What is the macroscopic (average) electric field inside matter when an external E field is applied? How is charge displaced when an electric field is applied? i.e. what are induced currents and densities What is the electric energy density inside matter? How do we relate these properties to quantum mechanical treatments of electrons in matter? Dielectrics

Microscopic picture of atomic polarisation in E field Change in charge density when field is applied Dielectrics E  (r) Change in electronic charge density Note dipolar character r No E field E field on - +  (r) Electronic charge density

Electrostatic potential of point dipole +/- charges, equal magnitude, q, separation a axially symmetric potential (z axis) a/2 r+r+ r-r- r q+q+ q-q- x z p 

Dipole Moments of Atoms Total electronic charge per atom Z = atomic number Total nuclear charge per atom Centre of mass of electric or nuclear charge distribution Dipole moment p = Zea

Electric Polarisation Electric field in model 1-D crystal with lattice spacing ‘a’  (x) x a

Electric Polarisation Expand electric field E x in same way (E y, E z = 0 by symmetry)  (x) x a

Electric Polarisation Apply external electric field and polarise charge density  (x) x a E

Electric Polarisation Apply external electric field and polarise charge density  (x) x a E

Polarisation P, dipole moment p per unit volume Cm/m 3 = Cm -2 Mesoscopic averaging: P is a constant vector field for a uniformly polarised medium Macroscopic charges are induced with areal density  p Cm -2 in a uniformly polarised medium Electric Polarisation p E P E P - + E

Contrast charged metal plate to polarised dielectric Polarised dielectric: fields due to surface charges reinforce inside the dielectric and cancel outside Charged conductor: fields due to surface charges cancel inside the metal and reinforce outside Electric Polarisation -- ++ E P -- --

Apply Gauss’ Law to right and left ends of polarised dielectric E Dep = ‘Depolarising field’ Macroscopic electric field E Mac = E + E Dep = E - P  o E + 2dA =  + dA  o E + =  +  o E - =  -  o E Dep = E + + E - = (  +  -  o E Dep = -P/  o P =  + =  - -- E P ++ E+E+ E-E-

Electric Polarisation Define dimensionless dielectric susceptibility  through P =  o  E Mac E Mac = E – P/  o  o E =  o E Mac + P  o E =  o E Mac +  o  E Mac =  o (1 +  )E Mac =  o  E Mac Define dielectric constant (relative permittivity)  = 1 +  E Mac = E /  E =  E Mac Typical values for  : silicon 11.8, diamond 5.6, vacuum 1 Metal:  →  Insulator:   (electronic part) small, ~5, lattice part up to 20

Electric Polarisation Rewrite E Mac = E – P/  o as  o E Mac + P =  o E LHS contains only fields inside matter, RHS fields outside Displacement field, D D =  o E Mac + P =  o  E Mac =  o E Displacement field defined in terms of E Mac (inside matter, relative permittivity  ) and E (in vacuum, relative permittivity 1). Define D =  o  E where  is the relative permittivity and E is the electric field

Non-uniform polarisation Uniform polarisation  induced surface charges only Non-uniform polarisation  induced bulk charges also Displacements of positive charges Accumulated charges ++-- P - + E

Non-uniform polarisation Charge entering xz face at y = 0: P x=0  y  z Charge leaving xz face at y =  y: P x=  x  y  z = (P x=0 + ∂P x /∂x  x)  y  z Net charge entering cube: (P x=0 - P x=  x )  y  z = -∂P x /∂x  x  y  z xx zz yy z y x Charge entering cube via all faces: - (∂P x /∂x + ∂P y /∂y + ∂P z /∂z)  x  y  z = Q pol  pol = lim (  x  y  z)→0 Q pol /(  x  y  z) - .P =  pol P x=  x P x=0

Non-uniform polarisation Differentiate - .P =  pol wrt time .∂P/∂t + ∂  pol /∂t = 0 Compare to continuity equation .j + ∂  /∂t = 0 ∂P/∂t = j pol Rate of change of polarisation is the polarisation-current density Suppose that charges in matter can be divided into ‘bound’ or polarisation and ‘free’ or conduction charges  tot =  pol +  free

Non-uniform polarisation Inside matter .E = .E mac =  tot /  o = (  pol +  free )/  o Total (averaged) electric field is the macroscopic field - .P =  pol .(  o E + P) =  free .D =  free Introduction of the displacement field, D, allows us to eliminate polarisation charges from any calculation

Validity of expressions Always valid:Gauss’ Law for E, P and D relation D =  o E + P Limited validity: Expressions involving  and  Have assumed that  is a simple number: P =  o  E only true in LIH media: Linear:  independent of magnitude of E interesting media “non-linear”: P =   o E +  2  o EE + …. Isotropic:  independent of direction of E interesting media “anisotropic”:  is a tensor (generates vector) Homogeneous: uniform medium (spatially varying  )

Boundary conditions on D and E D and E fields at matter/vacuum interface matter vacuum D L =  o  L E L =  o E L + P L D R =  o  R E R =  o E R  R = 1 No free charges hence .D = 0 D y = D z = 0 ∂ D x / ∂ x = 0 everywhere D xL =  o  L E xL = D xR =  o E xR E xL = E xR /  L D xL = D xR E discontinuous D continuous

Boundary conditions on D and E Non-normal D and E fields at matter/vacuum interface .D =  free Differential form ∫ D.dS =  free, enclosed Integral form ∫ D.dS = 0 No free charges at interface D L =  o  L E L D R =  o  R E R dSRdSR dSLdSL LL RR -D L cos  L dS L + D R cos  R dS R = 0 D L cos  L = D R cos  R D ┴ L = D ┴ R No interface free charges D ┴ L - D ┴ R =  free Interface free charges

Boundary conditions on D and E Non-normal D and E fields at matter/vacuum interface Boundary conditions on E from ∫ E.dℓ = 0 (Electrostatic fields) E L.dℓ L + E R.dℓ R = 0 -E L sin  L dℓ L + E R sin  R dℓ R = 0 E L sin  L = E R sin  R E || L = E || R E || continuous D ┴ L = D ┴ R No interface free charges D ┴ L - D ┴ R =  free Interface free charges ELEL ERER LL RR dℓLdℓL dℓRdℓR

Boundary conditions on D and E D L =  o  L E L D R =  o  R E R dSRdSR dSLdSL LL RR