what is the macroscopic (average) electric field inside matter when an external e field is applied?...

• 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

Dr(r) Change in electronic charge density

Note dipolar character

r

No E fieldE field on

- +

r(r) Electronic charge density

Electrostatic potential of point dipole• +/- charges, equal magnitude, q, separation a• axially symmetric potential (z axis)

-or1

r1

4

q)(

r

potential dipole point''

r4

cos p

r

cos

4

qa

cos2r

a

r

1

cos r

a

2r

a1

r

1

r

1

cos r

a

2r

a1r

cos r a2

arr

2o

2o

2

21

2

22

222

r

a/2

a/2

r+

r-rq+

q-

x

z

p

q

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

space all

el )d( Ze rr

0 if d )(

d )()( Ze a Ze

nucspace all

el

space all

elnucelnuc

rrrr

rrrrrr

space all

nuc )d( Ze rr

space all

el/nuc

space all

el/nuc

el/nuc )d(

d )(

rr

rrr

r

Electric Polarisation

Electric field in model 1-D crystal with lattice spacing ‘a’

r(x)

x5 1 0 1 5 2 0

0 .2

0 .4

0 .6

0 .8

1 .0

a

symmetry by b a

0 electrons) (nucleidensity average a

a

x2nsin b

a

x2ncos a a(x)

(x)density charge of expansion series Fourier

nn

o

nno

00


Expand electric field Ex in same way (Ey, Ez = 0 by symmetry)

r(x)

x5 1 0 1 5 2 0

0 .2

0 .4

0 .6

0 .8

1 .0

a

1n 2n

aa d

a d

a

2n

a

x2ncos

a

a

x2ncos d

a

2n

dx

dE(x).

d symmetry by c field ic)(macroscop average c

a

x2nsin d

a

x2ncos c cE(x)

nn

nn

n

n

nno

nno

oooo

el

00

E

Electric PolarisationApply external electric field and polarise charge density

r(x)

x5 1 0 1 5 2 0

0 .2

0 .4

0 .6

0 .8

1 .0

a

- - - - - - -

b c.f. b

aa 0 aa

a

x2nsin b

a

x2ncos a a(x)

charge dunpolarise forsymmetry c.f.

symmetry has charge onPolarisati

n'n

n'no

'o

'n

'n

'o

00

'

even

odd

E

Electric PolarisationApply external electric field and polarise charge density

r(x)

x5 1 0 1 5 2 0

0 .2

0 .4

0 .6

0 .8

1 .0

a

- - - - - - -

fieldc macroscopi the provide charges surface These

ends the at charges onpolarisati inducednow are there But

equation sPoisson' fromc a Since

dd c c.f. c

a

x2nsin d

a

x2ncos c c(x)E

beforesymmetry c.f.symmetry has charge onPolarisati

'o

'o

n'nn

'n

'n

'n

'o

0,0

00

'

evenodd

E

• Polarisation P, dipole moment p per unit volume Cm/m3 = Cm-2

• Mesoscopic averaging: P is a constant vector field for a uniformly polarised medium

• Macroscopic charges are induced with areal density sp Cm-2 in a uniformly polarised medium


p E

P E

P- + E

P.n

• 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


s- s+

E

P

s- s-

Electric Polarisation• Apply Gauss’ Law to right and left ends of polarised dielectric

• EDep = ‘Depolarising field’

• Macroscopic electric field EMac= E + EDep = E - P/o

E+2dA = s+dA/o

E+ = s+/2o

E- = s-/2o

EDep = E+ + E- = (s++ s-)/2o

EDep = -P/o P = s+ = s-

s-

E

P s+

E+E-

Electric PolarisationDefine dimensionless dielectric susceptibility c through

P = o c EMac

EMac = E – P/o

o E = o EMac + Po E = o EMac + o c EMac = o (1 + c)EMac = oEMac

Define dielectric constant (relative permittivity) = 1 + c

EMac = E / E = e EMac

Typical values for e: silicon 11.8, diamond 5.6, vacuum 1Metal: e →Insulator: e (electronic part) small, ~5, lattice part up to 20

Electric PolarisationRewrite EMac = E – P/o as

oEMac + P = oE

LHS contains only fields inside matter, RHS fields outside

Displacement field, D

D = oEMac + P = o EMac = oE

Displacement field defined in terms of EMac (inside matter,

relative permittivity e) 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


Charge entering xz face at y = 0: Px=0DyDz

Charge leaving xz face at y = Dy: Px=DxDyDz

= (Px=0 + ∂Px/∂x Dx) DyDz

Net charge entering cube: (Px=0 - Px=Dx ) DyDz = -∂Px/∂x DxDyDz

Dx

Dz

Dy

z

y

x

Charge entering cube via all faces:

-(∂Px/∂x + ∂Py/∂y + ∂Pz/∂z) DxDyDz = Qpol

rpol = lim (DxDyDz)→0 Qpol /(DxDyDz)

-.P = rpol

Px=DxPx=0


Differentiate -.P = rpol wrt time

.∂P/∂t + ∂rpol/∂t = 0

Compare to continuity equation .j + ∂r/∂t = 0

∂P/∂t = jpol

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

rtot = rpol + rfree


Inside matter

.E = .Emac = rtot/o = (rpol + rfree)/o

Total (averaged) electric field is the macroscopic field

-.P = rpol

.(oE + P) = rfree

.D = rfree

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 Drelation D = eoE + P

• Limited validity: Expressions involving e and

• Have assumed that is a simple number: P = eo Eonly true in LIH media:

• Linear: independent of magnitude of E interesting media “non-linear”: P = eoE + 2

eoEE + ….

• Isotropic: independent of direction of E interesting media “anisotropic”: is a tensor (generates vector)

• Homogeneous: uniform medium (spatially varying e)

Boundary conditions on D and E

D and E fields at matter/vacuum interface

matter vacuum

DL = oLEL = oEL + PL DR = oRER = oER R = 1

No free charges hence .D = 0

Dy = Dz = 0 ∂Dx/∂x = 0 everywhere

DxL = oLExL = DxR = oExR

ExL = ExR/L

DxL = DxR E discontinuous

D continuous


Non-normal D and E fields at matter/vacuum interface.D = rfree Differential form ∫ D.dS = rfree, enclosed Integral form

∫ D.dS = 0 No free charges at interface

DL = oLEL

DR = oRER

dSR

dSL

qL

qR

-DL cosqL dSL + DR cosqR dSR = 0

DL cosqL = DR cosqR

D┴L = D┴R No interface free charges

D┴L - D┴R = sfree Interface free charges


Non-normal D and E fields at matter/vacuum interface

Boundary conditions on E from ∫ E.dℓ = 0 (Electrostatic fields)

EL.dℓL + ER.dℓR = 0

-ELsinqLdℓL + ERsinqR dℓR = 0

ELsinqL = ERsinqR

E||L = E||R E|| continuous

D┴L = D┴R No interface free charges

D┴L - D┴R = sfree Interface free charges

EL

ER

qL

qRdℓL

dℓR


DL = oLEL

DR = oRER

dSR

dSL

qL

qR

interface at charges free of absence in tan

tan

cos E

sinE

cos E

sinE

cos D

sinE

cos D cos D

sinE sinE

R

L

R

L

RRR

RR

LLL

LL

LL

LL

L/RL/RL/R

RRLL

RRLL

oo

oED

what is the macroscopic (average) electric field inside matter when an external e field is applied?...

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