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2 Basic concepts This section introduces the magnetization M and the two magnetic fields B and H. These are vectors which are defined at every position r. Methods of calculating B and H are introduced. Units and dimensions in magnetism are discussed. 2.1 Moments and fields 2.1.1. Magnetization. This is the basic magnetic quantity in a solid. m I Fig 2.1 All magnetism is due to circulating currents m=IA (2.1) m magnetic dipole moment, I electric curent, A area vector. Units of m are A m2. The magnetisation M of a solid is the magnetic dipole moment per unit volume. M = m/V Units of M are A m-1. (2.2) 2.1.2. B and H fields B Fig 2.2. The magnetic field B produced by the current loop is divergenceless (solenoidal) B is known as the magnetic flux density or B-field. Units are Tesla (T) There are no magnetic 'poles' to act as sources or sinks of B (not like E) (Maxwell's Eqn) (2.3a) The vector operator means (∂/∂x, ∂/∂y, ∂/∂z). is the divergence (div) of a vector ∂/∂x + ∂/∂y + ∂/∂z It can be written in integral form as SB.dA = 0 (2.3b) The relation between magnetic flux density and current density j (units A m-2) in the steady state is given by another of Maxwell's equations which can also be written in point form or integral form. In point form, µoj (2.4a) Here µ0 is the magnetic constant µ0 = 4π 10-7 T m A-1. (2.5) is the rotation (curl) of a vector is given by the expression ex(∂By/∂z - ∂By/∂y) + ey(∂Bz/∂x - ∂Bx/∂z) + ez(∂Bx/∂y - ∂By/∂x) In integral form, I loopB.dl = µ0I loo p (2.4b) Equation 2.4 is known as Ampere's law. The difficulty wiTh (2.4a) is that the current density j is made up of contributions from currents in external circuits j0 (which we can measure) and from the atomic currents jm that create the magnetization of a solid (which we cannot measure). The relation between jm and M is jm From (2.4a) (2.6) µo(j0 + jm) µ0 - M)µoj0 Define µ0 - M)H', or B = µ0(H' + M) (2.7) Then we can retain Ampere's law for the field H', which depends only on the measurable currents j0. 'j0 loopH'.dl = I0 (2.8a) (2.8b) Eq 2.7 does not mean that H' is created only by the conduction currents. Any magnet creates an H-field both in the space around it, and within its own volume. We write H' = Hd + H0 (2.9) where H0 is created by the conduction currents j0 and Hd is the field created by the magnet. It is known as the stray field outside the magnet, and the demagnetizing field inside the magnet. In free space, the distinction between the B-field and the H-field is trivial, they are proportional, though measured in different units; from (2.7), B = µ0H' (2.10) hence the magnetic constant µ0 is known as the permeability of free space. Inside the magnet the B-field and the H-field are quite different, and oppositely directed. QuickTime™ and a GIF decompressor are needed to see this picture. Fig 2.3 Illustration of M, B and H' for a magnet. The relation (2.7) is illustrated at a point P. 2.1.2 External Fields. An external field H which can be created by conduction currents or other magnets or both creates a torque on a magnet as it lowers its energy by aligning with the field. Whenever H interacts with matter, µ0 comes into the equation. In free space = µ0 H µ0mH sin; = µ0mH = mB (2.11) = -µ0m H = -m B (2.12) The corresponding energy E = -µ0mH cos; m H Fig 2.4 A magnet in an external field The relation between the internal field in a magnet H' and the external field H is H' = Hd + H (2.13) Only in the case of a uniformly-magnetized ellipsoid is there a simple expression for Hd. Hd = - N M (2.14) N is generally a 3 x 3 matrix with Nxx + Nyy + Nzz = 1 , but it reduces to a scalar demagnetizing factor when M is along one of the principal axes, x, y, z. It is common practice to use a demagnetizing factor to obtain approximate internal fields in samples of other shapes, which may not be uniformly magnetized. N Examples. Long needle, M parallel to axis,a 0 Long needle, M perpendicular to 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 0 Nc = ( 1 - Na)/2 QuickTime™ and a GIF decompressor are needed to see this picture. Fig 2.5 Demagnetizing factors for general ellipsoids. For ellipsoids of revolution with axes (a,c,c) explicit formulae are Na = 2/(1-2){(1-2)-1/2 sinh-1[(1-2)1/2/] - 1} (2.15a) for prolate (cigar-shaped) ellipsoids with = c/a < 1 and Na = 2/(1-2){1 - (1-2)-1/2 sinh-1[(1-2)1/2/]} (2.15b) for oblate (cigar-shaped) ellipsoids with > 1. For nearly-spherical shapes ( ≈ 1) Na = 1/3 - 1/15( The state of magnetization of a sample depends on H, M = M(H). H is the independent variable, also known as ‘magnetizing force’. M M H H' Fig 2.6 Magnetization of a sphere in the external and internal fields The magnetization of a uniform soft ellipsoid adjusts itself so that H' = 0 up to saturation, hence the external susceptibility r defined as r = M/H is 1/N. (2.16) From (2.13) and (2.14) H' = H - NM (2.17) Usually in paramagnets and diamagnets, the difference between the external susceptibility (M/H) and the internal susceptibility (M/H') is negligible. Not so in ferromagnets. There 1/rint =1/r -N (2.18) A related quantity is the permeability, defined for a paramagnet, or a soft ferromagnet in small fields as µ = B/H (2.19) from (2.7) µ = µ0(1 + r) (2.20) The relative permeability µr = µ/µ0 = (1 + r) (2.21) 2.2 Magnetic field calculations From (2.2a) and (2.7), it follows that ’ Any divergence of M in the material produces a stray field. There are three main methods for calculating magnetic fields created by a magnet. ¨ Dipole sum* H(r) = 1/4π(r)r/r5 - M/r3]d3r ¡ Current sheet H(r) = 1/4πjm(r-r’)/|r-r’|3d3r’ + 1/4πjs(r-r’)/|r-r’|3d2r’ ¬ Pole distribution. H(r) = -1/4πM(r-r’)/|r-r’|3d3r’ + 1/4πM.n(r-r’)/|r-r’|3d2r’ P P r MdV P n js ¨ ¡ ¬ For uniform magnetization, only the surface contributions count, jm = 0, js = M Also m = -M = 0; s - M. . Here is the unit vector normal to the surface. * Equivalent expressions for the field of a dipole m are H = 1/4π[(mr)r/r5 - m/r3] ; H = 1/4πr3[2mcoser + msine]; (Hx, Hy, Hz) = (m/4πr3)(3sincoscos, 3sincossin, 3cos2 - 1). Potential can be exploited in these calculations. Provided there are no conduction currents present, H’ can be deduced from the the magnetic scalar potential m; since 0 for any scalar , H = m (2.22) The analogy is with electrostatics. There is supposed to be a distribution of positive and negative magnetic poles, which act as sources of H. There is a surface pole density s = M.,, and a volume pole density m = -The potential of at distance r from a magnetic pole p is p/4πr. The potential of a dipole m is m.r/4πr3.Units of mare amps. The potential for a general, nonuniform magnetiztion distribution is m = -1/4πM/|r-r’|d3r’ + 1/4πM.n/|r-r’|d2r’ (2.23) Then equation ¬ above follows from (2.22) and (2.23). Since = 0, it follows from (2.7) and (2.22) that msatisfies Poisson’s equation m= (2.24) More generally applicable is the magnetic vector potential A. There is no restriction regarding conduction currents. The flux density at any point can be written as B = A (2.25) This definition is consistent with the vector identity a 0, true for any vector a. Units of A are T m. The equipotentials of A are the 'lines of force'. The vector potential for a general, nonuniform magnetiztion distribution is A = µ0/4πM/|r-r’|d3r’ + µ0/4πM/|r-r’|d2r’ (2.26) Then equation ¡ above follows from (2.25) and (2.26), since jm = M js = M The vector identity a = (a) - a, and the Coulomb gauge a = 0 give Poisson’s equation for the vector potential A = -µ0(jm + j) (2.27) Boundary Conditions: At any interface, it follows from (2.2) and (2.4) that the perpendicular component of B and the parallel component of H are continuous (see Fig 2.3, 2.7). H|| Fig.2.7. Boundary conditions for B and H. 2.3 Units 2.3.1 SI Units We use SI throughout with the Sommerfeld convention (2.7). Engineers prefer the Kenelly convention, both are consistent, compatible SI units. B = µ0H + J (2.19) J is the polarization, with units of Tesla, like B. J = µoM. At room temperature, for iron Js = 2.16 T -1 M = 1.71 MA m The international system is based on five fundamental units kg, m, s, K, and A. Derived units include the newton (N) = kg.m/s2, joule (J) = N.m, coulomb (C) = A.s, volt (V) = J/C, tesla (T) = J/Am2 = Vs/m2, weber (Wb) = V.s = T.m2 and hertz (Hz) = s-1. Recognized multiples are in steps of 10±3, but a few exceptions are admited such as cm (10-2 m) and Å (10-10 m). Multiples of the meter are fm (10-15), pm (10-12), nm (10-9), µm (10-6), mm (10-3) m (10-0) and km (103). Flux density B is measured in telsa (also mT, µT). Magnetic moment is measured in A.m2 so the magnetization and magnetic field are measured in A/m. From (2.12) it is seen hat an equivalent unit for magnetic moment is J/T, so magnetization can also be expressed as J/(T.m3). , the magnetic moment per unit mass in J/(T.kg) is the quantity most usually measured in practice. µ0 is exactly 4π.10-7 T.m/A. The SI system has two compelling advantages for magnetism: (i) it is possible to check the dimensions of any expression by inspection and (ii) the units are directly related to the practical units of electricity. 2.3.2 cgs Units Much of the primary literature on magnetism is still written using cgs units, or a confusing mixture where large fields are quoted in tesla and small ones in oersted! Fundamental cgs units are cm, g and s. The electromagnetic unit of current is equivalent to 10 A. The electromagnetic unit of potential is equivalent to 10 nV. The electromagnetic unit of magnetic dipole moment (emu) is equivalent to 1 mA.m2. Derived units include the erg (10-7 J) so that an energy density such as K1 of 1 J/m3 is equivalent to 10 erg/cm3. The convention relating flux density and magnetization is B = H+ 4πM where the flux density or induction B is measured in gauss (G) and field H in oersted (Oe). Magnetic moment is usually expresed as emu, and magnetization is therefore in emu/cm3, although 4πM is frequently considerd a flux-density expression and quoted in kilogauss. µ0 is numerically equal to 1 G/Oe, but it is normally omitted from the equations. The most useful conversion factors between SI and cgs units in magnetism are B 1 T kG H 1 kA/m 12.57 (≈12.5) Oe m 1 J/T 1000 emu (BH)max1 MJ/m3 MG.Oe 1 J/T.kg 1 emu/g M 1 kA/m 1 emu/cm3 1 G 0.1 mT 1 Oe ≈A m-1 1 emu 1 mJ/T 1 MG.Oe kJ/m3 The dimensionless susceptibility M/H is a factor 4π larger in SI than in cgs. Table 2.1: Magnetic units. _________________________________________________________ SI relation cgs relation _________________________________________________________ B B M J H Hmd = µo(H + M) = µoH + J B = H + 4 πM = H + I =1T B = 10 kG = 1 kA/m M = 1 emu/cm3 =1T 4πM = 10 kG = 1 kA/m H = 4π Oe = -DM Hmd = - 4πD M = - N M (0 ≤ D ≤ 1) (0 ≤ D ≤ 1, 0 ≤ N ≤ 4π) 2 m = 1 J/T (A.m ) m = 1000 emu (erg/G) = 1 J/Tkg = 1 emu/g = ∂M/∂H = 1 = ∂M/∂H = 4π (BH)max = 1 MJ/m3 (BH)max = 40π MG.Oe E = -Vµ0H.M (J) E = -VH.M (erg) 2 2 = µ0m/4πr (A) m/r (Oe.cm) _________________________________________________________ Table 2.2: Energy units. ______________________________________________________ Unit J eV K _________________________________________________________________ J = 1 6.242 1018 7.243 1022 -19 eV = 1.6022 10 1 11.60 103 K = 1.381 10-23 8.617 10-5 1 -24 -5 T = 9.274 10 5.788 10 0.6717 mc2 = 8.188 10-14 0.5110 106 5.930 109 -21 -2 kJ/mole = 1.6605 10 1.036 10 120.3 kcal = 4184 2.611 1022 3.030 1026 -18 Ry= 2.180 10 13.606 157.9 103 1/cm = 1.986 10-23 12.389 10-5 1.4388 -34 -15 Hz = 6.6261 10 4.136 10 4.799 10-11 ______________________________________________________ 2.4 Dimensions In the SI system, the basic quantities are mass (m), length (l), time (t), charge (q) and temperature (). Any other quantity has dimensions which are a combination of the dimensions of these five basic quantities, m, l, t, q and . In any relation between a combination of physical properties, all the dimensions must balance. Table 2.3 Dimnsions Mechanical Quantity area volume velocity acceleration density energy momentum angular momentum moment of inertia force power pressure stress elastic modulus frequency diffusion coefficient viscosity (dynamic) viscosity (kinematic) Planck’s constant symbol A V v a m 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 1 0 1 unit J J.K-1 J.K-1.kg-1 J.K-1 W.m-1.K-1 J.mol-1.K-1 m l t q 1 2 -2 0 1 2 -2 0 0 2 -2 0 1 2 -2 0 1 1 -3 0 1 2 -2 0 k J.K-1 E p L I F p P S K D h l t 2 3 1 1 -3 2 1 2 2 1 2 -1 -1 -1 0 2 -1 2 2 0 0 -1 -2 0 -2 -1 -1 0 -2 -3 -2 -2 -2 -1 -1 -1 -1 -1 q 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 unit m2 m3 m.s-1 m.s-2 kg.m-3 J kg.m.s-1 kg.m2.s-1 kg.m2 N W Pa N.m-2 N.m-2 s-1 m2.s-1 N.s.m-2 m2.s-1 J.s Thermal Quantity enthalpy entropy specific heat heat capacity thermal conductivity Sommerfeld coefficient Boltzmann’s constant symbol H S C c 1 2 -2 0 -1 -1 -1 -1 -1 0 -1 Electrical Quantity current current density potential electromotive force capacitance resistance resistivity conductivity dipole moment electric polarization electric field electric displacement electric flux permitivity thermopower mobility symbol I j V C R p P E D S µ unit A A.m-2 V V F .m S.m-1 C.m C.m-2 V.m-1 C.m-2 C F.m-1 V.K-1 m2V-1s-1 m 0 0 1 1 -1 1 1 -1 0 0 1 0 0 -1 1 -1 l m 0 0 -1 0 1 1 1 0 1 1 0 0 1 1 1 0 l 0 -2 2 2 -2 2 3 -3 1 -2 1 -2 0 -3 2 0 t -1 -1 -2 -2 2 -1 -1 1 0 0 -2 0 0 2 -2 1 q 1 0 1 0 -1 0 -1 0 2 0 -2 0 -2 0 2 0 1 0 1 0 -1 0 1 0 1 0 2 0 -1 -1 1 0 t -1 -1 -1 -1 -1 -1 0 0 0 -1 -1 -1 -2 -2 -2 0 q 1 0 1 0 1 0 1 0 -1 0 -1 0 -2 0 0 0 -2 0 -1 0 1 0 1 0 0 0 0 0 0 0 -1 0 Magnetic Quantity magnetic moment magnetisation specific moment magnetic field strength magnetic flux magnetic flux density inductance susceptibility (M/H) permeability (B/H) magnetic polarisation magnetomotive force magnetic ‘charge’ energy product anisotropy energy exchange coefficient Hall coefficient symbol unit m A.m2 M A.m-1 A.m2.kg-1 H A.m-1 Wb B T L H µ H.m-1 J T F A qm A.m (BH) J.m-3 K J.m-3 A J.m-1 RH m3.C-1 2 -1 2 -1 2 0 2 0 1 0 0 1 -1 1 1 3 Examples: 1) Kinetic energy of a body; E = (1/2)mv2 [E] = [ 1, 2,-2, 0, 0] 2) Lorentz force on a moving charge; F = qvxB [F] = [ 1, 1,-2, 0, 0] 3) [m] = [ 1, 0, 0, 0, 0] [v2] = 2[ 0, 1,-1, 0, 0] [ 1, 2,-2, 0, 0] [q] = [ 0, 0, 0, 1, 0] [v] = [B] = [ 0, 1,-1, 0, 0] [ 1, 0,-1,-1, 0] [ 1, 1,-2, 0, 0] Domain wall energy w = AK (w is an energy per unit area) [w] = [EA-1] [AK] =1/2[ AK] = [ 1, 2,-2, 0, 0] [A]=1/2[ 1, 1,-2, 0, 0] -[ 1, 1,-2, 0, 0] []=1/2[ 1,-1,-2, 0, 0] = [ 1, 0,-2, 0, 0] [ 1, 0,-2, 0, 0] 4) Magnetohydrodynamic force on a moving conductor f = vxBxB (f is a force per unit volume) [f] = [ FV-1] = [ 1, 1,-2, 0, 0] -[ 0, 3, 0, 0, 0] [ 1,-2,-2, 0, 0] [] = [-1,-3, 1, 2, 0] [v] = [ 0, 1,-1, 0, 0] [B2] = 2[ 1, 0,-1,-1, 0] [ 1,-2,-2, 0, 0] 5) Flux density in a solid B = µ0(H + M). (Note that quantities added or subtracted in a bracket must have the same dimensions) [B]= [ 1, 0,-1,-1, 0] [µ0] = [ 1, 1, 0,-2, 0] [M],[H] = [ 0,-1,-1, 1, 0] [ 1, 0,-1,-1, 0] 6) Maxwell’s equation xH = j + dD/dt. [xH] = [Hr-1] = [ 0,-1,-1, 1, 0] -[ 0, 1, 0, 0, 0] = [ 0,-2,-1, 1, 0] [j] = [ 0,-2,-1, 1, 0] [dD/dt] = [Dt-1] = [ 0,-2, 0, 1, 0] -[ 0, 0, 1, 0, 0] = [ 0,-2,-1, 1, 0]