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Fields and Waves I. Lecture 15 Intro to Magnetic Fields 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. - PowerPoint PPT Presentation
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Fields and Waves I
Lecture 15Intro to Magnetic Fields
K. A. ConnorElectrical, Computer, and Systems Engineering Department
Rensselaer Polytechnic Institute, Troy, NYY. Maréchal
Power Engineering DepartmentInstitut National Polytechnique de Grenoble, France
April 22, 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 22, 2023 Fields and Waves I 3
Overview
General considerations on magnetic fields
Magnetostatics & Electrostatics• Similarities• Differences
Ampere’s Law Magnetic Vector Potential
April 22, 2023 Fields and Waves I 4
The Earth’s Magnetic Field
April 22, 2023 Fields and Waves I 5
Navy applications
Compute the magnetic signature of the ship
Reduce magnetic risk in real time
Field modulus under the ship
http://www.lmn.ensieg.inpg.fr/index/ind_bref.html
Closed loop degaussing
April 22, 2023 Fields and Waves I 6
Experimental meanshttp://www.lmn.ensieg.inpg.fr/index/ind_bref.html
April 22, 2023 Fields and Waves I 7
Maxwell’s Equations
Integral Form Differential Form B dS 0
D dS dv
H dl J dS d
dtD dS
E dl d
dtB dS
B 0
D
E B
t
H J D
t
0 0
Magnetostatics
April 22, 2023 Fields and Waves I 8
Introducing B and H fields
In air, HB 0
with 0 as a constant
netIsdjldH jH
Ampere’s Law
0B
0sdB
Integral form
• Magnetostatic form of Maxwell’s equations
• Calculate B and H fields from I and J
April 22, 2023 Fields and Waves I 9
jH
0B
Magnetostatics Electrostatics
0E
D
HB,
ED,
have curl (rotation) but no divergence (flux)
do not have curl (rotation) but have divergence (flux)
Maxwell’s equations
April 22, 2023 Fields and Waves I 10
Direction of E
E points away from q
and towards -q
multiple charges - use superposition
http://www.slcc.edu/schools/hum_sci/physics/tutor/2220/e_fields/
E-Fields
April 22, 2023 Fields and Waves I 11
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 22, 2023 Fields and Waves I 12
Example 1 Electrostatic or Magnetostatic?
April 22, 2023 Fields and Waves I 13
Example 1 Electrostatic or Magnetostatic?
April 22, 2023 Fields and Waves I 14
Example 1 Electrostatic or Magnetostatic?
April 22, 2023 Fields and Waves I 15
Example 1 Electrostatic or Magnetostatic?
Intro to Magnetic Fields
Some first applications
April 22, 2023 Fields and Waves I 17
Standard GeometriesTorus
Solenoid http://www.directindustry.fr/prod/lcr-electronics/
assemblage-de-cables-electriques-pour-applications-telecom-donnees-35095-214564.html#prod_214564
http://www.magasia.com.tw/inductor.html
April 22, 2023 Fields and Waves I 18
Standard Geometries
http://optical-components.globalspec.com/Industrial-Directory/wave_frontier_toroidal http://upload.wikimedia.org/wikipedia/commons/thumb/a/a1/
Electronic_component_inductors.jpg/676px-Electronic_component_inductors.jpg
http://cbdd.wsu.edu/kewlcontent/cdoutput/tr501/
page15.htm
April 22, 2023 Fields and Waves I 19
Standard Geometries
http://library.thinkquest.org/16600/advanced/ampere.shtml
http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/solenoid.html
http://www.cse.iitk.a
c.in/users/dheeraj/cs425/
lec01.html
http://www.irf.com/technical-info/guide/circuit.html http://www.amethyst-designs.co.uk/Product_Range/
Toroidal_transformers.php
http://detail.en.china.cn/provide/detail,1065931280.html
April 22, 2023 Fields and Waves I 20
References for Inductors
http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/imgmag/ http://library.thinkquest.org/16600/advanced/ampere.shtml http://neo.lcc.uma.es/cEA-web/population.htm http://www.hut.fi/~then/mytexts/radiohairiot.html
April 22, 2023 Fields and Waves I 21
Hand Wound Solenoid for Paperclip Launcher
April 22, 2023 Fields and Waves I 22
Not So Standard Torus
International Tokamak Experimental Reactor Planned for nuclear fusion research
Note standard person
http://www.plasma.inpe.br/LAP_Portal/LAP_Site/Text/Tokamak_Development.htm
Intro to Magnetic Fields
Calculating the B and H fields with Ampere’s law
April 22, 2023 Fields and Waves I 24
Ampere’s law
netIsdjldH
jH
Ampere’s law
HB 0
April 22, 2023 Fields and Waves I 25
Example 2
Three standard geometries for analytical magnetostatic calculations are shown on the next slide.
a. Use the right hand rule (thumb along the current direction, fingers for B) and determine the direction of B in each case.
b. All 3 geometries can best be analyzed in cylindrical coordinates. For each, determine whether B is a function of r, φ, and/or z. (Example from electric fields, E of cylindrically symmetric charge is only a function of r.)
Add up B-field for different segments - see what cancels and what adds up - use symmetry
April 22, 2023 Fields and Waves I 26
Example 2
Add up B-field for different segments - see what cancels and what adds up - use symmetry
April 22, 2023 Fields and Waves I 27
Example 2: Case of the solenoid
(c) Infinite tightly wound solenoid
(b) Tightly wound solenoid
(a) Looselywound solenoid
Ulaby
April 22, 2023 Fields and Waves I 28
Example 2
Recall that for the E field, source distributions that only depended on cylindrical radius (r), produced E fields that only depended on r and only had an r component. For the B field, source distributions that only depend on cylindrical r also produce B fields that only depend on r, but B has components in directions perpendicular to r.
April 22, 2023 Fields and Waves I 29
netIsdjldH
or
net00 IsdjldB
Approach similar to using Gauss’ Law, use symmetry to get B-field out of integral
Example: Consider an infinite wire solenoid
HB, sectional view
Ampere’s Law - solve for B & H
April 22, 2023 Fields and Waves I 30
Solenoid has current I through n turns/length
Find B
STEP 1: Choose path for integral - ldB
B
1
23
4
• Chosen paths are 1,2, 3 and 4 - they form a closed loop
Ampere’s Law - solve for B & H
April 22, 2023 Fields and Waves I 31
STEP 2: Evaluate ldB
• Segments 2 and 4 have ldB
0ldB
• Segment 3 has 0B
(will show later)
• Segment 1 has l
0zz lBdzBldB
arbitrary length
Ampere’s Law - Infinite Solenoid
April 22, 2023 Fields and Waves I 32
STEP 3: find Inet
• current passing through loop :
IlnI net
STEP 4: solve for B
net00 IsdjldB
IlnlB 0z z0 aInB ˆ
Ampere’s Law - Infinite Solenoid
April 22, 2023 Fields and Waves I 33
Example 3 – Ampere’s Law
April 22, 2023 Fields and Waves I 34
Example 3 – Ampere’s Law
April 22, 2023 Fields and Waves I 35
Example 3 – Ampere’s Law
April 22, 2023 Fields and Waves I 36
Example 3 – Ampere’s Law
April 22, 2023 Fields and Waves I 37
Using Ampere’s Law
Just like with Gauss’ Law, a great deal of symmetry is necessary to use Ampere’s Law to find B or H.
Simplify everything before attempting a solution. Applicability is limited, but this technique is still
very useful. There is an analog to using the electric potential,
although for B, it is a bit more complex since it involves a vector potential instead of a scalar potential. It is still easier since the vector potential is in the direction of the current.
Intro to Magnetic Fields
Magnetic vector potential
April 22, 2023 Fields and Waves I 39
A
In electrostatics: VE0E
In magnetostatics: 0B
• there is a math theorem that states :
0F
, for any F
• this means that you can find A
, with AB
Note: • more than one A
possible
• like A
V , can be easier to work with than the field
Magnetic Vector Potential,
April 22, 2023 Fields and Waves I 40
Example 4 – Magnetic Vector Potential
April 22, 2023 Fields and Waves I 41
Example 4 – Magnetic Vector Potential
April 22, 2023 Fields and Waves I 42
Example 4 – Magnetic Vector Potential
April 22, 2023 Fields and Waves I 43
Previously we used: netIsdjldH
Now we will look at the effect of
0sdB0B
This surface integral encloses the volume
0sdBdvB0B Recall,
Flux and Vector Potential,A
April 22, 2023 Fields and Waves I 44
Physical meaning
0... upleft right
sdBsdBsdBsdB
The flux is conservative:flux coming in = flux going out
April 22, 2023 Fields and Waves I 45
Physical meanings of conservative flux
• B
lines close on themselves (no beginning or end)• In contrast, static E
lines start & end on charges
Flux lines of circular coil
April 22, 2023 Fields and Waves I 46
BField
lines
Along sides: sdB
0sdB
left right
sdBsdBsdB
Define Flux, sdB , enters from left and leaves to the right
Concept of Flux Tubes (lines)
Incoming field
outgoing field
April 22, 2023 Fields and Waves I 47
Example of flux tubes (lines) cross section
The same flux passes through both surfaces since they are in the same flux tube
April 22, 2023 Fields and Waves I 48
A
is related to Flux
ldAsdAsdB
After some math….
Alternative way to find
FLUX
Flux and Vector Potential sdB
ldA
April 22, 2023 Fields and Waves I 49
Example 5 – Magnetic Vector Potential and flux
April 22, 2023 Fields and Waves I 50
Example 5 – Magnetic Vector Potential and flux