ECE 3318
Applied Electricity and Magnetism
Spring 2016
Prof. David R. Jackson
ECE Dept.
Notes 18
1
Example (cont.)
Find curl of E from a static point charge
z
 q 
E  rˆ 
2 
4


r
0


q
y
x
1
 E  rˆ
r sin 
   E sin   E 
1  1 Er   rE   ˆ 1    rE  Er 





  
  


 
r  sin  
r 
r  r
 

0
2
Example (cont.)
Note: If the curl of the electric field is zero for the field from a point charge, then
by superposition it must be zero for the field from any charge density.
This gives us Faraday’s law:
 E  0
(in statics)
3
Faraday’s Law in Statics
(Integral Form)
C
n̂
Stokes's theorem:
 E  d r      E   nˆ dS  0
C
Here S is any surface
that is attached to C.
S
Hence
 E  dr  0
C
4
Faraday’s Law in Statics
(Differential Form)
We show here how the integral form also implies the differential form.
Assume
 E  dr  0
C
We then have:
xˆ  curl E  lim
1
s 0 S
x

Cx
1
yˆ  curl E  lim
s 0 S
y

Cy
1
s 0 S
z

zˆ  curl E  lim
Hence
Cz
E  dr  0
E  dr  0
E  dr  0
 E  0
5
Faraday’s Law in Statics (Summary)
 E  0
Stokes’s theorem
Differential (point) form of Faraday’s law
Definition of curl
 E dr  0
Integral form of Faraday’s law
C
6
Path Independence and Faraday’s Law
The integral form of Faraday’s law is equivalent to
path independence of the voltage drop calculation.
C2
Proof:
B
A
 E  d r  0 (in statics)
C
C
Also,
C1
 E dr   E dr  E dr
C
C1
C2
Hence,
 E dr  0
C

 E dr  E dr
C1
C2
7
Summary of Path Independence
Path independence for VAB
Equivalent
 E dr  0
 E  0
C
Equivalent properties of an electrostatic field
8
Summary of Electrostatics
Here is a summary of the important equations
related to the electric field in statics.
  D  v
Electric Gauss law
 E  0
Faraday’s law
D  0 E
Constitutive equation
9
Faraday’s Law: Dynamics
Experimental Law (dynamics):
B
 E  
t
This is the general Faraday’s law in dynamics.
Michael Faraday*
(from Wikipedia)
*Ernest Rutherford stated: "When we consider the magnitude and extent of his discoveries
and their influence on the progress of science and of industry, there is no honour too great
to pay to the memory of Faraday, one of the greatest scientific discoverers of all time".
10
Faraday’s Law: Dynamics (cont.)
B
 E  
t
zˆ     E   
y
Assume a Bz that increases with time:
ˆ z t 
B  zB
Bz
0
t
dBz
0
dt
The changing magnetic field produces an electric field.
Experiment
Magnetic field Bz (increasing with time)
x
Electric field E
11
Faraday’s Law: Integral Form
B
 E  
t
Integrate both sides over an arbitrary open surface (bowl) S:
B 

S    E   nˆ dS  S   t   nˆ dS
Apply Stokes’ theorem for the LHS:
 B  ˆ
C E  d r  S   t   n dS
Note:
The right-hand rule
determines the direction of
the unit normal, from the
direction along C.
Faraday's law in integral form
12
Faraday’s Law (Experimental Setup)
We measure a voltage across a loop due to
a changing magnetic field inside the loop.
y
+
v(t) > 0
Open-circuited loop
x
Magnetic field B (Bz is increasing with time)
13
Faraday’s Law (Experimental Setup)
B
v  vAB   E  d r 
A
 E dr
y
+
A
C
v(t) > 0
-
B
Also
C
 B  ˆ
C E  d r   S  t   n dS
Bz

dS
t
S
x
S
(nˆ   zˆ )
So we have
Bz
v
dS
t
S
Note:
The voltage drop along the
PEC wire is zero.
14
Faraday’s Law (Experimental Setup)
Bz
v
dS
t
S
Assume
a uniform magnetic field for simplicity
Assume
(at least uniform over the loop area).
Bz
v
dS

t S
y
A
+
v(t) > 0
-
B
so
C
Bz
vA
t
x
A = area of loop
15
Faraday’s Law (Flux Form)
Bz
v
dS
t
S
Assume
Assume a stationary loop (not changing with time).
d
v   Bz dS
dt S
Define
   Bz dS
y
+
v(t) > 0
-
x
S
Then
d
v
dt
 = magnetic flux through loop in z direction
16
Faraday’s Law
Summary
d
v
Assume
dt
Bz
vA
t
General form
y
+
v(t)
-
Uniform field
x
   Bz dS
 = magnetic flux through loop in z direction
S
17
Lenz’s Law
This is a simple rule to tell us the polarity of the output voltage
(without having to do any calculation).
+
A
y
The voltage polarity is such that it
Assume
would set up a current flow that would
oppose the change in flux in the loop.
R
v(t) > 0
-
B
I
x
Note:
A right-hand rule tells us the direction of
the magnetic field due to a wire carrying
a current.
(A wire carrying a current in the z
direction produces a magnetic field in
the positive  direction.)
A = area of loop
Bz is increasing with time.
Bz
vA
0
t
18
Example: Magnetic Field Probe
A small loop can be used to measure the magnetic field (for AC).
+ v(t)
-
y
Bz
vA
t
Assume
C
Assume
x
Bz  B0 cos t   
  2 f
A = area of loop
Then we have
v   A B0 sin t   
a  radius  0
A  a
c/ f
2
At a given frequency, the output voltage is proportional to the strength of the magnetic field.
19
Applications of Faraday’s Law
Faraday’s law explains:
 How electric generators work
 How transformers work
Output voltage of generator
d
v
dt
Output voltage on secondary of transformer
Note: For N turns in a loop we have
d
vN
dt
20
The world's first electric generator!
(invented by Michael Faraday)
A magnet is slid in and out of the coil, resulting in a voltage output.
(Faraday Museum, London)
21
The world's first transformer!
(invented by Michael Faraday)
The primary and secondary coils are wound together on an iron core.
(Faraday Museum, London)
22
AC Generators
Diagram of a simple alternator (AC generator) with a rotating magnetic core
(rotor) and stationary wire (stator), also showing the current induced in the
stator by the rotating magnetic field of the rotor.
Bz  B0 sin t   
+
  angular velocity of magnet
-
so
dBz
v  t   NA
dt
v  t    NAB0  cos t   
or
z
A = area of loop
v  t   V0 cos t   
If the magnetic field from the magnet is constant but
the magnetic rotates at a fixed speed,
a sinusoidal voltage output is produced.
http://en.wikipedia.org/wiki/Alternator
23
AC Generators (cont.)
Generators at Hoover Dam
24
AC Generators (cont.)
Generators at Hoover Dam
25
Transformers
A transformer changes an AC signal from one voltage to another.
http://en.wikipedia.org/wiki/Transformer
26
Transformers
 High voltages are used for transmitting power over long distances
(less current means less conductor loss).
 Low voltages are used inside homes for convenience and safety.
http://en.wikipedia.org/wiki/Electric_power_transmission
27
Transformers (cont.)

v p t   N p
d
dt


vs  t   N s
d
dt
Ideal transformer
Hence
vs  t  N s

vp t  N p
Vs N s

Vp N p
(time domain)
(phasor domain)
http://en.wikipedia.org/wiki/Transformer
28
Transformers (cont.)
Ideal transformer (no losses):
v p  t  i p t   vs t  is t 
Hence
is  t  v p  t   vs  t  




i p  t  vs  t   v p  t  
1
so
is  t   N s 



ip t   N p 
(time domain)
1
Is  Ns 



Ip  Np 
1
(phasor domain)
29
Transformers (cont.)
Impedance transformation (phasor domain):
Z in 
Hence
Z in
Z out
Vp
Ip
Z out
Vs

Is
1
1
 Ns 
 Ns 
Ip 



2
N
 V p  I s  V p   N p 
 Np 
p 

   



 I V  I 
N




Ns
 s
 p  s  p  V Ns




p

N 
 Np 
 p
so
Z in  N p 


Z out  N s 
2
30
Transformers (cont.)
Impedance transformation
2
 Np 
Z in  
 ZL
 Ns 
N :1
Z in
ZL
31
Maxwell’s Equations
(Differential Form)
  D  v
B
 E  
t
B  0
D
 H  J 
t
D  0E
B  0 H
Electric Gauss law
Faraday’s law
Magnetic Gauss law
Ampere’s law
Constitutive equations
32
Maxwell’s Equations
(Integral Form)
 D  nˆ dS  Qencl
Electric Gauss law
B
C E  dr  S  t  nˆ dS
Faraday’s law
 B  nˆ dS  0
Magnetic Gauss law
D
C H  dr  iS  S t  nˆ dS
Ampere’s law
S
S
iS   J  nˆ dS (current through S )
S
33
Maxwell’s Equations (Statics)
In statics, Maxwell's equations decouple into two independent sets.
  D  v
B
 E  
t
B  0
D
 H  J 
t
  D  v
 E  0
B  0
 H  J
Electrostatics
Magnetostatics
v  E
J B
34
Maxwell’s Equations (Dynamics)
In dynamics, the electric and magnetic fields are coupled together.
Each one, changing with time, produces the other one.
Example:
A plane wave propagating through free space
B
E  
t
D
H 
t
E
J 0
z
H
power flow
E  xˆ cos t  kz 
From ECE 3317:
1
H  yˆ   cos t  kz 
 0 
k   0 0
 0  0 /  0
35