Fields and Waves I

1
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

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 Ulabys textbook.
3
Overview

• General considerations on magnetic fields
• Magnetostatics Electrostatics
• Similarities
• Differences
• Amperes Law
• Magnetic Vector Potential

4
The Earths Magnetic Field
5
Navy applications

• Compute the magnetic signature of the ship
• Reduce magnetic risk in real time

Closed loop degaussing
Field modulus under the ship
http//www.lmn.ensieg.inpg.fr/index/ind_bref.html
6
Experimental means
http//www.lmn.ensieg.inpg.fr/index/ind_bref.html
7
Maxwells Equations
Magnetostatics

• Integral Form

• Differential Form

0
0
8
Introducing B and H fields

• Magnetostatic form of Maxwells equations

• Calculate B and H fields from I and J

Integral form
Amperes Law
with m0 as a constant
In air,
9
Maxwells equations
Magnetostatics
Electrostatics
have curl (rotation) but no divergence (flux)
do not have curl (rotation) but have divergence
(flux)
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E-Fields
points away from q and towards -q
Direction of
multiple charges - use superposition
http//www.slcc.edu/schools/hum_sci/physics/tutor/
2220/e_fields/
11
B-Fields
wraps around

Direction of
http//encarta.msn.com/media_701504656_761566543_-
1_1/Right-Hand_Rule.html
multiple wires or segments - use superposition
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Example 1
Electrostatic or Magnetostatic?
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Example 1
Electrostatic or Magnetostatic?
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Example 1
Electrostatic or Magnetostatic?
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Example 1
Electrostatic or Magnetostatic?
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Intro to Magnetic Fields

• Some first applications

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Standard Geometries
Torus
Solenoid
http//www.directindustry.fr/prod/lcr-electronics/
assemblage-de-cables-electriques-pour-applications
-telecom-donnees-35095-214564.htmlprod_214564
http//www.magasia.com.tw/inductor.html
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Standard Geometries
http//cbdd.wsu.edu/kewlcontent/cdoutput/tr501/pag
e15.htm
http//upload.wikimedia.org/wikipedia/commons/thum
b/a/a1/Electronic_component_inductors.jpg/676px-El
ectronic_component_inductors.jpg
http//optical-components.globalspec.com/Industria
l-Directory/wave_frontier_toroidal
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Standard Geometries
http//hyperphysics.phy-astr.gsu.edu/hbase/magneti
c/solenoid.html
http//www.irf.com/technical-info/guide/circuit.ht
ml
http//www.amethyst-designs.co.uk/Product_Range/To
roidal_transformers.php
http//www.cse.iitk.ac.in/users/dheeraj/cs425/lec0
1.html
http//detail.en.china.cn/provide/detail,106593128
0.html
http//library.thinkquest.org/16600/advanced/amper
e.shtml
20
References for Inductors
http//hyperphysics.phy-astr.gsu.edu/hbase/magneti
c/imgmag/
http//library.thinkquest.org/16600/advanced/amper
e.shtml
http//neo.lcc.uma.es/cEA-web/population.htm
http//www.hut.fi/then/mytexts/radiohairiot.html
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Hand Wound Solenoid for Paperclip Launcher
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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
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Intro to Magnetic Fields

• Calculating the B and H fields with Amperes law

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Amperes law
Amperes law
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Example 2

• Three standard geometries for analytical
  magnetostatic calculations are shown on the next
  slide.
• Use the right hand rule (thumb along the current
  direction, fingers for B) and determine the
  direction of B in each case.
• All 3 geometries can best be analyzed in
  cylindrical coordinates. For each, determine
  whether B is a function of r, f, 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

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Example 2
Add up B-field for different segments - see what
cancels and what adds up - use symmetry
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Example 2 Case of the solenoid
(c) Infinite tightly wound solenoid
(b) Tightly wound solenoid
(a) Loosely wound solenoid
Ulaby
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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.
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Amperes Law - solve for B H
Approach similar to using Gauss Law, use
symmetry to get B-field out of integral
or
Example Consider an infinite wire solenoid
sectional view
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Amperes Law - solve for B H
Find
Solenoid has current I through n turns/length
STEP 1 Choose path for integral -

• Chosen paths are 1,2, 3 and 4 - they form a
  closed loop

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Amperes Law - Infinite Solenoid
STEP 2 Evaluate

• Segments 2 and 4 have

(will show later)

• Segment 3 has

arbitrary length

• Segment 1 has

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Amperes Law - Infinite Solenoid
STEP 3 find Inet

• current passing through loop

STEP 4 solve for
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Example 3 Amperes Law
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Example 3 Amperes Law
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Example 3 Amperes Law
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Example 3 Amperes Law
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Using Amperes Law

• Just like with Gauss Law, a great deal of
  symmetry is necessary to use Amperes 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.

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Intro to Magnetic Fields

• Magnetic vector potential

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Magnetic Vector Potential,
In electrostatics
In magnetostatics

• there is a math theorem that states

Note
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Example 4 Magnetic Vector Potential
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Example 4 Magnetic Vector Potential
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Example 4 Magnetic Vector Potential
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Flux and Vector Potential,
Previously we used
Now we will look at the effect of
Recall,
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Physical meaning
The flux is conservative
flux coming in flux going out
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Physical meanings of conservative flux
Flux lines of circular coil
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Concept of Flux Tubes (lines)
Field lines
Incoming field
outgoing field
Along sides
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Example of flux tubes (lines) cross section
The same flux passes through both surfaces since
they are in the same flux tube
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Flux and Vector Potential
After some math.
Alternative way to find FLUX
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Example 5 Magnetic Vector Potential and flux
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Example 5 Magnetic Vector Potential and flux