and the magnetic e¬ects of net currents are also negligible. The relations

now simplify considerably. There are no magnetic ¬elds and waves are no

longer supported. The result is called electrostatics, although slow time

dependence is not excluded.

13.7 Quasi-stationary electrostatics 341

13.7.1 The Poisson and Poisson“Boltzmann equations

Since the curl of the electric ¬eld E is now zero, E can be written as a pure

gradient of a potential

E = ’ grad φ. (13.45)

In a linear dielectric medium, where the dielectric constant may still depend

on position, we obtain from (13.15) and (13.17) the Poisson equation

div (µ grad φ) = ’ρ, (13.46)

which simpli¬es in a homogeneous dielectric medium to

ρ

∇2 φ = ’ . (13.47)

µ

Here, ρ is the density of free charges, not including the “bound” charges due

to divergence of the polarization. We drop the index 0 in ρ0 , as was used

previously.

In ionic solutions, there is a relation between the charge density and the

potential. Assume that at a large distance from any source terms, where

the potential is zero, the electrolyte contains bulk concentrations c0 of ionic

i

2 Electroneu-

species i, which have a charge (including sign) of zi e per ion.

trality prescribes that

c0 zi = 0. (13.48)

i

i

In a mean-¬eld approach we may assume that the concentration ci (r) of

each species is given by its Boltzmann factor in the potential φ(r):

zi eφ

ci (r) = c0 exp ’ , (13.49)

i

kTB

so that, with

ρ=F ci z i , (13.50)

i

where F is the Faraday constant (96 485.338 C), we obtain the Poisson“

Boltzmann equation for the potential:

zi F φ

div (µ grad φ) = ’F c0 zi exp ’ . (13.51)

i

RT

i

In the Debye“H¨ckel approximation the exponential is expanded and only

u

2 Note that concentrations must be expressed in mol/m3 , if SI units are used.

342 Electromagnetism

the ¬rst two terms are kept.3 Since the ¬rst term cancels as a result of

the electroneutrality condition (13.48), the resulting linearized Poisson“

Boltzmann equation is obtained:

F2 02

i ci z i

div (µ grad φ) = φ. (13.52)

RT

The behavior depends only on the ionic strength of the solution, which is

de¬ned as

1 02

I= i ci z i . (13.53)

2

Note that for a 1:1 electrolyte the ionic strength equals the concentration. In

dielectrically homogeneous media the linearized Poisson“Boltzmann equa-

tion gets the simple form

∇2 φ = κ2 φ, (13.54)

where κ is de¬ned by

2IF 2

2

κ= . (13.55)

µRT

The inverse of κ is the Debye length, which is the characteristic distance over

which a potential decays in an electrolyte solution.

The Poisson equation (13.46) or linearized Poisson“Boltzmann equation

(13.52) can be numerically solved on a 3D grid by iteration.4 In periodic

systems Fourier methods can be applied and for arbitrarily shaped systems

embedded in a homogeneous environment, boundary element methods are

available.5 The latter replace the in¬‚uence of the environment by a boundary

layer of charges or dipoles that produce exactly the reaction ¬eld due to the

environment. For simple geometries analytical solutions are often possible,

and in Sections 13.7.2, 13.7.3 and 13.7.4 we give as examples the reaction

potentials and ¬elds for a charge, a dipole and a charge distribution in a

sphere, embedded in a polarizable medium. Section 13.7.5 describes the

generalized Born approximation for a system of embedded charges.

The following boundary conditions apply at boundaries where the dielec-

tric properties show a stepwise change. Consider a planar boundary in the

x, y-plane at z = 0, with µ = µ1 for z < 0 and µ = µ2 for z > 0, without free

3 For dilute solutions the Debye“H¨ ckel approximation is appropriate. In cases where it is not

u

appropriate, the full Poisson“Boltzmann equation is also not adequate, since the mean-¬eld

approximation also breaks down. Interactions with individual ions and solvent molecules are

then required. In general, when κ’1 (see (13.55)) approaches the size of atoms, the mean-¬eld

approximation will break down.

4 A popular program to solve the PB equation on a grid is DELPHI, see Nicholls and Honig

(1991).

5 Ju¬er et al. (1991).

13.7 Quasi-stationary electrostatics 343

z

T T T T D2z

E2x

2 µ2 φ2 (0)

E T

E

TTTT

1 µ1 φ1 (0)

E1x D1z

Figure 13.1 Boundary conditions on a planar discontinuity: potential φ, tangential

electric ¬eld Ex and perpendicular displacement Dz are continuous.

charge density at the surface (Fig. 13.1). Since the potential is continuous

over the surface, its derivative along the surface, i.e., the component of the

electric ¬eld along the surface, is also continuous:

Ex (z ‘ 0) = Ex (z “ 0),

Ey (z ‘ 0) = Ey (z “ 0). (13.56)

Since there is no free charge at the boundary, there is no source for D, and

hence div D = 0. This implies that the ingoing ¬‚ux of D through one plane

of a cylindrical box equals the outgoing ¬‚ux through the other plane at the

other side of the boundary. Hence

Dz (z ‘ 0) = Dz (z “ 0). (13.57)

13.7.2 Charge in a medium

Our ¬rst example is a charge q in a cavity (µ0 ) of radius a in a homogeneous

dielectric environment with µ = µr µ0 . According to (13.25), the dielectric

displacement at a distance r > a equals

qr

D(r) = . (13.58)

4πr2 r

The ¬eld energy outside the cavity is

∞

q2

1 2 2

U¬eld = D (r)4πr dr = . (13.59)

2µ 8πµa

a