fields and waves i

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.


Overview

General considerations on magnetic fields

Magnetostatics & Electrostatics• Similarities• Differences

Ampere’s Law Magnetic Vector Potential


The Earth’s Magnetic Field


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



Experimental meanshttp://www.lmn.ensieg.inpg.fr/index/ind_bref.html



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


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


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


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




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




Example 1 Electrostatic or Magnetostatic?

Intro to Magnetic Fields

Some first applications


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


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


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



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

http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/imgmag/


http://neo.lcc.uma.es/cEA-web/population.htm

http://www.hut.fi/~then/mytexts/radiohairiot.html


Hand Wound Solenoid for Paperclip Launcher


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





Calculating the B and H fields with Ampere’s law


Ampere’s law

netIsdjldH

jH

Ampere’s law

HB 0


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


Example 2

Add up B-field for different segments - see what cancels and what adds up - use symmetry


Example 2: Case of the solenoid

(c) Infinite tightly wound solenoid

(b) Tightly wound solenoid

(a) Looselywound solenoid

Ulaby


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.


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


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


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


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


Example 3 – Ampere’s Law


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.


Magnetic vector potential


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,


Example 4 – Magnetic Vector Potential


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


Physical meaning

0... upleft right

sdBsdBsdBsdB

The flux is conservative:flux coming in = flux going out


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


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


Example of flux tubes (lines) cross section

The same flux passes through both surfaces since they are in the same flux tube


A

is related to Flux

ldAsdAsdB

After some math….

Alternative way to find

FLUX

Flux and Vector Potential sdB

ldA


Example 5 – Magnetic Vector Potential and flux

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waves iintroducing b

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