Magnetic vector potential For an electrostatic field We cannot therefore represent B by e.g. the gradient of a scalar since Magnetostatic field, try B is unchanged by

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 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

Torque on magnetic moment F F L/2 d   L/2 I B   F r r T F m L

Origin of permanent magnetic dipole moment non-zero net angular momentum of electrons Includes both orbit and spin Derive general expression via circular orbit of one electron radius: a charge: -e mass: m e speed: v ang. freq:  ang. momentum: L dipole moment: m Similar expression applies for spin. I a -e

Origin of permanent magnetic dipole moment Consider directions: m and L have opposite sense In general an atom has total magnetic dipole moment: ℓ quantised in units of h-bar, introduce Bohr magneton m L -e

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) If B perp to orbit (up), extra inwards Lorentz force: Approx: radius unchanged, ang. freq increased from  o to  -e +Ze -e B

Larmor frequency (  L ) balance equation when B ≠ 0 (mass) x (accel) = (electric force) + (extra force)  L is known as the Larmor frequency

Classical model for diamagnetism 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

Induced dipole moment 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

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 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 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 B appl B dip

Langevin analysis of paramagnetism As with polar dielectric media, the field B in the expressions should be the local field B loc but generally find B loc ≈ B

Uniform magnetisation Electric polarisation Magnetic polarisation 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 but note 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 volume, leaving net 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

Solenoid with magnetic core Recap, vacuum solenoid: With magnetic core (red), Ampere’s Law encloses two types of current, “conduction current” in the coils and“magnetisation current” on the surface of material:  r > 1:  M and I in same direction (paramagnetic)  r < 1:  M and I in opposite directions (diamagnetic) Substitute for  M :(see later) I L

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 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 z x MyMy I 1 I 2 I 3 I 1 -I 2 I 2 -I 3

Neighbouring elemental boxes 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

Upward and circulating currents 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

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 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

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

Ampere’s Law for H Often more useful to apply Ampere’s Law for H than for B Bound current in magnetic moments of atoms Free current in conduction currents in external circuits or metallic magnetic media IfIf L IfIf L IbIb

Magnetic Susceptibility  B Two definitions of magnetic susceptibility First M =  B B/  o is analogous to P =  o  E E B, field due to all currents, E, field due to all charges  B  r Au Quartz O 2 STP In this definition the diamagnetic susceptibility is negative and the relative permeability is less than unity c.f. D =  r  o E

Magnetic Susceptibility  M Second definition not analogous to P =  o  E E When  is much less than unity (all except ferromagnets) the two definitions are roughly equivalent B(T) H Am Para-, diamagnets Ferromagnet  ~ for Fe Hysteresis and energy dissipation

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

Faraday’s Law S B E dℓdℓ

I B(r)B(r) To establish steady current, cell must do work against Ohmic losses and to create magnetic field

Energy density in magnetic fields dℓdℓ daj

Energy density in magnetic fields

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 =  o jwhere j = j f + j M 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  induced in a circuit equals the rate of change of magnetic flux through the circuit dSdS B dℓdℓ

Displacement current Ampere’s Law Problem! 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 Continuity equation

Extending Ampere’s Law add term to LHS such that taking Div makes LHS also identically equal to zero: The extra term is in the bracket extended Ampere’s Law

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