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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
7/26/2019 2415 Web Lec 18 SourcesOfMagneticField II
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!oday"s !o#ics
• More a$out solenoids
• %iot&Sa'art law
• Magnetic (aterials
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)(#*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 $lue surface shown
here
• 0 B d I µ × =∫ l2
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-o(#act Muon Solenoid-MS
• May$e the $iggestsolenoid e'er9thestrong (agnetic .eld4! is inside thecentral solenoid,inner dia(eter :(,length 1;( !he .eldis there to cur'e the
#aths of #articles#roduced, to (easuretheir charge and(ass !he unit
weighs 12,5<< tons
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!he First Sign of )nti(atter
• -loud cha($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
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!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
•
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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 $ye'ery turn of the wire, total
current NI • )(#*re"s law
gi'es i((ediately
•
0 encl B d I µ × =∫ l
0
2
NI B
r
µ
π =
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-lic0er Buestion
• !he .eld outside atoroidal solenoid+
) Cecreases as 16r with distancefro( the center ofthe solenoid
% Cecreases as 16r 2
- Is Dero
•
B?
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-lic0er )nswer
• !he .eld outside atoroidal solenoid+
) Cecreases as 16r with distancefro( the center ofthe solenoid
% Cecreases as 16r 2
- Is Dero
•
B?
) surface s#anning the dotted circle will ha'e Derototal current #enetrating>as (uch u# as down, sothe )(#ere integral is Dero
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!oroids•
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!he %iot&Sa'art Law•
!he electric .eld fro(any distri$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 $its+
•
wea0er .eld
shown dar0e
0
2
ˆ
4
I d r dB
r
µ
π
×=
uurr l
2ˆ /kqr r
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!he %iot&Sa'art Law•
Field fro( a .nite lengthof straight wire+
• !he .eld at the #oint inthe .gure is inward, andhas (agnitude
• Using R E & y tanθ we .nd
•
RdB
r
dy
y
0
θ
I
0
2
ˆ
4
I d r dB
r
µ
π
×=
uurr l
2
1
0
2
sin
4
y
y
I dy
B r
µ θ
π =
∫
( )0
1 2cos cos
4
I B
µ θ θ
π = −
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-lic0er Buestion
• Su##ose we ha'e two current ele(ents+is at the origin, #ointing along the x&
a3is,
is at x E 1, y E <, z E < and #oints inthe z &direction
•
-all the force of 1 on 2
• Coes ?
) es
% @o
x
y
z
1 1 I d
rl
2 2 I d
rl
12 F r
12 21 F F = −
r r
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-lic0er )nswer•
Su##ose we ha'e two currentele(ents+is at the origin, #ointing along
the x&a3is,
is at x E 1, y E <, z E < and#oints in the z &direction
-all the force of 1 on 2
• Coes ?) es
% @o
x
y
z
Ahat if the two current ele(ents are Gust charged#articles (o'ing through s#ace? Ahat a$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
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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$'ious fro(sy((etry !he su( ofthe dℓ "s is 2π R, so the.eld is
•
R
x
r
θ
I
dBr
d uurl r
d
uur
l
( )
2 2
0 0 0
3/ 22 32 2
12 sin
4 2 2
I I I R R B R
r r R x
µ µ µ π θ
π = = =
+
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Ferro(agnetic Materials
• !he ato(s of Fe, -o and @i and rareearths are little (agnets+ in theinco(#letely .lled shell of electrons, the
electron s#ins line u#>and electrons arethe(sel'es (agnets
• !he ato(s also line u# (agnetically withtheir neigh$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?
•
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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
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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
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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, $ut0ee#s its (agnetiDation>do(ain walls don"t
(o'e easily
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7lectro(agnets
• 7lectro(agnets are solenoidswith iron inside to (agnify the(agnetic .eld
• In the electro(agnetic door$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
•