Physics 504, equations, and J

To review, in our original presentation of Maxwell's

equations, all and Jall represented all charges, both "free" and "bound". Upon separating them, "free" from

"bound", we have (dropping quadripole terms):

For the electric field

E called electric field

P called electric polarization is induced field

D called electric displacement is field of "free charges"

D = 0E + P For the magnetic field

B called magnetic induction (unfortunately)

M called magnetization is the induced field

H called magnetic field

H

=

1 ?0

B

-M

Then the two Maxwell equations with sources, Gauss for

E and Amp`ere, get replaced by

?D =

D

?H -

=J

t

Physics 504, Spring 2011 Electricity

and Magnetism

Shapiro

Poynting, Energy and Momentum in the fields, T? Poynting's Theorem

Dispersion Harmonic Fields

Energy in the Fields

The rate of work done by E&M fields on charged particles is:

qjvj ? E(xj, t),

or if we describe it by current density,

J ? E.

V

This must be the rate of loss of energy U in the fields

themselves, so

dU - = J ? E.

dt V

Now by Amp`ere's Law,

Physics 504, Spring 2011 Electricity

and Magnetism

Shapiro

Poynting, Energy and Momentum in the fields, T? Poynting's Theorem

Dispersion Harmonic Fields

D

J?E = ?H-

? E.

t

From the product rule and the cyclic nature of the triple product, we have for any vector fields

? (V ? W ) = W ? ( ? V ) - V ? ( ? W ),

so we may rewrite ? H ? E as

B - ? E ? H + H ? ? E = - E ? H - H ?

t where we used Faraday's law. Thus

D

B

? E ? H + E ? + H ? + J ? E = 0.

t

t

Let us assume the medium is linear without dispersion in electric and magnetic properties, that is B H and D E. Then let us propose that the energy density of the fields is

1 u(x, t) = E ? D + B ? H ,

2

Physics 504, Spring 2011 Electricity

and Magnetism

Shapiro

Poynting, Energy and Momentum in the fields, T? Poynting's Theorem

Dispersion Harmonic Fields

we have

u + J ? E + ? E ? H = 0.

t

As this is true for any volume, we may interpret this

equation, integrated over some volume V with surface V

as saying that the rate of increase in the energy in the

fields plus the energy of the charged particles plus the flux

of energy out of the volume is zero, that is, no energy is

created or destroyed. The flux of energy is then given by

the Poynting vector

Physics 504, Spring 2011 Electricity

and Magnetism

Shapiro

Poynting, Energy and Momentum in the fields, T? Poynting's Theorem

Dispersion Harmonic Fields

S = E ? H.

We have made assumptions which only fully hold for the vacuum, as we assumed linearity and no dispersion (the ratio of D to E independent of time).

Linear Momentum

So as not to worry about such complications, let's restrict our discussion to the fundamental description, or alternatively take our medium to be the vacuum, with the fields interacting with distinct charged particles (mj, qj at xj (t)). The mechanical linear momentum in some region of space is Pmech = j mj xj , so

dPmech = dt

Fj = qj E(xj) + vj ? B(xj)

j

j

= E + J ? B,

V

provided no particles enter or leave the region V . Let us postulate that the electromagnetic field has a linear momentum density

1 g = c2 E ? H = 0E ? B.

Physics 504, Spring 2011 Electricity

and Magnetism

Shapiro

Poynting, Energy and Momentum in the fields, T? Poynting's Theorem

Dispersion Harmonic Fields

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