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Fields and Waves I
Lecture 16Faraday’s Law
K. A. ConnorElectrical, Computer, and Systems Engineering Department
Rensselaer Polytechnic Institute, Troy, NY
Y. MaréchalPower Engineering Department
Institut National Polytechnique de Grenoble, France
April 19, 2023 Fields and Waves I 2
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.
April 19, 2023 Fields and Waves I 3
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
April 19, 2023 Fields and Waves I 4
Ampere’s Law
Maxwell’s Equations:
jH
0B Integral form
netIsdjldH
0sdB
HB 0
Ampere’s Law
jH
0B
Magnetostatics Electrostatics
0E
D
April 19, 2023 Fields and Waves I 5
Direction of B
B
wraps around j
I &
Use right-hand rule• thumb along
• fingers are in
j
I &
B
multiple wires or segments - use superposition
http://encarta.msn.com/media_701504656_761566543_-1_1/Right-Hand_Rule.html
B-Fields
April 19, 2023 Fields and Waves I 6
Magnetic Flux & Magnetic Vector Potential
ldAsdAsdB
Alternative way to find
FLUX
BA
Magnetic FLUX
Magnetic Vector potential definition:
Flux definition:
April 19, 2023 Fields and Waves I 7
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
April 19, 2023 Fields and Waves I 8
Field Due To Long Straight Wire
H dl I enclosed
B rI
o
2
First, determine the magnetic field due to a long straight wire carrying a current I. (See Example 5-5 of Ulaby)
I
Line for Ampere’s Law
BI
ro
2
http://www.ee.surrey.ac.uk/Workshop/advice/coils/terms.html
April 19, 2023 Fields and Waves I 9
Long Straight Wire
BI
r
A
ro z
2
The magnetic vector potential can be determined from first principles or from the magnetic field. We will do the latter.
AI
r constzo 2ln
Specifying the zero reference will determine this constant
From the curl expression
Note that the vector potential is always in the direction of the current
April 19, 2023 Fields and Waves I 10
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.
April 19, 2023 Fields and Waves I 11
Magnetic Field Direction1
3
2
4
At point 1 (point 3 has to opposite sign):
B
I
da
I
daC E N TE R
oy
oy
2
2
B
I
d da a
I
da aLE F T
ox y
ox y
2
1
2
1
2 42 2
B
I
d da a
I
da aR IG H T
ox y
ox y
2
1
2
1
2 42 2
B
I
daTO TA L
oy
2
April 19, 2023 Fields and Waves I 12
Magnetic Field Direction1
3
2
4
At point 2:
B
I
d da a
I
da aC E N TE R
ox y
ox y
2
2
1
2
1
2 22 2
B
I
d da a
I
da aLE F T
ox y
ox y
2 4
2
5
1
5 102
2 2
B
I
daR IG H T
oy
2
B TO TA L ?
Faraday’s Law and dynamic fields
Faraday’s Law
April 19, 2023 Fields and Waves I 14
Faraday’s Law comes from Maxwell’s equation:
t
BE
sdB
dt
dldE
In electrostatics, we used: 0ldE
Applications:
• inductors• transformers• motors• generators• noise
Faraday’s Law
April 19, 2023 Fields and Waves I 15
Faraday’s Law - concept of EMF
sdBdt
d
dt
dldEVemf
is the electromotive force
Time varying flux through a coil
ld
sd
The emf is similar to a VOLTAGE
sdBdt
dldE
Use right hand rule for sd
and ld
Orientation issues :
ld
sd
April 19, 2023 Fields and Waves I 16
Faraday’s Law – various types of EMF
dt
sdBsd
t
BsdB
dt
d
dt
dVemf
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
What does the flux derivative means ?
April 19, 2023 Fields and Waves I 17
dl
12
coil
Faraday’s Law : dynamic field experiment
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)
1 H
function generator
~
50
10 cos(t) V1 MHz IIIII
B
solenoid
April 19, 2023 Fields and Waves I 18
Transformers : Faraday’s law with dynamic fields
A huge range in sizeshttp://www.meppi.com/Products/Transformers/Power/Pages/Core-formTransformers.aspx
http://www.transformerfactory.com/e1-model-small-power-transformer-1va-70a.html
http://en.ferilex.eu/transformers.html
April 19, 2023 Fields and Waves I 19
Faraday’s Law: moving loop experiment
A loop falls through the magnetic field between two pole faces at a constant velocity, u0.
B
side view of loop
magnets
SIDE VIEWFRONT VIEW
magnet face
loopu
B
t = t1FRONT VIEW
uB
t = t2
u
B
FRONT VIEWt = t3
A current is flowing in the loop as it pass through the magnets
April 19, 2023 Fields and Waves I 20
Generators : Faraday’s Law Hoover Dam
http://isu.indstate.edu/jspeer/conservation/ http://www.wenzelontheweb.de/Hoover%20Dam.htm
Faraday’s Law and dynamic fields
Dynamic fields
April 19, 2023 Fields and Waves I 22
dl
12
coil
Example 1
1 H
function generator
~
50
10 cos(t) V1 MHz IIIII
B
solenoid
For the solenoid, inside and 0 outside
• n is the number of turns per unit length• a is the radius of the solenoid and the coil
zanIB ˆ0
April 19, 2023 Fields and Waves I 23
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?
April 19, 2023 Fields and Waves I 24
Example 1
April 19, 2023 Fields and Waves I 25
Example 1
April 19, 2023 Fields and Waves I 26High impedance Output
Faraday’s Law and Lenz’s law
Low impedance Output
l
12
IIIIIB
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.”
Ulaby
Faraday’s Law and dynamic fields
Moving conductors
April 19, 2023 Fields and Waves I 28
Previous example had: 0t
B
In moving loop example: 0t
B
,but sd
changes with time
)(tsd
is a function of t
dt
sdBsd
t
BsdB
dt
dldE
0
Faraday’s Law for moving loop
dt
sdB
dt
dldEVemf
April 19, 2023 Fields and Waves I 29
sI B
B-field into page
sI
sI
sI
u
sliding bar
At time t2 = t1 + t
sI z h
sd
B
ld
u
At time t1
sI sliding bar
Faraday’s Law for moving loop
April 19, 2023 Fields and Waves I 30
ldBulduB
)()(
Approximate flux derivative as:
uhBt
zhB
t
sdB
t
u
sdt
BldBusdB
dt
dldE
Alternate expression
dt
sdB
A general form:
sI z h
Faraday’s Law for moving loop
t
April 19, 2023 Fields and Waves I 31
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.
B
side view of loop
magnets
SIDE VIEWFRONT VIEW
magnet face
loopu
B
t = t1FRONT VIEW
uB
t = t2
u
B
FRONT VIEWt = 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?
April 19, 2023 Fields and Waves I 32
Example 2
April 19, 2023 Fields and Waves I 33
Example 2
Faraday’s Law and dynamic fields
Inductances
April 19, 2023 Fields and Waves I 35
Two types of Inductances:• self inductance - e.g. inductors
• mutual inductance - e.g. transformers
Self Inductance:
• coil of wire with 1I , creates B
• wire loop intersects sdB
• this creates sdBdt
dfme
...
Inductances
http://www.gaussbusters.com/ppm93.html
April 19, 2023 Fields and Waves I 36
Inductor
Geometric parameters for a solenoidal inductor
http://www3.telus.net/chemelec/Calculators/Helical-Coil-Calc.htm
April 19, 2023 Fields and Waves I 37
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.
April 19, 2023 Fields and Waves I 38
Example 3
April 19, 2023 Fields and Waves I 39
Self inductance
Two ways to calculate the inductance:
• Calculate the emf then use = L dI/dt.
• Calculate the total flux linkage and use L = Total Flux / I
Things to remember :
or
The flux linkage, N• only if all loops intersect same flux• not true for finite solenoid and will need:
April 19, 2023 Fields and Waves I 40
Self inductance
IB
I ,I
L is independent of I
L depends on materials (through ) and geometry (like C)
2NL , because• x N, because NB
• x N, because N
Note: To calculate L, don’t need Faraday’s Law just need:I
L
thus
April 19, 2023 Fields and Waves I 41
Coil 1 Coil 2
1I
1B1B
Mutual Inductance: Current through Coil 1 induces e.m.f. in Coil 2
Mutual inductance
April 19, 2023 Fields and Waves I 42
1
2121 I
L
2121 sdB
Mutual Inductance
Also,dt
dILemf 1
212
And, 2112 LL
where,
Mutual inductance
Coil 1 Coil 2
1I
1B1B