B = curl A. (13.29)

338 Electromagnetism

The de¬nitions are not unique: the physics does not change if we replace φ by

φ’‚f /‚t and simultaneously A by A’ grad f , where f is any di¬erentiable

function of space and time. This is called the gauge invariance. Therefore

the divergence of A can be chosen at will. The Lorentz convention is

1 ‚φ

div A + = 0, (13.30)

c2 ‚t

implying (in a vacuum) that

1 ‚2

∇ ’ 2 2 φ = ’ρ0 /µ0 ,

2

(13.31)

c ‚t

1 ‚2

∇ ’ 2 2 A = ’μ0 j 0 .

2

(13.32)

c ‚t

13.5 Waves

The Maxwell equations support waves with the velocity of light in vacuum,

as can be seen immediately from (13.31) and (13.32). For example, a linearly

polarized electromagnetic plane wave in the direction of a vector k, with

wave length 2π/k and frequency ω/2π, has an electric ¬eld

E(r, t) = E 0 exp[i(k · r ’ ωt)], (13.33)

where E 0 (in the polarization direction) must be perpendicular to k, and a

magnetic ¬eld

1

B(r, t) = k — E(r, t). (13.34)

ω

The wave velocity is

ω

= c. (13.35)

k

The wave can also be represented by a vector potential:

i

A(r, t) = E(r, t). (13.36)

ω

The scalar potential φ is identically zero. Waves with A = 0 cannot exist.

The vector

Σ=E—H (13.37)

is called the Poynting vector. It is directed along k, the direction in which

the wave propagates, and its magnitude equals the energy ¬‚ux density, i.e.,

the energy transported by the wave per unit of area and per unit of time.

13.6 Energies 339

13.6 Energies

Electromagnetic ¬elds “contain” and “transport” energy. The electromag-

netic energy density W of a ¬eld is given by

W = 1 D · E + 1 B · H. (13.38)

2 2

In vacuum or in a linearly polarizable medium with time-independent per-

mittivities, for which D = µE and B = μH, the time dependence of W

is

dW

= ’ div Σ ’ j · E. (13.39)

dt

Proof Using the time-independence of µ and μ, we can write

™ ™ ™

W =E·D+H ·B

(to prove this in the case of tensorial permittivities, use must be made of

™

the fact that µ and μ are symmetric tensors). Now we can replace D by

™

curl H ’j (see (13.20)) and B by ’ curl E (see (13.19)) and use the general

vector equality

div (A — B) = B · curl A ’ A · curl B. (13.40)

Equation (13.39) follows.

Equation (13.39) is an energy-balance equation: ’ div Σ is the energy

¬‚owing out per unit volume due to electromagnetic radiation; ’j · E is

the energy taken out of the ¬eld by the friction of moving charges and

dissipated into heat (the Joule heat). Note that this equation is not valid if

permittivities are time-dependent.

For quasi-stationary ¬elds, where radiation does not occur, the total ¬eld

energy of a system of interacting charges and currents can also be expressed

in terms of the source densities and potentials as follows:

+ j · A) dr,

1

U¬eld = W (r) dr = 2 (ρφ (13.41)

where integration of W is over all space, while the sources ρ and j are

con¬ned to a bounded volume in space.

Proof Quasi-stationarity means that the following equations are valid:

E = ’ grad φ,

div D = ρ,

B = curl A,

340 Electromagnetism

curl H = j.

Using the ¬rst two equations we see that

D · E dr = ’ D · grad φ dr = φ div D dr = ρφ dr.

The third integral follows from the second by partial integration, whereby

the integral over the (in¬nite) boundary vanishes if the sources are con¬ned

to a bounded volume. Similarly

B · H dr = ’ ( curl A) · H dr = A · curl H dr = j · A dr.

The reader is invited to check the partial integration result by writing the

integrand ount in all coordinates.

This equation is often more convenient for computing energies than the

¬eld expression (13.38).

Both expressions contain a self-energy for isolated charges that becomes

singular for delta-function point charges. This self-energy is not taken into

account if the Coulomb interaction energy between a system of point charges

is considered. Thus for a set of charges qi at positions r i , the total interaction

energy is

’ φself ),

1

Uint = i qi (φi (13.42)

i

2

where

qj

1

φi ’ φself = . (13.43)

|r j ’ r i |

i

4πµ

j=i

This is indeed the usual sum of Coulomb interactions over all pairs:

qi qj

1

Uint = . (13.44)

|r j ’ r i |

4πµ

i<j

1

The factor in (13.42) compensates for the double counting of all pairs.

2

13.7 Quasi-stationary electrostatics

In the vast majority of molecular systems of interest, motions are slow