current loop in constant magnetic field
��X
O
𝜃
𝜔
�� ∙ ��𝑎=𝐵 𝑑𝑎𝑐𝑜𝑠 (𝜔 𝑡 )
Sinusoidal voltage change
Amplitude of voltage depends on B, A and POWER GENERATOR
Note on: Faraday’s Law / Lenz’s Law
vBLemf
Moving rod in constant magnetic field
��
dBvemf
X
X
X
X
X
X
X
X
X
X
X
X
vL
Which end of the rod is positive and which end is negative?
+
-
+
-
?
ELEC 3105 Basic EM and Power
Engineering
Displacement Current
Displacement Current
Capacitor
WIRE
enclosedIdH
So far we have the following expression for Ampere’s law.
Certainly there is no problem in evaluating the integral. The path shown encloses an area A through which the wire cuts.
We are in fact in the process of charging the capacitor through the current I of the wire.
IdH
For surface A chosen
Capacitor
WIRE
enclosedIdH
So far we have the following expression for Ampere’s law.
Certainly there is a problem in evaluating the integral. The path shown encloses an area A’ through which the wire does not cut. Yet the integral value evaluated here for the surface that does not contain the wire A’ and the surface of the previous slide A which does contain the wire must be the same since the surfaces A’ and A are arbitrary.
We are in fact in the process of charging the capacitor through the current I of the wire.
0
dHFor path A’ chosen
Both integrals should give the same result??????
Displacement Current
Capacitor
WIRE
IdH
For different surfaces
Capacitor
WIRE
0
dH
We have a problemDisplacement Current
A A
adt
DadJdH
Solution to the problem
Between the capacitor plates we have a changing electric flux density. This changing electric flux density is equivalent to a current density.
t
DJ
D
Displacement Current
A A
adt
DadJdH
Solution to the problem
A A
adt
DadJadH
t
DJH
Displacement Current
Ampere’s law in integral form
Ampere’s law in derivative form
Apply Stoke’s theorem
Ampere’s Law in derivative form: Valid for all points in space
t
DJH
Displacement Current
Figure 8.1 Application of Ampere’s law to a circuit with an �air-filled capacitor and time-varying current.
NOW it might make sense, if
i = dD/dt Sp
Taken from ELEC 3909
t
DJH
�
Ratio Jc to JDSome materials are neither good conductors nor perfect dielectrics, so that both conduction current and displacement current exist. A model for a poor conductor or lossy dielectric is shown below. Assume a time dependence of ejwt for the electric field.
E
s e
Then
Remember this result (for ELEC 3909)
Ratio Jc to JDA circular cross-section conductor of radius 1.5 mm caries a current ic = 5.5 sin(4E10t)(m A). What is the amplitude of the displacement current density if s = 35 MS/m and r = 1.
E
s e
Then
Remember this result (for ELEC 3909)
79.9
369
104
75.3E
EE
E
wJ
J
D
c
s
2
2
386.779.9
35.1/65.5
m
AE
E
EEJ D
m
Faraday Homopolar Generator
B
a
V
The circular metal disk rotates at (rad/s) in a uniform flux density B. Sliding contacts connect a voltmeter to the disk. What voltage is indicated on the meter.Solution: One radial element of the disk is examined. A general point on the radial element has velocity:
ˆrv
r
Faraday Homopolar GeneratorIn steady state the magnetic force on the free charges in the disk will equal the
electric force induced by charge migration. We can then equate electric and magnetic forces to solve for the electric field in the rotating disk.
ˆrv me FF
qvBqE
)ˆ( rrBE
aa
wrBdrdEV00
Faraday Homopolar Generator
B
a
V
The circular metal disk rotates at (rad/s) in a uniform flux density B. Sliding contacts connect a voltmeter to the disk. What voltage is indicated on the meter.
aa
wrBdrdEV00
2
2BaV
Super-Matrix Formulation
H
E
H
E
ro
ro
mm
01
10
tje
E
cE
r
r
m
01
10
HjZo
'FFBeeeR
rJC
Eo
qpn
EzjkzjGjqpoqpn
qpn
E zn
zjGjqpoqpn
qpno
o
r
r neeR
rJC
m
m
m
1
1
Solve for allowed Frequencies
Super-Matrix Formulation
E
cE
r
r
m
01
10
Wav
elen
gth
Tabulation Index
Band Gap
Resonator States
Super-Matrix FormulationW
avel
engt
h
Tabulation Index
Band Gap
Resonator States
Axial magnetic field