r2

where b and c are constants to be determined from the boundary conditions

at r = a:

(i) φ is continuous:

φcav (a) = φmed (a); (13.79)

13.7 Quasi-stationary electrostatics 347

(ii) D is continuous in the radial direction:

dφcav dφmed

µ0 =µ . (13.80)

dr dr

a a

Applying these conditions gives straightforwardly

μ 2(µr ’ 1)

b=’ . (13.81)

4πµ0 a3 2µr + 1

The term br cos θ = bz in the cavity potential is simply the potential of

a homogeneous electric ¬eld in the z-direction. This is the reaction ¬eld

resulting from the polarization in the medium, in the direction of the dipole,

and with magnitude ’b:

μ 2(µr ’ 1)

ERF = (13.82)

4πµ0 a3 2µr + 1

The energy of the dipole in the reaction ¬eld is

μ2 2(µr ’ 1)

1

= ’ μERF = ’

URF . (13.83)

8πµ0 a3 2µr + 1

2

We can now specify the potential anywhere in space:

r 2(µr ’ 1)

μ cos θ 1

’3 (r ¤ a),

φcav (r) = (13.84)

r2 a 2µr + 1

4πµ0

μ cos θ 3

(r ≥ a).

φmed (r) = (13.85)

r2 2µr + 1

In the presence of ions in the medium, a similar reasoning can be followed

as we used in deriving (13.72) for a single charge. The result is

μ 2(µ ’ 1)

ERF = , (13.86)

4πµ0 a3 2µ + 1

with

κ2 a2

µ = µr 1+ . (13.87)

2(1 + κa)

The e¬ect of the ionic strength is to increase the e¬ective dielectric constant

and therefore the screening of the dipolar ¬eld, just as was the case for

the reaction potential of a charge. But the extra screening has a di¬erent

dependence on κ.

Let us ¬nally consider “ both for a charge and for a dipole “ the special

case that either µ = ∞ or κ = ∞ (these apply to a conducting medium). In

348 Electromagnetism

this case

q

φRP = ’ , (13.88)

4πµ0 a

μ

= . (13.89)

E RF

4πµ0 a3

The potential and the ¬eld of the source now vanish at the cavity boundary

and are localized in the cavity itself.

13.7.4 Charge distribution in a medium

Finally, we consider a charge distribution in a spherical cavity (µ0 ) of radius

a, centered at the origin of the coordinate system, embedded in a homo-

geneous dielectric environment with µ = µr µ0 and possibly with an inverse

Debye length κ. There are charges qi at positions r i within the cavity

(ri < a). There are two questions to be answered:

(i) what is the reaction potential, ¬eld, ¬eld gradient, etc., in the cen-

ter of the cavity? This question arises if Coulomb interactions are

truncated beyond a cut-o¬ radius rc and one wishes to correct for

the in¬‚uence of the environment beyond rc (see Section 6.3.5 on page

164);

(ii) what is the reaction ¬eld due to the environment at any position

within the sphere? This question arises if we wish to ¬nd energies

and forces of a system of particles located within a sphere but em-

bedded in a dielectric environment, such as in molecular dynamics

with continuum boundary conditions (see Section 6.2.2 on page 148).

The ¬rst question is simple to answer. As is explained in the next section

(Section 13.8), the potential outside a localized system of charges can be

described by the sum of the potentials of multipoles, each localized at the

center of the system of charges, i.e., at the center of the coordinate system.

The simplest multipole is the monopole Q = i qi ; it produces a reaction

potential ¦RP at the center given by (13.72), replacing the charge in the

center by the monopole charge. The reaction potential of the monopole is

homogeneous and there are no reaction ¬elds. The next multipole term is

the dipole μ = i qi r i , which leads to a reaction ¬eld ERF at the center

given by (13.86) and (13.87). The reaction ¬eld is homogeneous and there

is no reaction ¬eld gradient. Similarly, the quadrupole moment of the dis-

tribution will produce a ¬eld gradient at the center, etc. When the system

is described by charges only, one needs the ¬elds to compute forces, but

¬eld gradients are not needed and higher multipoles than dipoles are not

13.7 Quasi-stationary electrostatics 349

s qim = ’ (µr ’1) a q

T (µr +1) s

a2 /s θ

T

sq

a

sT

r

µ0 µ = µr µ 0

Figure 13.2 A source charge q in a vacuum sphere, embedded in a dielectric medium,

produces a reaction potential that is well approximated by the potential of an image

charge qim situated outside the sphere.

required. However, when the system description involves dipoles, one needs

the ¬eld gradient to compute forces on the dipole and the reaction ¬eld

gradient of the quadrupole moment of the distribution would be required.

The second question is considerably more complicated. In fact, if the sys-

tem geometry is not spherical, full numerical Poisson (or linearized Poisson-

Boltzmann) solutions must be obtained, either with ¬nite-di¬erence methods

on a grid or with boundary-element methods on a triangulated surface. For

a charge distribution in a sphere (µ0 ), embedded in a homogeneous dielectric

environment (µ = µr µ0 ), Friedman (1975) has shown that the reaction ¬eld

is quite accurately approximated by the Coulomb ¬eld of image charges out-

side the sphere. The approximation is good for µr 1. We shall not repeat

the complete derivation (which is based on the expansion of the potential of

an excentric charge in Legendre polynomials) and only give the results.

Consider a charge q, positioned on the z-axis at a distance s from the

center of a sphere with radius a (see Fig. 13.2). The potential inside the

sphere at a point with spherical coordinates (r, θ) is given by the direct

Coulomb potential of the charge plus the reaction potential ¦R (r, θ):

∞

q n+1 rs n

(1 ’ µr )