Magnetic vector potential For an electrostatic field E.d 0 E - x E x 0 We cannot therefore represent B by e.g. the gradient of a scalar since x B o j (rhs not zero) Magnetostatic field, try also .B 0 always (.E ) o BxA .B . x A 0 x B x x A (see later) B is unchanged by A' A x A' x A x A 0 5). Magnetic Phenomena Electric polarisation (P) - electric dipole moment per unit vol. Magnetic polarisation (M) - magnetic dipole moment per unit vol. M magnetisation Am-1 c.f. P polarisation Cm-2 Element magnetic dipole moment m When all moments have same magnitude & direction M=Nm N number density of magnetic moments Dielectric polarisation described in terms of surface (uniform) or volume (non-uniform) bound charge densities By analogy, expect description in terms of surface (uniform) or volume (non-uniform) magnetisation current densities Definitions • Electric polarisation P(r) Magnetic polarisation M(r) P(r). nˆ jP (r). nˆ dt 1 M(r) r x j(r) 2 P(r) t jM (r) x M(r) 0 jP (r) p r (r)dr allspace p electric dipole moment of localised charge distribution 1 m r x j(r) dr 2 all space m magnetic dipole moment of localised current distribution Magnetic moment of current loop For a planar current loop m = I A z A m2 z unit vector perpendicular to plane 1 m r x j(r) dr 2 all space 1 (r - a) 2 ˆ r x I r d dr 2 all space 2a ˆ a a zˆ I 2 ( r a ) r dr d 4a all space a zˆ I 4 a 2 zˆ I A 4a r x ˆ a zˆ A a 2 (r a) ˆ j I (r 2 a2 ) ˆ I 2a Cs-1 m-2 Magnetic moment and angular momentum • Magnetic moment of a group of electrons m • Charge –e mass me j(r ) qi v i (r ri ) v5 i v4 r5 O 1 m qi r x v i (r ri ) dr 2 i all space r4 r3 v3 1 m qi ri x v i 2 i v1 r1 r2 v2 i me ri x v i angular momentum -e m 2me -e i 2m L e i L i L total angular momentum i Force and torque on magnetic moment Fi qi v i x B F Lorentz force (r ) v(r ) x B(r ) dr continuous distributi on of current all space j(r ) x B(r ) dr all space Bk (r ) Bk (0) r.Bk (0) ... F m.B(0) suggests Um -m.B(0) F U Um -m.B(0) c.f. Up -p.E(0) Torque T r x j(r ) x B(r ) dr m x B(0) all space Torque on magnetic moment F F I x B c.f. q v x B Torque T r x F L ˆ T r x F 2 ILB sin T 2 T IA B sin Tˆ mxB /2 F L/2 d L/2 F r m T r L I B F ˆ L L dU 2 F.t d 2 F sin d 2 2 L 2 ILB sin d 2 U IAB sin d IAB cos m.B Origin of permanent magnetic dipole moment non-zero net angular momentum of electrons -e Includes both orbit and spin a Derive general expression via circular orbit of one electron I q 1 v I radius: a 2 2 a charge: -e qv I q -e mass: me 2 a speed: v 2 qva eva e a ang. freq: m Ia 2 2 2 2 ang. momentum: L dipole moment: m e 2 L L mea m 2me Similar expression applies for spin. Origin of permanent magnetic dipole moment e L m 2me m and L have opposite sense Consider directions: L -e m In general an atom has total magnetic dipole moment: e e m i L 2me i 2me e ℓ quantised in units of h-bar, introduce Bohr magneton B 2m e L 1 0,1,2 L z m m , 1,..., m 1B 0,1,2 m z m B m , 1,..., Diamagnetic susceptibility (r < 1) Characterised by r < 1 In previous analysis of permanent magnetic dipole moment, m = 0 when net L = 0: now look for induced dipole moment Applied magnetic field causes small change in electron orbit, leading to induced L, hence induced m Consider force balance equation when B = 0 (mass) x (accel) = (electric force) 1 2 2 2 Ze Ze 2 meo a ωo 2 3 4oa 4omea -e +Ze B If B perp to orbit (up), extra inwards Lorentz force: Approx: radius unchanged, ang. freq increased from o to ev B eaB -e Larmor frequency (L) balance equation when B ≠ 0 (mass) x (accel) = (electric force) + (extra force) 2 Ze me 2a eaB 2 4oa 1 2 Ze eB 3 2me 4omea eB o o L 2me 2 L is known as the Larmor frequency quadratic in me Z B oa3 2 Classical model for diamagnetism • Pair of electrons in a pz orbital m B a -e m = o + L |ℓ| = +meLa2 m = -e/2me ℓ v -e v x B v -e -e v x B = o - L |ℓ| = -meLa2 m = -e/2me ℓ Electron pair acquires a net angular momentum/magnetic moment Induced dipole moment Increase in ang freq increase in ang mom (ℓ) increase in magnetic dipole moment: e m meLa2 2me B -e m eB 2 e e 2a 2 e 2a 2 a m me B m B 2me 4me 4me 2me Include all Z electrons to get effective total induced magnetic dipole moment with sense opposite to that of B e2 m Zao2 B 6me ao2 : mean square radius of electron orbit ~ 10-27 for Z 12 B 1T c.f. 1B 9.274.10 -24 Am2 Critical comments on last expression Although expression is correct, its derivation is not formally correct (no QM!) It implies that ℓ is linear in B, whereas QM requires that ℓ is quantised in units of h-bar Fortunately, full QM treatment gives same answer, to which must be added any paramagnetic-contribution m2 e2 2 M N Zao B 3kT 6me p2 P N o E 3kT everything is diamagnetic to some extent Paramagnetic media (r > 1) analogous to polar dielectric alignment of permanent magnetic dipole moment in applied magnetic field B Bappl Bdip Bappl An aligned electric dipole opposes the applied electric field; But here the dipole field adds to the applied field! Other than that, it is completely analogous in thermal effect of disorder etc., hence use Langevin analysis again Langevin analysis of paramagnetism Up p.E Um m.B approximat ion when U kT Np 2 Nm2 P E M B 3kT 3kT P o EE B M B o small w hen U not small kT mB kT M Nm coth kT mB oNm2 Np 2 E B 3 okT 3kT As with polar dielectric media, the field B in the expressions should be the local field Bloc but generally find Bloc ≈ B Uniform magnetisation Electric polarisation p i C.m -2 i P ( Cm ) 3 V m I z x y IyΔz I M xyΔz x Magnetic polarisation M m i i V A.m2 -1 (Am ) 3 m Magnetisation is a current per unit length For uniform magnetisation, all current localised on surface of magnetised body (c.f. induced charge in uniform polarisation) Surface Magnetisation Current Density Symbol: M ; a vector current density but note units: Am-1 Consider a cylinder of radius r and uniform magnetisation M where M is parallel to cylinder axis Since M arises from individual m, (which in turn arise in current loops) draw these loops on the end face Current loops cancel in volume, leaving net surface current. M m Surface Magnetisation Current Density magnitude M = M but for a vector must also determine its direction M M n̂ M is perpendicular to both M and the surface normal M M nˆ c.f. b P.nˆ Normally, current density is “current per unit area” in this case it is “current per unit length”, length along the Cylinder - analogous to current in a solenoid. Solenoid with magnetic core Recap, vacuum solenoid: Bv ac onI I With magnetic core (red), Ampere’s Law L encloses two types of current, “conduction current” in the coils and“magnetisation current” on the surface of material: B.d I o encl BL o nL I ML B o nI M rB v ac r > 1: M and I in same direction (paramagnetic) r < 1: M and I in opposite directions (diamagnetic) Substitute for M : B o nI M (see later) Non-uniform magnetisation A rectangular slab of material in which M is directed along y-axis only but increases in magnitude along the x-axis only z I1-I2 I2-I3 My x I1 I2 I3 As individual loop currents increase from left to right, there is a net “mag current” along the z-axis, implying a “mag current density” which we will call jM z Neighbouring elemental boxes dx dx Consider 3 identical element boxes, centres separated by dx If the circulating current on the central box is My dy Then on the left and right boxes, respectively, it is My My My dx dy and My dx dy x x Upward and circulating currents My My 1 M M dx My dx My dy y y 2 x x The “mag current” is the difference in neighbouring circulating currents, where the half takes care of the fact that each box is used twice! This simplifies to M My My 1 2 y dx dy dxdy jMz dxdy jMz 2 x x x Non-uniform magnetisation A rectangular slab of material in which M is directed along -x-axis only but increases in magnitude along the y-axis only My z -Mx z y x jMz My x I1-I2 I2-I3 x jMz Mx y Total magnetisation current || z Similar analysis for x, y components yields I1 I2 I3 My Mx jM x y jM M z Magnetic Field Intensity H Recall Ampere’s Law B.d o Iencl or B o j Recognise two types of current, free and bound B o j o jf jM o jf M B M jf H jf o B where H M or B o H M o Electric Magnetic B D oE P H .D f H jf o M Ampere’s Law for H Often more useful to apply Ampere’s Law for H than for B H jf H.d S jf .d S s s hence H.d I enclf ree Bound current in magnetic moments of atoms Free current in conduction currents in external circuits or metallic magnetic media If Ib If L L vacuum B v ac onI f c.f. H nI f core o nI f M M n' Ib B Magnetic Susceptibility B • Two definitions of magnetic susceptibility • First M = BB/o is analogous to P = oEE B, field due to all currents, E, field due to all charges B M B B o H M oH BB o o 1 or B H r oH r c.f. D = roE 1 B 1 B B r Au -3.6.10-5 0.99996 Quartz -6.2.10-5 0.99994 O2 STP +1.9.10-6 1.000002 In this definition the diamagnetic susceptibility is negative and the relative permeability is less than unity Magnetic Susceptibility M • Second definition not analogous to P = o E E M MH B o H M o H MH or B o 1 M H r oH r 1 M When is much less than unity (all except ferromagnets) the two definitions are roughly equivalent 1.5 B(T) -500 0 +500 -1.5 H Am-1 Ferromagnet ~ 150-5000 for Fe Hysteresis and energy dissipation 1 1 1 Para-, diamagnets Boundary conditions on B, H For LIH magnetic media B = oH (diamagnets, paramagnets, not ferromagnets for which B = B(H)) .B 0 B.d S 0 B1cos1 S B2cos 2 S 0 B1 B2 H.d I H1sin1 L H2sin 2 L I encl f ree 0 H1|| H2|| 2 B2 S H .d 1 - H1 sin 1 1 A B1 2 B 1 1 1 enclf ree 1B 2 1 dℓ1 C A H .d H1 2 A 2 B I enclfree H2 dℓ2 2 H2 sin 2 2 Boundary conditions on B, H H||1 H||2 H1sin1 H2sin 2 B 1 B 2 B1cos1 B2cos 2 r1 oH1cos1 r2 oH2cos 2 H1sin1 H2sin 2 r1 oH1cos1 r2 oH2cos 2 tan 1 r1 tan 1 r1 c.f. tan 2 r2 tan 2 r2 Faraday’s Law E.d 0 E electrosta tic field d B E.d dt S t .dS Faraday' s Law S B.dS E B B E.d x E . d S .d S d B.d S S S t B x A A xE x t t t A E electrosta tic time - varying field t dℓ S Faraday’s Law B(r) I To establish steady current, cell must do work against Ohmic losses and to create magnetic field Energy density in magnetic fields Potential difference .d Power supplied to dv . j d da . j dv Total power j.E dW A E . j dv dt all space t Joule heating j E A j. work to establish magnetic field t dW A j. dv dt all space t 1 A x B . dv t o all space A E t j da dℓ Energy density in magnetic fields dW 1 dt o A x B. dv t all space .(axb) b.( xa) - a.( xb) A A B. x x B dv - . o all space t t 1 B 1 A B. dv x B .dS o all space t o S t 1 1 d B.B dv 2o dt all space W UM 1 2 1 2o B.B dv c.f. all space B.H dv c.f. U E 1 2 o 2 E.E dv in vacuum all space D.E dv in magnetic or dielectric media Time variation Combining electrostatics and magnetostatics: (1) .E = /o where = f + b (2) .B = 0 “no magnetic monopoles” (3) x E = 0 “conservative” (4) x B = oj where j = jf + jM Under time-variation: (1) and (2) are unchanged, (3) becomes Faraday’s Law (4) acquires an extra term, plus 3rd component of j Faraday’s Law of Induction emf x induced in a circuit equals the rate of change of magnetic flux through the circuit x B.d S x E.d t B E.d t B.dS B E.dS t .dS Stokes' Theorem B E t dS dℓ E.d 0 in general, only electrosta tic fields for which E C x 0 so E no longer representi ble simply by E Displacement current B o j j .j 1 1 o B Ampere’s Law . B 0 Problem! o Continuity equation .j 0 for non - steady currents t Steady current implies constant charge density so Ampere’s law consistent with the Continuity equation for steady currents Ampere’s law inconsistent with the continuity equation (conservation of charge) when charge density time dependent Extending Ampere’s Law add term to LHS such that taking Div makes LHS also identically equal to zero: .E o .E o j ? 1 o B . j .? 0 o.E . o E .j t t t E 1 j o B The extra term is in the bracket t o extended Ampere’s Law E B o j o o t Types of current j E B o j o o t j jf jM jP Total current P jP t jM x M k M = sin(ay) k j i jM = curl M = a cos(ay) i • Polarisation current density from oscillation of charges in electric dipoles • Magnetisation current density variation in magnitude of magnetic dipoles in space/time