Tue Thur 13:00-14:15 (S103) Ch. 2 Magnetostatics Ki-Suk Lee Class Lab. Materials Science and Engineering Nano Materials Engineering Track Goal of this class Goal of this class Goal of this class Goal of this class Goal of this chapter 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 elementary quantity in solid-state magnetism is the magnetic moment m. 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 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 2.1 The magnetic dipole moment 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 Amp`ere, 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 2.1 The magnetic dipole moment 2.1 The magnetic dipole moment 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 The magnetic dipole moment 2.1.1 Fields due to electric currents and magnetic moments the Biot–Savart law 2.1 The magnetic dipole moment 2.1.1 Fields due to electric currents and magnetic moments 2.1 The magnetic dipole moment 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. 2.1 The magnetic dipole moment 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. 2.1 The magnetic dipole moment 2.1 The magnetic dipole moment The field falls off rapidly, as the cube of the distance from the magnet. It has axial symmetry about m. 2.1 The magnetic dipole moment 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 2.1 The magnetic dipole moment 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. 2.2.1 The B-field 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 Syst`eme 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 Equation (2.19) establishes the dimensions of B, but the magnitude of the tesla depends on the definition of the ampere. the force per metre = = 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 = 2.2.1 The B-field 2.2.1 The B-field 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: In a material medium, where jc is the conduction current in electrical circuits and jm is the Amp`erian magnetization current associated with the magnetized medium. 2.2.3 The H-field Amp`ere’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.