2415 web lec 18 sourcesofmagneticfield ii

7/26/2019 2415 Web Lec 18 SourcesOfMagneticField II

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Sources of Magnetic Field II

Physics 2415 Lecture 18

Michael Fowler, Ua



!oday"s !o#ics

• More a$out solenoids

• %iot&Sa'art law

• Magnetic (aterials



)(#*re"s Law+ eneral -ase• )(#*re"s Law states that for

any (agnetic .eld generated $ya steady /ow of electricalcurrents, if we ta0e an ar$itraryclosed #ath in s#ace andintegrate around it, then

where now I is the total netcurrent /owing across anysurface ha'ing the #ath ofintegration as its $oundary,such as the $lue surface shown

here

• 0 B d I µ × =∫ l2



-o(#act Muon Solenoid-MS

• May$e the $iggestsolenoid e'er9thestrong (agnetic .eld4! is inside thecentral solenoid,inner dia(eter :(,length 1;( !he .eldis there to cur'e the

#aths of #articles#roduced, to (easuretheir charge and(ass !he unit

weighs 12,5<< tons



!he First Sign of )nti(atter

• -loud cha($er trac0 of a#article, 1=;2

• !he central line is a lead#late !here is a downward

#er#endicular % .eld• !he tighter cur'ature in

the to# half (eans aslower #article>so it ca(e

fro( $elow !he cur'aturethen tells us it"s a#ositi'ely charged #article>it"s a #ositron, anantielectron

-ould it $e a #roton? @o>for theo$ser'ed cur'ature, the #roton would

ha'e to $e 'ery slow, and would create(any water dro#lets in this cloud



!oroidal Solenoid

• !his is doughnut sha#ed+ ta0ean ordinary solenoid, $end itaround so the ends (eet, andthe line along the center of thesolenoid $eco(es a circle !hisgeo(etry can contain a #las(awithout needing end (irrors

• Ae show here a cross section,

li0e cutting a doughnut fortoasting+ the current is into thescreen at the outside, co(ingout on the inside

•



Field of !oroidal Solenoid

• Using )(#*re"s law to .nd the.eld inside the solenoid+

• Fro( sy((etry, the .eld#oints anticloc0wise as shown

• !he white dashed #ath is acircle of radius r !he graysurface co'ering it is cut $ye'ery turn of the wire, total

current NI • )(#*re"s law

gi'es i((ediately

•

0 encl B d I µ × =∫ l

0

2

NI B

r

µ

π =



-lic0er Buestion

• !he .eld outside atoroidal solenoid+

) Cecreases as 16r with distancefro( the center ofthe solenoid

% Cecreases as 16r 2

- Is Dero

•

B?



-lic0er )nswer

• !he .eld outside atoroidal solenoid+

) Cecreases as 16r with distancefro( the center ofthe solenoid

% Cecreases as 16r 2

- Is Dero

•

B?

) surface s#anning the dotted circle will ha'e Derototal current #enetrating>as (uch u# as down, sothe )(#ere integral is Dero



!oroids•



!he %iot&Sa'art Law•

!he electric .eld fro(any distri$ution ofcharges can $e found $yadding or integrating

ter(s fro( each$it of charge

• !he (agnetic analog isthe %iot&Sa'art law+ for

any collection of wirecurrents, add together.elds fro( current $its+

•

wea0er .eld

shown dar0e

0

2

ˆ

4

I d r dB

r

µ

π

×=

uurr l

2ˆ /kqr r



!he %iot&Sa'art Law•

Field fro( a .nite lengthof straight wire+

• !he .eld at the #oint inthe .gure is inward, andhas (agnitude

• Using R E & y tanθ we .nd

•

RdB

r

dy

y

0

θ

I

0

2

ˆ

4

I d r dB

r

µ

π

×=

uurr l

2

1

0

2

sin

4

y

y

I dy

B r

µ θ

π =

∫

( )0

1 2cos cos

4

I B

µ θ θ

π = −



-lic0er Buestion

• Su##ose we ha'e two current ele(ents+is at the origin, #ointing along the x&

a3is,

is at x E 1, y E <, z E < and #oints inthe z &direction

•

-all the force of 1 on 2

• Coes ?

) es

% @o

x

y

z

1 1 I d

rl

2 2 I d

rl

12 F r

12 21 F F = −

r r



-lic0er )nswer•

Su##ose we ha'e two currentele(ents+is at the origin, #ointing along

the x&a3is,

is at x E 1, y E <, z E < and#oints in the z &direction

-all the force of 1 on 2

• Coes ?) es

% @o

x

y

z

Ahat if the two current ele(ents are Gust charged#articles (o'ing through s#ace? Ahat a$out @ewton"s

!hird Law? It turns out that the total (o(entu( of thetwo #articles is not conserved + there is (o(entu(

1 1 I d rl

2 2 I d

rl

12 F r

12 21 F F = −

r r



Field on the )3is of a-urrent Loo#

• !he 'ectors andare always at rightangles, and adding the

contri$utions fro( theHs going around thecircle, only contri$utionsalong the a3is sur'i'e

also o$'ious fro(sy((etry !he su( ofthe dℓ "s is 2π R, so the.eld is

•

R

x

r

θ

I

dBr

d uurl r

d

uur

l

( )

2 2

0 0 0

3/ 22 32 2

12 sin

4 2 2

I I I R R B R

r r R x

µ µ µ π θ

π = = =

+



Ferro(agnetic Materials

• !he ato(s of Fe, -o and @i and rareearths are little (agnets+ in theinco(#letely .lled shell of electrons, the

electron s#ins line u#>and electrons arethe(sel'es (agnets

• !he ato(s also line u# (agnetically withtheir neigh$ors

• )ll this lining u# is fully e3#lained $yuantu( (echanics and cannot $ee3#lained otherwise

•

So why isn"t e'ery #iece of iron (agnetic?

•



Co(ains

• Jecall the electric .eld was astore of energy>we"ll see thisis also true of the (agnetic.eld !his (eans (aterials willtend to arrange the(sel'es to(ini(iDe the energy in the(agnetic .eld

•

For e3a(#le, a single crystal ofiron will rearrange its ato(orientation into do(ains+ the(agnetic .eld is then fars(aller

•

C i i 7 t l



Co(ains in 73ternalFields

In an e3ternal (agnetic .eld, the do(ain structurechanges+ do(ains #ointing with the a##lied .eldgrow !he (aterial $eco(es (agnetiDed

%ar0hausen eKect+ (o'e a(agnet ra#idly near (agnetic(aterial, and the (o'ing

do(ain walls (a0e a noise asthey (o'e #ast i(#urities, etcUseful for assessing (aterial

uality

http://en.wikipedia.org/wiki/Barkhausen_effect





ard and Soft MagneticMaterials

• Ae"'e seen how the do(ains can grow andshrin0 in res#onse to e3ternal .elds ow readilythis ha##ens for a #articular (aterial de#ends onhow easily the do(ain walls can (o'e !hey canget hung u# on lattice defects or i(#urities, etc

• ) soft (agnet iron has do(ain walls that (o'eeasily It readily $eco(es (agnetiDed in ane3ternal .eld, $ut can easily lose that(agnetiDation if su$Gect to 'i$ration

• ) #er(anent (agnet is less easy to #roduce, $ut0ee#s its (agnetiDation>do(ain walls don"t

(o'e easily



7lectro(agnets

• 7lectro(agnets are solenoidswith iron inside to (agnify the(agnetic .eld

• In the electro(agnetic door$ell,#ressing the $utton closes thecircuit, the (agnet #ulls the $arand s(all ha((er forward toring the $ell>and also to $rea0

the circuit, which #asses alongthe $ar to a contact at the to# !he cycle re#eats as long as the$utton is #ressed

•

2415 web lec 18 sourcesofmagneticfield ii

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