Ch. 2 Magnetostatics Ki-Suk Lee

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We begin with magnetostatics, the classical physics of the magnetic fields, forces and energies associated with distributions of magnetic material and steady electric currents. The concepts presented here underpin the magnetism of solids.
Magnetostatics refers to situations where there is no time dependence.
2.1 The magnetic dipole moment
The local magnetization M(r ) fluctuates on an atomic scale – dots represent the atoms. The mesoscopic average, shown by the dashed line, is uniform.
The elementary quantity in solid-state magnetism is the magnetic moment m.
the local magnetization M(r,t ) which fluctuates wildly on a subnanometre scale, and also rapidly in time on a subnanosecond scale.
But for our purposes, it is more useful to define a mesoscopic average <M(r,t )> over a distance of order a few Nanometres, and times of order a few microseconds to arrive at a steady, homogeneous, local magnetization M(r). The time-averaged magnetic moment δm in a mesoscopic volume δV is
Continuous medium approximation The representation of the magnetization of a solid by the quantity M(r) which varies smoothly on a mesoscopic scale.
The concept of magnetization of a ferromagnet is often extended to cover the macroscopic average over a sample:
According to Ampère, a magnet is equivalent to a circulating electric current; the elementary magnetic moment m can be represented by a tiny current loop. If the area of the loop is A square metres, and the circulating current is I amperes, then
Magnetic moment and magnetization are axial vectors. They are unchanged under spatial inversion, r →−r, but they do change sign under time reversal t →−t.
normal polar vectors such as position, force, velocity and current density, which change sign on spatial inversion but not necessarily on time reversal.
Strictly speaking, axial vectors are tensors; They can be written as a vector product of two polar vectors, as in (2.4), but their three independent components can be manipulated like those of a vector.
2.1.1 Fields due to electric currents and magnetic moments
the Biot–Savart law
2.1.1 Fields due to electric currents and magnetic moments
calculate the field created by a magnetic moment associated with a small current loop, firstly at the centre, and secondly at a distance r much greater than the size of the loop.
To simplify the second calculation, we choose a square loop of side δl r and first evaluate the field at two special positions, point ‘A’ on the axis of the loop and point ‘B’ in the broadside position.
The field falls off rapidly, as the cube of the distance from the magnet. It has axial symmetry about m.
Faraday represented magnetic fields using lines of force. (The basic idea dates back to Descartes.) The lines provide a picture of the field by indicating its direction at any point; its magnitude is inversely proportional to the spacing of the lines. The direction of the field of a point dipole relative to the normal to r is known as ‘dip’.
the differential equation for the line of force
c is a different constant for each line
The field of the magnetic moment has the same form as that of an electric dipole p = qδl formed of positive and negative charges±q which are separated by a small distance δl. The vector p is directed from −q to +q.
Hence the magnetic moment m may somehow be regarded as a magnetic dipole; its associated magnetic field is called the magnetic dipole field.
One uses Cartesian coordinates, with m in the z-direction
another resolves the field into components parallel to r and m:
2.2 Magnetic fields
The magnetic field that appears in the Biot–Savart law and in Maxwell’s equations in vacuum is B, but the hysteresis loop of Fig. 1.3 traced M as a function of H. It is time to explain why we need these two magnetic fields, with different units and dimensions.
Fields with this property are said to be solenoidal; the lines of force all form continuous loops.
Gauss’s theorem:
magnetic flux flowing out of the surface S through area dA
an alternative name for the B-field is magnetic flux density
2.2.1 The B-field
Flux has a named (but little used) unit of its own, the weber, abbreviated to Wb. A unit equivalent to T is Wb /m2. The other synonym for B is magnetic induction.
Sources of the B-field are: (i) electric currents flowing in conductors; (ii) moving charges (which constitute an electric current); and (iii) magnetic moments (which are equivalent to current loops).
Time-varying electric fields are also a source of magnetic fields, and vice versa, but we are restricting our attention to magnetostatics
2.2.1 The B-field
Sources of the B-field are: (i) electric currents flowing in conductors;
(ii) moving charges (which constitute an electric current);
(iii) magnetic moments (which are equivalent to current loops).
2.2.1 The B-field
All the sources of B are moving charges, but B itself interacts with charges only when they move. The fundamental relation between the fields and the force f exerted on a charged particle is the Lorentz expression
The electric and magnetic fields can therefore be expressed in terms of the basic quantities of mass, length, time and current. The units of these four quantities in the Système International (SI) are the kilogram, metre, second and ampere. A coulomb is an ampere second.
Hence the units of E and B are, respectively, newtons per coulomb (N C−1), and N C−1 m−1 s. The latter reduces to kg s−2A−1, or tesla. Units and dimensions are discussed in Appendix A.
2.2.1 The B-field
The ampere is then defined as the current flowing in conductors in vacuum which produces a force of precisely 2 × 10−7 N m−1 when the two conductors are 1 metre apart
=
2.2.2 Uniform magnetic field
Structures that produce a uniform magnetic field in their bore: (a) a long solenoid, (b) Helmholtz coils and (c) a Halbach cylinder.
An infinitely long solenoid creates a uniform field that is parallel to its axis in the bore, and zero everywhere outside.
Helmholtz coils are a pair of matched coaxial coils whose separation is equal to their radius a (Fig. 2.6(b)). The field is uniform, with zero second derivative at the centre. It is given by
2.2.3 The H-field
Now we come to the H-field, also known as the magnetic field strength or magnetizing force. It is an indispensable auxiliary field whenever we have to deal with magnetic or superconducting material. The magnetization of a solid reflects the local value of H. The distinction between B and H is trivial in free space. They are simply related by the magnetic constant μ0:
where jc is the conduction current in electrical circuits and jm is the Ampèrian magnetization current associated with the magnetized medium.
In a material medium,
Ampère’s law
where Ic is the total conduction current threading the path of the integral. The new field is no longer divergenceless, but has sources and sinks associated with nonuniformity of the magnetization.
2.2.3 The H-field
where Ic is the total conduction current threading the path of the integral. The new field is no longer divergenceless, but has sources and sinks associated with nonuniformity of the magnetization.
We can imagine that H, like the electric field E, arises from a distribution of positive and negative magnetic charge qm. The field emanating from a single charge would be
Units of qm are A m.
2.2.3 The H-field
magnetic charges do offer a mathematically convenient way of representing the H-field, and some force and field calculations become much simpler if we make use of them. Charge avoidance is a useful principle in magnetostatics.
Any magnet will produce an H-field both in the space around it and within its own volume. We can write the field as the sum of two contributions
where Hc is created by conduction currents and Hm is created by the magnetization distributions of other magnets and of the magnet itself.
The second contribution is known as the stray field outside a magnet or as the demagnetizing field within it. It is represented by the symbol Hd
2.2.3 The H-field
Units of H, like those of M, areAm−1. One tesla is equivalent to 795 775 A m−1 (or approximately 800 kA m−1).
Inside the magnet the B-field and the H-field are quite different, and oppositely directed. H is also oppositely directed to M inside the magnet, hence the name ‘demagnetizing field’.
The field lines of H appear to originate on the horizontal surfaces of the magnet, where a magnetic charge of density σm = M · en resides; en is a unit vector normal to the surface. The H-field is said to be conservative (∇ × H = 0), whereas the B-field, whose lines form continuous closed loops, is solenoidal (∇ · B = 0).
When considering magnetization processes, H is chosen as the independent variable, M is plotted versus H, and B is deduced from (2.33). The choice is justified because it is possible to specify H at points inside the material in terms of the demagnetizing field, acting together with the fields produced by external magnets and conduction currents.
2.2.4 The demagnetizing field
It turns out that in any uniformly magnetized sample having the form of an ellipsoid the demagnetizing field Hd is also uniform. The relation between Hd and M is
Along the principal axes of the ellipsoid, Hd and M are collinear and the principal components of N in diagonal form (Nx ,Ny ,Nz) are known as demagnetizing factors. Only two of the three are independent because the demagnetizing tensor has unit trace:
2.2.4 The demagnetizing field
2.2.5 Internal and external fields
The external field H', acting on a sample that is produced by steady electric currents or the stray field of magnets outside the sample volume, is often called the applied field. The sample itself makes no contribution to H'.
2.2.5 Internal and external fields
Ways of measuring magnetization with no need for a demagnetizing correction: (a) a toroid, (b) a long rod and (c) a thin plate.
a sphere for which the magnetization is uniform
2.2.6 Susceptibility and permeability The simplest materials are linear, isotropic and homogeneous (LIH). For magnetism, this means that the susceptibility or applied field is small and a small uniform magnetization is induced in the same direction as the external field:
where χ’ is a dimensionless scalar known as the external susceptibility. Where χ is the internal susceptibility.
For single crystals, the susceptibility may be different in different crystallographic directions, and M = χH becomes a tensor relation with χij a symmetric second-rank tensor, which has up to three independent components in the principal-axis coordinate system.
2.2.6 Susceptibility and permeability
Permeability is related to susceptibility. It is defined in the internal field. In LIH media the permeability μ is given by
The relative permeability
Consider the example of an isotropic, soft ferromagnetic sphere with high permeability and no hysteresis.
M is zero in the multidomain state that exists before the field is applied.
The ideal soft material has
a hard ferromagnetic sphere
Generally, magnetic media are not linear, isotropic and homogeneous but nonlinear and hysteretic and often anisotropic and inhomogeneous as well!
Then B, like M, is an irreversible and nonsingle-valued function of H, represented by the B(H) hysteresis loop deduced from the M(H) loop
2.3 Maxwell’s equations
Maxwell’s equations in a material medium are expressed in terms of all four fields:
In order to solve problems in solid-state physics we need to know the response of the solid to the fields. The response is represented by the constitutive relations
portrayed by the magnetic and electric hysteresis loops and the current– voltage characteristic. The solutions are simplified in LIH media, where
2.4 Magnetic field calculations
magnetic moments currents magnetic charge
They yield identical results for the field in free space outside the magnetized material but not within it.
Alternative approaches for calculation of the magnetic field outside a uniformly magnetized cylinder
AmperianDirect Coulombian
C.f. Internal magnetic field of a dipole
The two models for a dipole (current loop and magnetic poles) give the same predictions for the magnetic field far from the source. However, inside the source region they give different predictions. The magnetic field between poles (see figure for Magnetic pole definition) is in the opposite direction to the magnetic moment (which points from the negative charge to the positive charge), while inside a current loop it is in the same direction (see the figure to the right). Clearly, the limits of these fields must also be different as the sources shrink to zero size. This distinction only matters if the dipole limit is used to calculate fields inside a magnetic material.[4]
If a magnetic dipole is formed by making a current loop smaller and smaller, but keeping the product of current and area constant, the limiting field is
Unlike the expressions in the previous section, this limit is correct for the internal field of the dipole.[4][9]
https://en.wikipedia.org/wiki/Magnetic_moment#cite_note-Brown62-4
2.4 Magnetic field calculations The second approach considers the equivalent current distributions in the bulk and at the surface of the magnetized material:
Using the Biot–Savart law (2.5) and adding the effects of the bulk and surface contributions to the current density,
2.4 Magnetic field calculations
The third approach uses the equivalent distributions of magnetic charge in the bulk and at the surface of the magnetized material:
2.4.1 The magnetic potentials
Vector potential
The flux density invariably satisfies
where A is a magnetic vector potential. Units of A are Tm. There is substantial latitude in the choice of vector potential for a given field.
B = (0,0,B)
A = (0,xB,0)
A = (-yB,0,0)
c.f. some useful vector calculus
The definition of A is not unique, because it is permissible to add on the gradient of any arbitrary scalar function
Gauge transformation
A useful gauge is the Coulomb gauge, where f is chosen so that
A convenient expression for A in the Coulomb gauge is
It does not matter that the definition of A is not unique, because the observed effects depend on the magnetic field, not on the potential from which it is mathematically derived.
The expression for the field due to a distribution of currents obtained by integrating the Biot–Savart law (2.5)
At large distances the vector potential for a magnetic moment m equivalent to a current loop is
Ampère’s law
we see that in the Coulomb gauge, the vector potential satisfies Poisson’s equation
Scalar potential When the H-field is produced only by magnets, and not conduction currents, it too can be expressed in terms of a potential. The field is then conservative, and Ampère’s law (2.28) becomes
so the scalar potential satisfies Poisson’s equation:
By solving Poisson’s equation:
Magnetostatic calculations are easier with the scalar pmotential, but it should be understood that it is only permissible to use it for problems where no conduction currents are present.
2.4.2 Boundary conditions The perpendicular component of B is continuous.
The parallel component of H is continuous.
2.4.2 Boundary conditions
If medium 1 is air and medium 2 has a high permeability, this shows that the lines of magnetic field strength inside a highly permeable medium tend to lie parallel to the interface, whereas in air they tend to lie perpendicular to the interface. This is the reason why soft iron acts as a magnetic mirror
The situation is reversed if the iron is replaced by a sheet of superconductor. Ideally, the superconductor is a perfect diamagnet into which flux does not penetrate. It follows that B is parallel to the surface, and the image is repulsive.
2.4.2 Boundary conditions
2.4.3 Local magnetic fields
The question then arises: ‘What is the value of the local magnetic field Hloc at a point in a solid?’ The point may be an atomic site.
The calculation of B at any point r may be carried out in principle by replacing the integral (2.51) by a sum over atomic point dipoles mi .
Region 1 : a continuum Region 2 : the Lorentz cavity, where the atomic-scale structure is taken into account
a is the interatomic separation.
The Lorentz cavity is chosen to be spherical. The field due to region 1 can be evaluated from the distribution of surface charges σm = M · en on the inner and outer surfaces.
The field H2 produced by the atoms contained within the cavity is evaluated as a dipole sum
2.4.3 Local magnetic fields
2.5 Magnetostatic energy and forces
There are two main contributions to the energy of a ferromagnetic body:
1. Atomicscale electrostatic effects like exchange or single-ion anisotropy, and magnetostatic effects.
2. The magnetostatic effects, which involve the self-energy of interaction of the body with the field it creates by itself, as well as the interaction of the body with steady or slowly varying external magnetic fields, are considered here.
Exchange and other electrostatic effects are the subject of Chapter 5.
The magnetostatic interactions are rather weak compared to the short- range exchange forces responsible for ferromagnetism, but they are important in ferromagnets nonetheless because the domain structure and magnetization process depends on them. It is the long-range nature of the dipole–dipole interaction, varying as r^−3, that allows these weak interactions to determine the magnetic microstructure.
Magnetic fields do no work on electric currents or moving charges because the magnetic part of the Lorentz force ( j × B) per unit volume or (v × B) per unit charge is always perpendicular to the motion. We cannot associate a potential energy function with the magnetic force.
Let us first consider a small rigid magnetic dipole m in a pre-existing steady field B.
Taking the reference state at θ = 0, where θ is the angle between m and B, and integrating the torque gives the ‘potential energy’
We assume that turning the magnetic dipole has no effect either on its moment or on the sources of B.
Zeeman energy
there is no net force on a magnetic moment in a uniform field; the ‘potential energy’ does not depend on position. However, if B is nonuniform, the energy of the dipole does depend on its position.
The energy εm is minimized for a ferromagnet or paramagnet by this force tending to pull the material into a region where the field is greatest, but a diamagnet is pushed out to a region where the field is smallest.
the mutual interaction of two parallel dipoles
Nose-to-tail broadside
Reciprocity
The two dipoles are an example of the reciprocity theorem, a useful result in magnetostatics, which states that the energy of interaction of two separate distributions of magnetization M1 and M2 producing fields H1 and H2 is
The reciprocity theorem is used to simplify magnetic energy calculations such as the interaction of a magnetic medium with a read head, for example.
2.5.1 Self-energy
We consider the energy of a body with magnetization M(r) in a magnetic field. The result is different according to whether the field in question is an external field H or the demagnetizing field Hd created by the body itself.
Here we discuss the second case, considering first a small moment δm at a point inside the macroscopic magnetized body.
The factor of ½ which always appears in expressions for the self-energy is needed to avoid double counting because each element δm contributes as a field source and as a moment.
2.5.1 Self-energy
2.5.1 Self-energy
We have used the handy result that for a magnet in its own field, when no currents are present,
where the integral is again over all space.
Far from the magnet, , so the integral over a surface of infinite radius is zero.
2.5.1 Self-energy
This shows that the energy associated with a permanent magnet can be either associated with the integral of H2d over all space (2.79), or with the integral of −Hd · M over the magnet (2.78), but not both. These are alternative ways of regarding the same energy term.
The expression (2.78) assumes the magnetization is known in advance, so we can evaluate the magnetostatic energy in the field produced by the magnetization configuration. In practice, the magnetization tends to adopt a configuration which minimizes its self-energy. For an ellipsoid, there may be uniform magnetization along the axis where N is smallest.
2.5.2 Energy associated with a magnetic field
In free space,
An expression for the energy associated with a static magnetic field may be obtained by considering an inductor L consisting of a current loop which creates a flux
The electromotive force (emf) developed in a circuit
The power needed to maintain a current I in the inductor is
Integrating from 0 to I gives an expression for the energy associated with the inductor:
The same energy can be associated with the field in space created by the current in the inductor.
In free space,
This is actually a general statement irrespective of whether the field is created by electric currents or magnetic material (2.79).
When designing magnetic circuits that include permanent magnets, the aim is usually to maximize the energy associated with the field created by the magnet in the space around it.
where the indices o and i indicate integrals over space outside and inside the magnet.
The integral on the left is the one to be maximized.
The energy product:
It is twice the energy stored in the stray field of the magnet.
The energy stored in the field outside the magnet
2.5.3 Energy in an external field
They exhibit hysteresis, so that the energy needed to prepare a state described by B and H depends on the path followed.
When the magnetization is uniform, this expression becomes
The increment of work δw done to produce a small change in flux δ
More generally, we would love to have an expression for the energy of the magnetization distribution M(r) in the external, applied field H' which is supposedly undeformed by the presence of the magnetic material.
The basic constitutive relation for the material is
The applied field H' is supposed to be created by some external current distribution j' .
The work associated with the energy changes of the magnetized body alone
a term associated with the H-field in empty space
The first integral is zero
The second integral is the contribution to the magnetostatic self-energy
by reciprocity
This expression relates the magnetic energy to the self-energy and the constitutive relation M = M(H).
The energy increment per unit volume
The energy expended to magnetize a sample is related to its anisotropy energy, including shape anisotropy, since the magnetization process in the external field depends on the orientation of the sample.
the energy associated with the
applied field
the hysteresis energy loss per cycle
An expression for the energy needed to magnetize an LIH paramagnetic material in an external field may be deduced. Here the moment is induced by the field, according to (2.40). Hence, from (2.93)
2.5.4 Thermodynamics of magnetic materials
HX represents some external action on the system, and X is a state variable
The internal energy per unit volume
The system is usually defined by fixing one variable in each of the (T,S) and (HX,X) pairs. Four thermodynamic potentials can be defined by fixing two variables experimentally and leaving the other two variables free.
When T is fixed, as is often the case,
When the magnetization is uniform
At thermodynamic equilibrium
The changes in F and G at constant temperature are associated with the areas under the reversible H(M) or −M(H) curves.
In the case of an LIH medium, the changes of Helmholtz free energy and Gibbs free energy on magnetizing the medium are
The spontaneous magnetization of a ferromagnet falls with increasing temperature. The fall becomes precipitous just below the Curie point, where the entropy of the spin system increases rapidly as the spin moments become disordered.
In this temperature range, large entropy changes can be produced by modest applied fields. The entropy and magnetization of the ferromagnet are obtained as partial derivatives of the Gibbs free energy
the specific heat of magnetic origin
Moreover, from the second derivatives of the four thermodynamic potentials, four Maxwell relations are obtained.
2.5.5 Magnetic forces
Forces in thermodynamics are related to the gradient of the free energy, which represents the ability of the system to do work.
When M is uniform and independent of H‘
The Kelvin force
2.5.5 Magnetic forces
A general expression for the force density when M is not independent of H is
Note that H in this expression is the internal field, not the applied field H’ , and υ = 1/d, where d is the density.
When this is independent of density, as it is for dilute solutions or suspensions of magnetic particles, the first term is zero and the force Fm is given by the Kelvin expression for a paramagnet with H’ = H. The demagnetizing field is negligible in dilute paramagnetic solutions, but in more concentrated samples such as ferrofluids, the first term takes care of the dipole–dipole interactions.
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Ch. 2 Magnetostatics Ki-Suk Lee