1 Maxwells equations in matter(integrate with next section)

Recall Maxwells equations in general:

(0)E = , B = 0,E = tB, (B/0) = J+ t(0E)

1.1 Free charge.

Maxwells equations in matter divide the net chargedensity into free charge f and bound charge b: = f + b.

We could define Ef to be the electric field causedby the distribution of free charge, define Eb to bethe electric field caused by the distribution of boundcharge, and say that

Ef = f , Eb = b,E = Ef +Eb.

Such a division of the electric field is not very useful,however, because there is not a general, obvious con-stitutive relation specifying Eb in terms of E. Eb isdetermined by the global distribution of free charge,whereas a constitutive relation naturally relates lo-cal, physical quantities.

We need a quantity which locally defines the re-sponse to an overall electromagnetic field due to re-arrangement of the bound charges. This quantity isP, the polarization or the dipole moment per unitvolume. The average polarization within a small re-gion is minus (0 times) the average electric field inthat region resulting from rearrangement of chargesoriginally lying within the region. [Really need toshow that this is well-defined.]

We assume that b = P, where P is called thepolarization or the dipole moment per unit volume.

1.2 Polarization.

The dipole moment of a pair of equal and oppositecharges q equals the charge separation p = qs,

where s is the displacement vector from the negativeto the positive charge. More generally, polarizationis charge separation per unit volume. P n is thesurface charge density due to charge separation; ap-plying Gausss theorem to the charge that remainsinside a body gives that b = P. Note thatthe polarization does not generally equal the electricfield that would be caused by the bound charge inan electrostatic environment, because the curl of thedipole moment is not necessarily zero.

1.3 Electric displacement

Since b = f ,(0E+P) = f , i.e. (D) = f ,where D := (0E+P) is the electric displacement.

1.4 Linear medium

In a linear medium we assume that the polarization,and hence the electric displacement, is a linear func-tion of the electric field:

P = E (0E),

where E

is called the electric susceptibility (zero in

a vacuum). So

D = E,

where := 0(I+E

) is called the permittivity of the

material (0 in a vacuum).

2 Maxwells equations in matter(integrate with previous sec-tion)

Names of quantities:

P =: polarizationM =: magnetization = volume density of mag-netic momentb =: bound charge densityJb =: bound current density

Definitions of auxiliary fields:

D := (0E)+P = electric field produced by freecharges

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H := (B/0) M = magnetic field strength(versus magnetic induction B).

Definitions of bound charge and current:

b := PJb := M+ tP

Definition of free charge and current:

f + b := Jf + Jb := J

Using these definitions in the microscopic inhomoge-nous Maxwell equations

t(0E) (B/0) = J, (0E) =

to eliminate (0E) and (B/0) converts them to themacroscopic inhomogenous Maxwell equations

tDH = Jf , D = f .

For closure, Maxwell needs you to specify the cur-rent. The macro equations are a framework for spec-ifying a closure for a portion of the current: thebound part.

The generic linear constitutive relation is

D = E,H = 1 B,

where

=: permittivity ,

= 0 (1 + m) =: permeabilityin general are functions of time, position, and wavevector.

The polarization P is minus the electric field due tobound charges (i.e. the electric field canceled by thebound charges). D is the electric field due to freecharges. In general,

P = e (0E),where e is electric susceptibility. Similarly, mag-netic moment is generally given by

M = m H,

where m is volume magnetic susceptibility.

Maxwell specifies Ohms law :

J(k, ) = (k, ) E(k, ).

Micro Maxwell says

tB+E = 0, B = 0,tE c2B = J/0, E = /0,t+ J = 0.

Reference: http://en.wikipedia.org/wiki/Maxwells_equations

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