Transcript Fields and Waves I Lecture - Rensselaer Polytechnic Institute

Fields and Waves I
Lecture 16
Faraday’s Law
K. A. Connor
Electrical, Computer, and Systems Engineering Department
Rensselaer Polytechnic Institute, Troy, NY
Y. Maréchal
Power Engineering Department
Institut National Polytechnique de Grenoble, France
These Slides Were Prepared by Prof. Kenneth A. Connor Using
Original Materials Written Mostly by the Following:
 Kenneth A. Connor – ECSE Department, Rensselaer Polytechnic
Institute, Troy, NY
 J. Darryl Michael – GE Global Research Center, Niskayuna, NY
 Thomas P. Crowley – National Institute of Standards and
Technology, Boulder, CO
 Sheppard J. Salon – ECSE Department, Rensselaer Polytechnic
Institute, Troy, NY
 Lale Ergene – ITU Informatics Institute, Istanbul, Turkey
 Jeffrey Braunstein – Chung-Ang University, Seoul, Korea
Materials from other sources are referenced where they are used.
Those listed as Ulaby are figures from Ulaby’s textbook.
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Overview
 Review
• Ampere’s Law
• Magnetic Flux
• Magnetic Vector Potential
 Faraday’s Law
• EMF
• Induced Voltage/Current
• Moving Magnet or Loop
 Inductance
• Self Inductance
• Mutual Inductance
 Quiz review
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Ampere’s Law
Maxwell’s Equations:
 
 H  j

B  0
Ampere’s Law
 
 
 H  dl   j  ds  I net
Integral form
 
 B  ds  0
Magnetostatics


B  0  H
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Electrostatics
 
 H  j

 E  0

B  0

 D  
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B-Fields
Direction of

B

B
wraps around
I
&

j

j
Use right-hand rule
• thumb along I &
• fingers are in

B
http://encarta.msn.com/media_701504656_7615
66543_-1_1/Right-Hand_Rule.html
multiple wires or segments - use superposition
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Magnetic Flux & Magnetic Vector Potential
Magnetic Vector potential definition:
 
 A  B
Flux definition:
 
 
 
   B  ds     A  ds   A  dl
Alternative
way to find
FLUX
Magnetic
FLUX
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Example – Field Due To Several Wires
What is the direction
of B and A at the 4
indicated points?
z-direction is
into the page
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Field Due To Long Straight Wire
First, determine the magnetic field due to a long straight wire
carrying a current I. (See Example 5-5 of Ulaby)
Line for Ampere’s Law
I
 
 H  dl  I enclosed
B 2r
o
I
o I
B 
2r
http://www.ee.surrey.ac.uk/Workshop/advice/coils/terms.html
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Long Straight Wire
The magnetic vector potential can be determined from first
principles or from the magnetic field. We will do the latter.
o I
Az
B 

2r
r
o I
Az  
lnr   const
2
From the curl expression
Specifying the
zero reference
will determine
this constant
Note that the vector potential is
always in the direction of the current
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Magnetic Vector Potential Direction
All currents are in the z-direction and,
thus, the vector potential will also be
in the z-direction.
Its sign is arbitrary, since we are free
to select the reference potential point
anywhere. That is, we could chose all
potentials to be positive or all to be
negative.
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1
Magnetic Field Direction
2
At point 1 (point 3 has to opposite sign):

 2I
 I
BCENTER   o a y   o a y
2d
d

BLEFT 

BRIGHT 
4
3
o I
1
 1
 o I
a x 
a y  
 a x  a y

2
2 
2
2  4d
2 d  d

o I
1
 1
 o I


a

a
a x  a y

x
y 
2
2 
2
2  4d
2 d  d




o I
BTOTAL  
a y
2d
 
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1
Magnetic Field Direction
2
At point 2:

 I
BRIGHT  o a y
2d

o I
BLEFT 
2 4d 2  d 2
 

BCENTER 
4
3
o I
1 
 2
a x 
a y  
2a x  a y


5
5  10d

o 2 I
1
 1
 o I


a

a
a x  a y

x
y 
2
2 
2
2  2d
2 d  d




BTOTAL  ?
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Faraday’s Law and dynamic fields
Faraday’s Law
Faraday’s Law
In electrostatics, we used:
 
 E  dl  0
Faraday’s Law comes from Maxwell’s equation:


B
 E  
t
Applications:
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 
d  
 E  d l    B  ds
dt
• inductors
• transformers
• motors
• generators
• noise
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Faraday’s Law - concept of EMF

ds
Time varying flux through a coil

dl
Vemf
 
d
d  
  E  dl       B  ds is the electromotive force
dt
dt
The emf is similar to a VOLTAGE

ds
Orientation issues :
 
d  
 E  d l    B  ds
dt
Use right hand rule for
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
ds
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
and dl

dl
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Faraday’s Law – various types of EMF
What does the flux derivative means ?
Vemf






d
d
B 
ds 
      B  ds   
 ds   B  
dt
dt
dt 
 t
The emf may come from:
•A dynamic field and a stationary loop
•A moving loop in a static field
•Both moving loop and dynamic field
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Faraday’s Law : dynamic field experiment
dl
50 W
10 cos(wt) V
1 MHz
~
B
I
1 H
solenoid
function generator
I
I
I
coil
I
1
2
Assume that we hook up the experiment as shown where the 1
micro Henry inductor is connected across the output of the
function generator and monitor the output of the generator using
one scope channel.
Then, place a coil facing the inductor and connect it to the other
scope channel. An induced voltage is observed at the ends of the
pickup coil (in phase with the generator)
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Transformers : Faraday’s law with dynamic fields
http://www.transformerfactory.com/e1-model-small-power-transformer-1va-70a.html
 A huge range in sizes
http://www.meppi.com/Products/Transformers/Power/Pages/Core-formTransformers.aspx
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http://en.ferilex.eu/transformers.html
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Faraday’s Law: moving loop experiment
A loop falls through the magnetic field between two pole faces at a
constant velocity, u0.
side view of loop
loop
u
B
B
u
B
B
magnets
SIDE VIEW
u
magnet face
FRONT VIEW
t = t1
FRONT VIEW
t = t2
FRONT VIEW
t = t3
A current is flowing in the loop as it pass through the magnets
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Generators : Faraday’s Law
http://www.wenzelontheweb.de/Hoover%20Dam.htm
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Hoover Dam
http://isu.indstate.edu/jspeer/conservation/
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Faraday’s Law and dynamic fields
Dynamic fields
Example 1
dl
50 W
B
10 cos(wt) V
1 MHz
I
1 H
~
I
solenoid
function generator
For the solenoid,

B  0nIaˆz
I
I
coil
I
1
2
inside and 0 outside
• n is the number of turns per unit length
• a is the radius of the solenoid and the coil
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Example 1
a.
Circuit analysis. What are the current I and voltage V through the
inductor?
b.
What is the flux,  = B  ds, through the loop? Do this analytically
and then obtain a numerical value for n = 1560 and solenoid radius a =
2.5 mm. Pay attention to the signs/direction of dl and ds.
c.
What is the emf induced around the loop? Again do an analytical
calculation, but then plug in the numbers from above.
1) At t=0+, does a scope read V1 - V2 > or < 0?
2) If the clip leads were connected through a low impedance, which
way would current flow at t=0+?
d.
Sketch emf and  vs time. What is the flux when the emf is largest?
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Example 1
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Example 1
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Faraday’s Law and Lenz’s law
Previous result : t=0+, flux
decreasing, I as shown
Lenz’s law : “The current in the
loop is always in such a direction
as to oppose the change of
magnetic flux that produced it.”
B
I
I
I
I
I
l
1
2
Low impedance Output
Ulaby
High impedance Output
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Faraday’s Law and dynamic fields
Moving conductors
Faraday’s Law for moving loop

 

 ds 
 B 
d

 ds   B  
 E  dl    B  ds   
dt
dt 
 t
0

B
0
Previous example had:
t

B
In moving loop example:
 0 ,but
t
Vemf
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
ds changes with time

 ds (t ) is a function of t

 

d
ds
  E  dl       B 
dt
dt
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Faraday’s Law for moving loop
Is
Is

B
Is

ds
sliding
bar
B-field
into
page

B

u

u

dl
Is
At time t1
Is
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At time t2 = t1 +
Dt
sliding
bar
Is
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Dz
h
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Faraday’s Law for moving loop
Is
Approximate flux derivative as:



t
 
D  B  ds
Dt
Dz
h
Dt
Dz
 B  h 
 B  h  u
Dt

u
A general form:

  
 ds
 
   B  (u  dl )   (u  B )  dl
B
dt


 


Alternate
d
B 


 E  dl    B  ds   u  B   dl    ds
expression
dt
t
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Example 2
A loop falls through the magnetic field between two pole faces at a constant
velocity, u0. Assume that the magnetic field is B0 between the pole faces and
that the fringe fields are 0.
side view of loop
loop
u
B
B
u
B
B
magnets
SIDE VIEW
u
magnet face
FRONT VIEW
t = t1
FRONT VIEW
t = t2
FRONT VIEW
t = t3
Plot the flux through the loop,  =B  ds, as a function of time.
Calculate the emf around the loop for all times by derivation of the flux.
Calculate the emf around the loop for all times indicated using the uxB.
If the loop is connected across a low impedance output, will the current
be in the clockwise direction, 0, or in the counter-clockwise direction?
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Example 2
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Example 2
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Faraday’s Law and dynamic fields
Inductances
Inductances
Two types of Inductances:
• self inductance - e.g. inductors
• mutual inductance - e.g. transformers
Self Inductance:
• coil of wire with
I1
• wire loop intersects
, creates
 
 B  ds

B
d  
• this creates e.m. f .    B  ds
dt
http://www.gaussbusters.com/ppm93.html
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Inductor
Geometric parameters for a
solenoidal inductor
http://www3.telus.net/chemelec/Calculators/Helical-Coil-Calc.htm
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Example 3 : Solenoid Inductance
Consider a solenoid with N turns, length l , and radius a .
Assume the current is sinusoidal with a frequency f and ignore
fringing effects.
a. What emf,  E  dl is induced around the solenoid (include all
turns)?
b. The "voltage" across an inductor is the emf (with care taken
about signs). Find the solenoid inductance by substituting the
absolute value of the emf in part b. for the voltage in V = L
dI/dt.
c. What is the flux linkage through all N turns?
d. Calculate L = Flux/I and compare with your answer to part c.
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Example 3
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Self inductance
Two ways to calculate the inductance:
•
Calculate the emf then use = L dI/dt.
or
•
Calculate the total flux linkage and use L = Total Flux / I
Things to remember :
The flux linkage,
  N 
• only if all loops intersect same flux
• not true for finite solenoid and will need:
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  
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Self inductance
Note: To calculate L, don’t need Faraday’s Law just need: L 
LN
2
• x N, because
, because

B  I  ,   I

BN

I
• x N, because     N 

is independent of I
thus L 
I
L depends on materials (through ) and geometry (like C)
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Mutual inductance

B1

B1
I1
Coil 1
Mutual Inductance:
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Coil 2
Current through Coil 1
induces e.m.f. in Coil 2
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Mutual inductance
L21 
21
I1
where,


21   B1  d s2
Mutual Inductance
dI 1
Also, emf 2  L21 
dt
And,
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
B1

B1
I1
Coil 1
Coil 2
L12  L21
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