Download Chapter 2: Magnetostatics

Chapter 2: Magnetostatics
1.
The Magnetic Dipole Moment
2.
Magnetic Fields
3.
Maxwell’s Equations
4.
Magnetic Field Calculations
5.
Magnetostatic Energy and Forces
Comments and corrections please: [email protected]
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Further Reading:
• David Jiles Introduction to Magnetism and Magnetic Materials, Chapman and Hall 1991; 1997
A detailed introduction, written in a question and answer format.
• Stephen Blundell Magnetism in Condensed Matter, Oxford 2001
A new book providing a good treatment of the basics
• Amikam Aharoni Theory of Ferromagnetism, Oxford 2003
Readable, opinionated phenomenological theory of magnetism
• William Fuller
The classic text
Brown Micromagnetism, 1949
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1. The Magnetic Dipole Moment
The magnetic moment m is the elementary quantity in solid state magnetism.
Define a local moment density - magnetization - M(r,t) which fluctuates wildly on a
sub-nanometer and a sub-nanosecond scale.
Define a mesoscopic average magnetization
!m = M!V
The continuous medium approximation
M can be the spontaneous magnetization Ms within a ferromagnetic domain
A macroscopic average magnetization is the domain average
M = !iMiVi/ !iVi
M (r)
Ms
The mesoscopic average magnetization
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dl
r
m
m = IA
1/2 (r"l)
O
I
m =1/2# r"j(r)d3r
A magnetic moment m is equivalent to a current loop
m =1/2# r"j(r)d3r = 1/2# r"Idl = I# dA = m
Inversion
Space
Time
Polar vector
-j
j
Axial vector
M
-M
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1.1 Field due to electric currents and magnetic moments
Biot-Savart Law
B
Unit of B - Tesla
Unit of µ0
T/Am-1
j
!
Right-hand corkscrew
µ0=4$ 10-7 T/Am-1
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1.1 Field due to electric currents and magnetic moments
Field at center of current loop
Dipole field far from current loop
- lines of force
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1.1 Field due to electric currents and magnetic moments
BA = 4(µ0Idl/4$r2)sin%
sin%= dl/2r
A
%
r
&
B
Idl
m
At a general position,
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2. Magnetic Fields
2.1 The B-field
'.B = 0
dA
Gauss’s theorem
Flux: d( = BdA
Unit Weber (Wb)
Flux quantum (0 = 2.07 1015 Wb
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The B-field
Sources of B
" electric currents in conductors
" moving charges
I
" magnetic moments
" time-varying electric fields. Not in magnetostatics
r
B
' x B = µ0 j
ex
ey
Ampere’s law.
Good for very symmetric
current paths.
ez
)/)x )/)y )/)z
Bx
BY
BZ
B = µ0I/2$r
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The B-field
Forces:
F = q(E + v x B)
expression.
Lorentz
gives dimensions of B and E.
pT
1E-6
he
at
t
So
le
no
id
Fi
el
d
Ea
rt
h'
s
ne
ta
ry
pa
rp
te
In
rs
te
In
1E-9
la
lla
te
H
an
1E-12
rS
rt
ea
in
ra
B
um
H
an
um
H
1E-15
Sp
ce
ac
e
The field at a distance 1 m from a wire
carrying a current of 1 A is 0.2 µ*
1E-3
µT
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Su
rf
Pe
ac
rm
e
Su a
p e ne
H r c nt
yb o M
P u rid n d a g
u n
Ex l s e Ma c t i e t
g
pl M n n g
os ag et M
ag
iv ne
e t
ne
Fl
t
ux
C
om
N
eu
pr
es
tr
on
si
on
St
ar
M
ag
ne
ta
r
The force between two parallel wires
each carrying one ampere is precisely
2 10-7 N m-1.
1
T
1000
1E6
1E9
1E12
1E15
MT
10
Typical values of B
Human brain 1 fT
Earth 50 µT
Helmholtz coils 0.01 Am-
Electromagnet 1 T
Magnetar
1012 T
Superconducting magnet 10 T
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2.2 Uniform magnetic fields.
Long solenoid B =µ0nI
Helmholtz coils B =(4/5)3/2µ0NI/a
a
Halbach cylinder B =µ0M ln(r2/r1)I
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2.3 The H- field.
In free space B = µ0H
' x B = µ0(jc + jm)
'.H = - '.M
Coulomb approach to calculate H
H = qmr/4$r3 qm is magnetic charge
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The H- field.
H = Hc + Hm
Hm is the stray field outside the magnet and
the demagnetizing field inside it
B = µ0(H + M)
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2.4 The demagnetizing field
The H-field in a magnet depends on the magnetization M(r) and on the shape of the magnet.
Hd is uniform in the case of a uniformly-magnetized ellipsoid. Tensor relation:
Hd = - N M
A constraint on the values of N when M lies along one of the principal axes, x, y, z, is
Nx + Ny + Nz = 1
• It is common practice to use a demagnetizing factor to obtain approximate internal fields in
samples of other shapes (bars, cylinders), which may not be quite uniformly magnetized.
N
• Examples.
Long needle, M parallel to the long axis, a
0
Long needle, M perpendicular to the long axis
1/2
Sphere
1/3
Thin film, M parallel to plane
Thin film, M perpendicular to plane
1
Toroid, M perpendicular to r
0
General ellipsoid of revolution
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Nc = ( 1 - Na)/2
15
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2.5 External and internal fields
H = H’ + Hd
Inernal field
applied field
H
demag field
H’ - N M
For a powder sample Np = (1/3) + f(N - 1/3)
f is the packing fraction
H’
H’
H’
Ways of measuring magnetization with no need for a demag correction
toroid
long rod
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thin film
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H’
H’
Magnetization of a sphere, and a cube
The state of magnetization of a sample depends on H, ie M = M(H). H is the independent
variable.
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2.6 Susceptibility and permeability
Simple materials are linear, isotropic and homogeneous (LIH)
M = "’H’
"’ is external susceptibility
M = "H
" is internal susceptibility
It follows that from H = H’ + Hd that
1/+ = 1/+’ - N
For typical paramagnets and diamagnets + ! 10-5 to 10-3, so the difference
between + and +’ can be neglected.
In ferromagnets, + is much greater; it diverges as T , TC but +’ never exceeds 1/N.
M
M
M
H0
Ms /3
H
H’
H'
H
Magnetization curves for a ferromagnetic sphere, versus the external and internal fields. "’=3
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• A related quantity is the permeability, defined for a paramagnet, or a soft ferromagnet in
small fields as
µ = B/H.
Since B = µ0(H + M), it follows that µ = µ0(1 + +r).
The relative permeability µr= µ/µ0 = (1 + +)
µ0 is the permeability of free space.
•In practice it is much easier to measure the mass of a sample than its volume. Measured
magnetisation is usually - = M/., the magnetic moment per unit mass (. is the density).
Likewise the mass susceptibility is defined as +m = +/ .
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2. Maxwell’s Equations
In electrostatics, there is also an auxiliary field, D. D = %0E + P
(J is defined as the ‘magnetic polarization’ J = µ0M )
Maxwell’s equations in a material medium are expressed in terms of the four fields
In magnetostatics there is no time-dependence of B. D or #
Conservation of charge '.j = -)./)t. In a steady state )./)t = 0
Magnetostatics: '.j = 0;
'.B = 0
'xH = j
Constituent relations: j = j(E);
P = P(E);
M = M(H)
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Hysteresis
spontaneous magnetization
remanence
coercivity
virgin curve
initial susceptibility
major loop
The hysteresis loop shows the irreversible, nonlinear response of a ferromagnet to an
internal magnetic field M = M(H). It reflects the arrangement of the magnetization in
ferromagnetic domains.
The B = B(H) loop is deduced from the relation B = µ0(H + M).
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3 Magnetic Field Calculations
In magnetostatics, the sources of magnetic field are
i) current-carrying conductors and
ii) magnetic material
Biot-Savart law
-------Dipole sum
Amperian approach-currents Coulomb approach-magnetic charge
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a) Dipole integral
Integrate over the magnetization distribution M(r)
Compensates the divergence at the origin
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a) Amperian approach
Integrate over the equivalent currents j(r)
jm = $ x M
and
jms = M x en
Evaluate from the Biot-Savart law.
Zero for a uniform distribution of M
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a) Coulomb approach
Use the equivalent distribution of magnetic charge
#m = -$.M
and
#ms = M.en
Evaluate from the Biot-Savart law.
Zero for a uniform distribution of M
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4.1 The magnetic potentials
a) Vector potential for B
Maxwell’s =n
'.B =0
Now '.('xA) = 0 hence
B='xA
A is the magnetic vector potential. Units T m.
Latitude in the choice of A: (0, 0, Bz) can be represented by
(0, xB,0), (-yB, 0, 0) or (1/2yB, 1/2xB, 0)
The gradient of any scalar f(r) can be added to A since 'x'f= 0
B is unchanged by any transformation A ,A’ known as a
gauge transformation.
Coulomb gauge: choose f( r) so that '.A
then A = (1/2)B x r
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Vector potential for B
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b) scalar potential for H
When the H-field is produced only by magnets, and not by
conduction currents, it can be expressed in terms of a potential.
The field is conservative, ' x H = 0
Since ' x ' f( r) = 0 for any scalar, we can express H as
H = -'/m
Units of /m are Amps.
'.(H + M) = 0
'2 /m = -.m
Hence
where .m = - '.M
The potential due to a charge qm is /m = qm /4$r
A dipole m has potential m.r/4$r3
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4.2 Boundary conditions!
At any interface, it follows from Gauss’s law
#SB.dA = 0
that the perpendicular component of B is continuous.
It follows from from Ampère’s law
#loopH.dl = I0 = 0
(there are no conduction currrents on the surface)
that the parallel component of H is continuous.
Since B = ' x A
#SB.dA = #loopA.dl (Stoke’s theorem)
If dollows that the parallel component of A is continuous.
The scalar potential is continuous /m1 = /m2
.
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Boundary conditions!
In LIH media, B = µ0 µr H
B1en = B2en
H1en = µr2/µr1 H2en
Hence field lies ! perpendicular to the surface of soft iron
but parallel to the surface of a superconductor.
Diamagnets produce weakly repulsive images
Paramagnets produce weakly attractive images
Images in a ferromagnet (a) and a superconductor (b)
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4.3 Local magnetic fields!
Hloc = H1 + H2
#
$
H1 = -NM + (1/3)M2
H2 is evaluated as a dipole sum.
H2 =!
1
Generally H2 =f M
Here f
1; it depends on the crystal lattice
f = 0 for a cubic lattice.
Dipole interactions are source of an intrinsic anisotropy contribution.
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5. Magnetostatic Energy and Forces
Magnetostatic (dipole-dipole) forces are long-ranged, but weak. They determine
the magnetic microstructure.
M ! 1 MA m-1, µ0Hd ! 1 T, hence µ0HdM ! 106 J m-3
Atomic volume ! (0.2 nm)3; equivalent temperature ! 1 K.
Products BH, BM, µ0H2, µ0M2 are all energies per unit volume.
Magnetic forces do no work on moving charges F = q(vxB) or currents F = j x B)
No potential energy associated with the magnetic force.
"
0=mxB
In a non-uniform field, F = -'Um
U = #mBsin&’d&’
Um = -m.B
B
!
F = m.'B
m
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Interaction of two dipoles:
m1
B21
m2
Up= -m1B21 = -m2B12
B12
Up =-(1/2)(m1B21 + m2B12)
Reciprocity theorem:
H2
M1
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H1
M2
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5.1 Self-energy
Energy of a body in the field Hd it creates itself.
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5.2 Energy associated with a magnetic field
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Energy product -#i µ0B.Hd d3r
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5.2 Energy in an external field
For hysteretic material, B " µH. The energy needed
to prepare a state depends on the path followed.
The work done to produce a small flux change is
1W = -%I1t = I1(. By Ampere’s law, I = #loopHdl.
1W = #loop1(Hdl.
1W = #1BHd3r
It would be better to have an expression for the
energy of M( r) in the external, applied field H’,
because we don’t know what H( r) is like throughout
the body. The real H-field is the one in Maxwell’s
equations
H = H’ + Hd
The constitutive relation is M = M(H) nor M = M(H’)
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Energy in an external field
The applied field H’ is created by some current distribution j’
'.H’ = 0
' x H’ = j’
The field created by the body satisfies
'.Hd = - '.M
' x Hd = 0
B = µ0(H + M) = µ0(H’ + Hd + M)
The magnetic work 1W’ = #1B(H’ + Hd) d3r Subtract the term µ0 #1H’H’d3r for space
Energy change due to the body is 1W’ = #(1B H’ - µ0 1H’H) d3r
=0
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Energy in an external field
B
M
a)
#HdB
H
!
b)
H
#µ0H’dM
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5.4 Thermodynamics of magnetic materials
M
dU = HxdX + dQ
dQ = TdS
!G
2F
Four thermodynamic potentials
" !F
-2G
U(X,S)
E(HX,S)
F(X,T) = U - TS
dF = HdX - SdT
G(HX,T) = F- HXX
dG = -XdH - SdT
Magnetic work is H1B or µ0H’1M
H’
S = -()G/)T)H’ µ0 M = -()G/)H’)T’
Maxwell relations
dF = µ0H’dM - SdT
dG = -µ0MdH’ - SdT
()S/)H’)T’ = - µ0()M/)T)H’ etc.
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