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Magnetic Materials
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Magnetic Material
Magnetic field in materials is due to tiny electriccurrent loops
or magnetic dipoles . At normaltemperatures the magnetic moments of
the
dipoles are randomly oriented so that the netmagnetic moment is
zero.Magnetization = Magnetic moment per unitvolume
When an e ternal magnetic field is applied thedipoles align and
the material develops amagnetization.
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ypes of magnetic materials
ty. permeabilirelative theisandty
permeabiliabsolutetheisHere
HHH)1()H(B
ic. paramagnetisit positive,isitIf
c.diamagnetiismaterialthenegative,isIf
litysusceptibimagnetictheiswhereH
r
r !m!m!
m
mm
==+=+=
=
H
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%aramagnetic material in Magnetic &ield
Atomic magnets are oriented in directionof *$ currents in ad
acent loops cancelgiving rise to a surface current
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&orce on a magnetic dipole
&orce on a current carrying loop is
A
*
"ero.isloopcurrentclosedaonforcetheHence)( Bl d I F =
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A current loop e periences a force in aninhomogeneous
&ield
y
z
3 A
*4
=
=
! ,,!
!!
#
),,!(#)!,,!(#
B$%&' sides&n
dy z B
j I
dy y B j I dy y B j I F
y
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"ipole in 6nhomogeneous Magnetic &ield
y
z
3 A
*4
=
=
! ,!,!
!!
#
),!,!(#),,!(#
&$%'B sides&n
dz y B
k I
dz z Bk I dz z Bk I F
z
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"ipole in 6nhomogeneous Magnetic &ield
Assume that the derivatives are constant at the)oundaries of
small loops
i Im B
Bm z
Bk y
B j y
Bi I
z B
i z
Bk
y
Bi
y B
j I
dz z B
jdz y B
k I F
x x x
z x y x
y z
#and ! as
)(###
)##()##(
)#()#(! ! ,,!,!,!
==
=+
+=
+
+=
=
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or8ue on a current loop
A dipole e periences a tor8ue in a constantmagnetic field
or8ue is zero (hen the magnetic moment isparallel to the field.
o )ring the dipole to aposition (hich ma9es an angle (ith
themagnetic field$ one has to do (or9.
Bm=
Bm-U
mB
d mBd W
=
=
==
dipoleof nergy*otential
)cos1(
sin!
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:ector %otential of a current loop
= r l d I A !
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:ector %otential of a current loop
y
z
%
3
dl
r
dA
&torelative positionr
ldfrom*of distance plane"/in 0 pointfield
planey/in 0isloopcurrent
==
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y
z
%
3
dl
r
dA
dl;
Ad Ad
Al || d Ad
of component/yadds but
of component/0cancelshichelement w
oppositellysymmetricaa,ldeachorcomponent./"nohas,since
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:ector %otential of a current loop
d r
ar
ar
ar
ak I
d r
ar a
r ak
I
d ar ar a
k I
d a
k I l d I
A
)coscos1(1cos#
)cos
1(1
cos#
)cos(cos#
cos#
!
!
-1
!
!
!-1
!
!
!!
++=
+
+=
==
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:ector %otential of a current loop
r mr
k r m
k r mx
r x
k Ia
d r ax
ak I
d
r
xa
r
ax
r
a
r
ak I
A
xr
=
===
=
++=
!
!!!
!
!
.
!
!
#sin##
cos#
)coscos
1(1
cos#
coscos 2se
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:ector %otential of magnetized su)stance
6f M is the magnetization$ a volume d containsMd dipoles. he
vector potential at a point % isgiven )y
=
=
d r r M
d r r
r r M r A
1
)()(
!
!
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:ector %otential of magnetized su)stance
+=
=
=
S d M r r
d M r r
d r r
M d M r r
r A
r r M M
r r r r M
11.
11
.)(
11
2se
!
!
sd d =
surfacevol
vv
used havewestep,lastIn the
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*ound currents
n M r K
M r J
dS r r r K
d r r r J
S d M r r
d M r r
b
b
surfa !
b
"#l
b
#)(
)(
)()(
11)r '(
!!
!
=
=
+
=
+
=
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&ree and *ound 4u r rents
>et current is due to actual transport ofcharges (hile )ound
current is due tomagnetization.
Ampere;s a( for magnetized materialcontains contri)ution due to
)oth types ofcurrent and is$
b#und fr!! J J J +=
)(
)(
!
!!
M J
J J J B
f
b f
+=
+==
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&ields *$ , and M
"efine ,#field )y
M H
I
J H
!n l#s!d fr!!
f
=
=
=
'lso
ldH
ly,3uivalent
)(
M B H =!
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' ample
An infinitely long cylinder of radius ? contains afrozen in
magnetization M=9r$ (here r is distancefrom a is and no free
current. &ind *$ ,$ M.
#4##4#
#
##1
currentvolumeBound#
k$r k kr n M K k
r M
r M
r M J
k kr M
$r $r b
z z b
====
==
=
==
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6nfinitely long cylinder
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Magnetized cylinder @contd.
5 #
cylinder outside !B
/B!H!
alone.curl byspecifiedis ,!
!
!
!
kr
l d H
z M
M z
=
==
===
=
=
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A uniformly magnetized 4ylinder
sin#! M n M K
M J b
b
==
==
n
M
z
?
?sin
r l d #
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' ample B arge %iece of uniformly magnetizedmaterial from (hich
a sphere is scooped out.
* 0
M H
M B B
H
M
M B
H
M B B
M
M B
H
1
,! holetheof centreat the6ince
ionmagneti"at
uniformof sphereawithvoidtheill
oid,without vmaterialor the
!
!
!
!
!
!
!!
!
!!
+===
=
=
=
=
