Chapter 2: Magnetostatics - Trinity College, Dublin · PDF fileChapter 2: Magnetostatics 1....

Dublin January 2007 1

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]


Further Reading:

• David Jiles Introduction to Magnetism and Magnetic Materials, Chapman and Hall 1991; 1997A detailed introduction, written in a question and answer format.

• Stephen Blundell Magnetism in Condensed Matter, Oxford 2001A new book providing a good treatment of the basics

• Amikam Aharoni Theory of Ferromagnetism, Oxford 2003Readable, opinionated phenomenological theory of magnetism

• William Fuller Brown Micromagnetism, 1949The classic text


1. The Magnetic Dipole Moment

M (r)

Ms

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

The mesoscopic average magnetization


I

m m = IA

A magnetic moment m is equivalent to a current loop

O

r

dl

1/2 (r"l)

m =1/2# r"j(r)d3r

m =1/2# r"j(r)d3r = 1/2# r"Idl = I# dA = m

-M MAxial vector

j-jPolar vector

TimeSpaceInversion


1.1 Field due to electric currents and magnetic moments

Biot-Savart Law

!j

B

Right-hand corkscrew

Unit of B - Tesla

Unit of µ0 T/Am-1

µ0=4$ 10-7 T/Am-1



Field at center of current loop

Dipole field far from current loop

- lines of force



A

B

Idl

%

BA = 4(µ0Idl/4$r2)sin%sin%= dl/2r

At a general position,

&

r

m


2. Magnetic Fields 2.1 The B-field

'.B = 0

dA

Flux: d( = BdA

Unit Weber (Wb)

Flux quantum (0 = 2.07 1015 Wb

Gauss’s theorem


The B-field

Sources of B

" electric currents in conductors

" moving charges

" magnetic moments

" time-varying electric fields. Not in magnetostatics

' x B = µ0 j Ampere’s law.

Good for very symmetric

current paths.

B = µ0I/2$rBZBYBx

)/)z)/)y)/)x

ezeyex

I

r

B


The B-field

Forces:

F = q(E + v x B) Lorentz

expression.

gives dimensions of B and E.

The force between two parallel wires

each carrying one ampere is precisely

2 10-7 N m-1.

The field at a distance 1 m from a wirecarrying a current of 1 A is 0.2 µ*

1E-15 1E-12 1E-9 1E-6 1E-3 1 1000 1E6 1E9 1E12 1E15

MT

Mag

netar

Neu

tron S

tar

Exp

losi

ve F

lux

Com

press

ion

Puls

e M

agnet

Hyb

rid M

agnet

Super

conduct

ing M

agnet

Per

man

ent M

agnet

Hum

an B

rain

Hum

an H

eart

Inte

rste

llar

Spac

e

Inte

rpla

netar

y S

pace

Ear

th's

Fie

ld a

t the

Surf

ace

Sole

noid

pT µT T


Typical values of B Human brain 1 fT

Magnetar 1012 T

Superconducting magnet 10 T

Electromagnet 1 T

Helmholtz coils 0.01 Am-

Earth 50 µT


Long solenoid B =µ0nI

a

Helmholtz coils B =(4/5)3/2µ0NI/a

Halbach cylinder B =µ0M ln(r2/r1)I

2.2 Uniform magnetic fields.


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


The H- field. H = Hc + Hm

Hm is the stray field outside the magnet and

the demagnetizing field inside it

B = µ0(H + M)


2.4 The demagnetizing fieldThe 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, isNx + Ny + Nz = 1

• It is common practice to use a demagnetizing factor to obtain approximate internal fields insamples 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 0Thin film, M perpendicular to plane

1Toroid, M perpendicular to r 0

General ellipsoid of revolution Nc = ( 1 - Na)/2


2.5 External and internal fields

H = H’ + Hd

Inernal field applied field demag field

H ≈ 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 thin film


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 independentvariable.


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

H H'Ms /3 H’ H

M

H0

Magnetization curves for a ferromagnetic sphere, versus the external and internal fields. "’=3


• A related quantity is the permeability, defined for a paramagnet, or a soft ferromagnet insmall 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. Measuredmagnetisation is usually - = M/., the magnetic moment per unit mass (. is the density).

Likewise the mass susceptibility is defined as +m = +/ .


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)


Hysteresis

The hysteresis loop shows the irreversible, nonlinear response of a ferromagnet to aninternal magnetic field M = M(H). It reflects the arrangement of the magnetization inferromagnetic domains.

The B = B(H) loop is deduced from the relation B = µ0(H + M).

coercivity

spontaneous magnetization

remanence

major loop

virgin curveinitial susceptibility


3 Magnetic Field Calculations

--------

In magnetostatics, the sources of magnetic field are

i) current-carrying conductors and

ii) magnetic materialBiot-Savart law

Dipole sum Amperian approach-currents Coulomb approach-magnetic charge


a) Dipole integral Integrate over the magnetization distribution M(r)

Compensates the divergence at the origin


a) Amperian approach Integrate over the equivalent currents j(r)

Zero for a uniform distribution of M

jm = $ x M and jms = M x en

Evaluate from the Biot-Savart law.


a) Coulomb approach Use the equivalent distribution of magnetic charge

Zero for a uniform distribution of M

#m = -$.M and #ms = M.en

Evaluate from the Biot-Savart law.


4.1 The magnetic potentials

a) Vector potential for B

Maxwell’s =n '.B =0

Now '.('xA) = 0 hence B = ' x A

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


Vector potential for B


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

Hence '2 /m = -.m where .m = - '.M

The potential due to a charge qm is /m = qm /4$r

A dipole m has potential m.r/4$r3


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

.


Boundary conditions!

Images in a ferromagnet (a) and a superconductor (b)

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


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.


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, µ0M

2 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 = m x B

Um = -m.B

In a non-uniform field, F = -'Um

F = m.'BU = #mBsin&’d&’

!

"

m

B


Interaction of two dipoles:

m1

B21

m2

B12Up= -m1B21 = -m2B12

Up =-(1/2)(m1B21 + m2B12)

Reciprocity theorem:

M1

H1 H2

M2


5.1 Self-energy Energy of a body in the field Hd it creates itself.


5.2 Energy associated with a magnetic field


Energy product -#i µ0B.Hd d3r


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’)


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


Energy in an external field

B

H

M

Ha) b)

!

#HdB #µ0H’dM


5.4 Thermodynamics of magnetic materials

M

H’

G!

F!"

dU = HxdX + dQ

dQ = TdS

Four thermodynamic potentials

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

dF = µ0H’dM - SdT

dG = -µ0MdH’ - SdT

S = -()G/)T)H’ µ0 M = -()G/)H’)T’

Maxwell relations

()S/)H’)T’ = - µ0()M/)T)H’ etc.

2F

-2G

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