Magnetism in Matter 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

Magnetism in Matter Electric polarisation P(r) Magnetisation M(r) p electric dipole moment of m magnetic dipole moment of localised charge distribution localised current distribution

Magnetic moment of current loop a  For a planar current loop m = I A z A m 2 z unit vector perpendicular to plane

Magnetic moment and angular momentum Magnetic moment of a group of electrons m Charge –e mass m e O v1v1 r1r1 v4v4 v3v3 v2v2 v5v5 r5r5 r4r4 r3r3 r2r2

Force and torque on magnetic moment

Diamagnetic susceptibility Induced magnetic dipole moment when B field applied Applied field causes small change in electron orbit, inducing L,m Consider force balance equation when B = 0 (mass) x (accel) = (electric force) -e B  L is the Larmor frequency

Diamagnetic susceptibility Pair of electrons in a p z orbital  =  o +  L  |ℓ| = +m e  L a 2  m = -e/2m e  ℓ  =  o -  L  |ℓ| = -m e  L a 2  m = -e/2m e  ℓ a v -e m -e v x B v -e m -e v x B B Electron pair acquires a net angular momentum/magnetic moment

Diamagnetic susceptibility Increase in ang freq  increase in ang mom (  ℓ ) Increase in magnetic dipole moment: Include all Z electrons to get effective total induced magnetic dipole moment with sense opposite to that of B -e B m

Paramagnetism Found in atoms, molecules with unpaired electron spins Examples O 2, haemoglobin (Fe ion) Paramagnetic substances become weakly magnetised in an applied field Energy of magnetic moment in B field U m = -m.B U m = J for a moment of 1  B aligned in a field of 1 T U thermal = kT = J at 300K >> U m U m /kT= Boltzmann factors e -Um/kT for moment parallel/anti-parallel to B differ little at room temperature This implies little net magnetisation at room temperature

Ferro, Ferri, Anti-ferromagnetism Found in solids with magnetic ions (with unpaired electron spins) Examples Fe, Fe 3 O 4 (magnetite), La 2 CuO 4 When interactions H = -J m i.m j between magnetic ions are ( J) >= kT Thermal energy required to flip moment is Nm.B >> m.B N is number of ions in a cluster to be flipped and U m /kT > 1 Ferromagnet has J > 0 (moments align parallel) Anti-ferromagnet has J < 0 (moments align anti-parallel) Ferrimagnet has J < 0 but moments of different sizes giving net magnetisation

Uniform magnetisation Electric polarisation Magnetisation I zz yy xx 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 Units: A m -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 interior, leaving only net (macroscopic) surface current M m

Surface Magnetisation Current Density magnitude  M = M but for a vector must also determine its direction  M is perpendicular to both M and the surface normal 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. M MM

Surface Magnetisation Current Density Solenoid in vacuum With magnetic core (red), Ampere’s Law integration contour encloses two types of current, “conduction current” in the coils and “magnetisation current” on the surface of the core  > 1:  M and I in same direction (paramagnetic)  < 1:  M and I in opposite directions (diamagnetic)  is the relative permeability, c.f.  the relative permittivity Substitute for  M B I L

Magnetisation Macroscopic electric field E Mac = E Applied + E Dep = E - P  o Macroscopic magnetic field B Mac = B Applied + B Magnetisation B Magnetisation is the contribution to B Mac from the magnetisation B Mac = B Applied + B Magnetisation = B +  o M Define magnetic susceptibility via M =  B B Mac /  o B Mac = B +  B B Mac E Mac = E - P  o = E - E Mac B Mac (1-  B ) = B E Mac (1+  ) = E Diamagnets B Magnetisation opposes B Applied  B < 0 Para, Ferromagnets B Magnetisation enhances B Applied  B > 0  B  Au Quartz O 2 STP

Magnetisation Rewrite B Mac = B +  o M as B Mac -  o M = B LHS contains only fields inside matter, RHS fields outside Magnetic field intensity, H = B Mac /  o - M = B/  o = B Mac /  o -  B B Mac /  o = B Mac (1-  B ) /  o = B Mac /  o c.f. D =  o E Mac + P =  o  E Mac  = 1/(1-  B )  = 1 +  Relative permeability Relative permittivity

Non-uniform Magnetisation Rectangular slab of material with M directed along y-axis M increases in magnitude along x-axis Individual loop currents increase from left to right There is a net current along the z-axis Magnetisation current density z x MyMy I 1 I 2 I 3 I 1 -I 2 I 2 -I 3

Non-uniform Magnetisation Consider 3 identical element boxes, centres separated by dx If the circulating current on the central box is Then on the left and right boxes, respectively, it is dx

Non-uniform Magnetisation Magnetisation 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

Non-uniform Magnetisation Rectangular slab of material with M directed along x-axis M increases in magnitude along y-axis z x MyMy I 1 I 2 I 3 I 1 -I 2 I 2 -I 3 z y -M x x Total magnetisation current || z Similar analysis for x, y components yields

Types of Current j Polarisation current density from oscillation of charges as electric dipoles Magnetisation current density from space/time variation of magnetic dipoles M = sin(ay) k k i j j M = curl M = a cos(ay) i Total current

Magnetic Field Intensity H Recall Ampere’s Law Recognise two types of current, free and bound

Magnetic Field Intensity H  D/  t is displacement current postulated by Maxwell (1862) to exist in the gap of a charging capacitor In vacuum D =  o E and displacement current exists throughout space

Boundary conditions on B, H 1 2 B 1 B 2 22 11 S For LIH magnetic media B =  o H (diamagnets, paramagnets, not ferromagnets for which B = B(H)) 1 2 H2H2 H1H1 22 11 dℓ1dℓ1 dℓ2dℓ2 C AB I enclfree

Boundary conditions on B, H

Energy density in magnetic fields dℓdℓ daj

Energy density in magnetic fields