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
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Magnetostatic (dipole-dipole) forces are long-ranged, but weak. They determine the magnetic microstructure. M 1 MA m-1. , i.
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Dublin January 2007^
Chapter 2: Magnetostatics 1. The Magnetic Dipole Moment2. Magnetic Fields3. Maxwell’s Equations4. Magnetic Field Calculations5. Magnetostatic Energy and Forces Comments and corrections please:^ [email protected]
Dublin January 2007^ 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 Brown^ Micromagnetism
, 1949 The classic text
Dublin January 2007^ m^ m^ = I A I A magnetic moment m is equivalent to a current loop
d l r 1/2 (r"l) O^3 m^ =1/2#^ r"j(r)dr (^3) m =1/2# r"j(r)dr = 1/2# r" I dl =^ I #^ d A^ =^ m -MM Axial vector
j-j Polar vector
TimeSpace Inversion
Dublin January 2007^
1.1 Field due to electric currents and magnetic moments
Biot-Savart Law^ B j^!^ Right-hand corkscrew
Unit of B - Tesla-1Unit of^ μT/Am^0 -7^ -1 μ=4$^10 T/Am^0
Dublin January 2007^
1.1 Field due to electric currents and magnetic moments^ A
% B Idl (^2) B = 4(μIdl/4$r)sin%A (^0) sin%= dl/2r At a general position,
r & m
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2. Magnetic Fields^ 2.1 The B-field^ '. B^ =^^0 dA
Flux: d(^ = BdAUnit Weber (Wb)^15 Flux quantum^ (= 2.07 10^0 Gauss’s theorem^ Wb
Dublin January 2007^
The B-field^ Forces:^ F^ = q( E^ +^ v^ x^ B)^ Lorentzexpression.gives dimensions of^ B^ and
E. The force between two parallel wireseach carrying one ampere is precisely-7 -12 10N m. 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
MagnetarPulse MagnetHybrid MagnetSuperconducting MagnetExplosive Flux CompressionPermanent MagnetNeutron Star 1E6 1E9 1E12^ 1E15 MT Human BrainHuman HeartInterstellar SpaceInterplanetary SpaceEarth's Field at the SurfaceSolenoid μ T^ pT^ T
Dublin January 2007^ Typical values of^ B^
Human brain 1 fT (^12) Magnetar 10T Electromagnet 1 TSuperconducting magnet 10 T Earth 50^ μT -Helmholtz coils 0.01^ Am
Dublin January 2007^
2.3 The H- field.^ In free space
B^ =^ μ H^0 ' x^ B^ =^ μ( j +^ j )^0 c^ m '. H^ = -^ '. M^ Coulomb approach to calculate H^ H^ = q r /4$m
(^3) r qis magnetic chargem^
Dublin January 2007^
The H- field.^ H^ =^ H+ Hc^
m H is the stray field^ outside the magnet and m the^ demagnetizing field^ inside it B = μ( H + M ) 0
- Dublin January
Dublin January 2007^
2.5 External and internal fields^ H
=^ H’ + Hd Inernal field applied field^ demag field H H’ - N M For a powder^ sample^ N = (1/3) + f( p^
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
Dublin January 2007^
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 ’ +^ H
that d (^) 1/+ = 1/+’ -^ N For typical paramagnets and diamagnets
-5^ -3 +! 10 to 10, so the difference between^ +^ and^ +’ can be neglected.In ferromagnets,^ +^ is much greater; it diverges as T
,^ Tbut^ +’ never exceeds 1/C^
N.
M M H H' Ms^ /^3 H ’^
H M H (^0) Magnetization curves for a ferromagnetic sphere, versus the external and internal fields.
Dublin January 2007^
-^ A related quantity is the^ permeability
, defined for a paramagnet, or a soft ferromagnet in small fields as
μ =^ B / H. Since^ B^ = μ( H^ +^ M ), it follows that μ = μ^0
(1 +^ +). 0 r The relative permeability μ= μ/μr
= (1 +^ +)^ μis the^ permeability of free space 0 0
- 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^