1

Chapter 4. Electrostatics of Macroscopic Media

4.1 Multipole Expansion

Approximate potentials at large distances

Fig 4.1

We consider the potential in the far-field region (see Fig. 4.1 where | | ) due to a

localized charge distribution ( for ). If the total charge is q, it is a good

approximation to treat the charge distribution as a point charge, so

. Even if q is

zero, the potential does not vanish, but it decays much faster than . We will discuss more

details about how the potential behaves in the far-field region.

Electric dipole

We begin with a simple, yet exceedingly important case of charge distribution. Two equal and

opposite charges separated by a small distance form an electric dipole. Suppose that +q and –q

are separated by a displacement vector d as shown in Figure 4.2, then the potential at x is

(

)

[(

)

(

)

]

x'x

x' x

x

x'd 3

)(x'

a

d

+q

-q

x

r+

r- Fig 4.2. An electric dipole consists of two

equal and opposite charges +q and –q

separated by a displacement d.

(4.1)

2

In the far-field region for | | ,

[(

) (

)]

This reduces to the coordinate independent expression

where is the electric dipole moment. For the dipole p along the z-axis, the electric fields

take the form

{

From this, we can obtain the coordinate independent expression

where is a unit vector.

(4.2)

(4.3)

(4.4)

(4.5)

Fig 4.3. Field of an electric dipole

3

Multipole expansion

We can expand the potential due to the charge distribution

∫

| |

using Eq. 3.68

| | ∑

∑

In the far-field region, . Then we find

∑

[∫

]

We can rewrite the equation

∑

where the coefficients

∫

are called multipole moments. This is the multipole expansion of in powers of . The first

term ( ) is the monopole contribution ( ); the second ( ) is the dipole ( );

the third is quadrupole; and so on.

Monopole moment or total charge q ( √ :

∫

Electric dipole moment p ( linear combinations of ):

∫

Quadrupole moment tensor ( linear combinations of ):

∫(

)

(3.68)

(1.12)

(4.6)

(4.7)

(4.8)

(4.9)

(4.10)

(4.11)

4

The expansion of in rectangular coordinates

[

∑

]

Energy of a charge distribution in an external field

If a localized charge distribution is placed in an external potential , the

electrostatic energy of the system is

∫

If is slowly varying over the region of , we can expand it in a Taylor series

∑

∑

(

)

∑ ( )

Then, the energy takes the form

∑

4.2 Polarization and Electric Displacement in Macroscopic Media

Dielectrics

Properties of an ideal dielectric material

It has no free charges. Instead, all charges are attached to specific atoms or molecules.

Electric fields can induce only small displacements from their equilibrium positions.

In a macroscopic scale, the effects of the electric fields can be visualized as a

displacement of the entire positive charge in the dielectric relative to the negative

charge. The dielectric is said to be polarized.

Electric Polarization

If an electric field is applied to a medium composed of many atoms and molecules, each atom

or molecule forms a dipole pi due to the field induced displacements of the bound charges (see

Fig. 4.4). Typically, this induced dipole moment is approximately proportional to the field:

(4.12)

(4.13)

(4.14)

(4.15)

(4.16)

5

where is called atomic polarizability. These little dipoles are aligned along the direction of

the field, and the material becomes polarized. An electric polarization P is defined as dipole

moment per unit volume:

∑

is a volume element which contains many atoms, yet it is infinitesimally small in the

macroscopic scale. N is the number of atoms per unit volume and is the average dipole

moment of the atoms.

Bound charges

The dipole moment of is , so the total electric potential (see Eq. 4.3) is

∫

| |

We can rewrite this equation as

∫

| |

Integrating by parts gives

{∫ [

| |] ∫

| | }

Using the divergence theorem

{∫

| | ∫

| | }

+

+

++

++

+

+++

dV

pi

i

idV

pP1

E

Fig 4.4. An external electric field

induces electric polarization in a

dielectric medium.

(4.17)

(4.18)

(4.19)

(4.20)

(4.21)

6

where is a surface element and n is the normal unit vector. Here we define surface and

volume charge densities:

and

Then, the potential due to the bound charges becomes

∫

| | ∫

| |

Electric displacement

When a material system includes free charges as well as bound charges , the total charge

density can be written:

And Gauss’s law reads

With the definition of the electric displacement D,

Equation 4.26 becomes

When an averaging is made of the homogeneous equation, , the same equation

holds for the macroscopic, electric field E. This means that the electric field is still derivable

from a potential in electrostatics. Equations 4.28 and 4.29 are the two electrostatic equations in

the macroscopic scale.

+

+

+

+ b

b

(4.22)

(4.23)

(4.24)

Fig 4.5. Origin of bound

charge density.

(4.25)

(4.26)

(4.27)

(4.28)

(4.29)

7

Electric susceptibility, permittivity, and dielectric constant

For many substances (we suppose that the media are isotropic), the polarization is proportional

to the field, provided E is not too strong:

The constant is called the electric susceptibility of the medium. The displacement D is

therefore proportional to E,

where is electric permittivity and is called the dielectric constant or

relative electric permittivity.

Boundary conditions on the field vectors

Consider two media, 1 and 2, in contact as shown in Fig. 4.6. We shall assume that there is a

surface charge density . Applying the Gauss’s law to the small pill box S, we obtain

This leads to

i.e.,

Thus the discontinuity in the normal component of D is given by the surface density of free

charge on the interface.

The line integral of around the path L must be zero:

This gives

i.e,

Thus the tangential component of the electric field is continuous across an interface.

n21

1

2

D2

E1D1

E2

LS

S l

(4.30)

(4.31)

(4.33)

Fig 4.6. Boundary conditions on the

field vectors at the interface between

two media may be obtained by

applying Gauss’s law to surface S and

integrating around the path L.

(4.32)

(4.34)

(4.35)

(4.36)

(4.37)

8

4.3 Boundary-Value Problems with Dielectrics

If the dielectrics of interest are linear, isotropic, and homogeneous, (Eq. 4.31), where

is a constant characteristic of the material, and we may write

Since still holds, the electric field is derivable from a scalar potential , i.e.,

, so that

Thus the potential in the dielectric satisfies the Poisson’s equation; the only difference between

this equation and the corresponding equation for the potential in vacuum is that replaces

(vacuum permittivity). In most cases of interest dielectrics contains no charge, i.e., . In

those circumstances, the potential satisfies Laplaces equation throughout the body of dielectric:

An electrostatic problem involving linear, isotropic, and homogeneous dielectrics reduces,

therefore, to finding solutions of Laplace’s equation in each medium and joining the solutions in

the various media by means of the boundary conditions. We treat a few examples of the

various techniques applied to dielectric media.

Point charge near a plane interface of dielectric media

We consider a point charge q embedded in a semi-infinite dielectric a distance d away from a

plane interface ( ) that separates the first medium from another semi-infinite dielectric

as shown in Fig. 4.7. From Eqs. 3.34 and 3.37, we obtain the boundary conditions:

{

| |

| |

| |

zq

d

2 1

x

(4.38)

(4.39)

(4.40)

Fig 4.7.

(4.41)

9

We apply the method of images to find the potential satisfying these boundary conditions (see

Fig. 4.8). For the potential in the region , we locate an image charge q’ at . Then

the potential at a point described by cylindrical coordinates is

(

)

where

√ and √

For the potential in the region , we locate an image charge q’’ at . Then the

potential at a point is

Fig 4.8. (a) The potential for is due to q and an image charge q’ at . (b) The potential for

is due to an image charge q’’ at .

The first two boundary conditions in Eq. 4.41 are for the tangential components of the electric

field:

(

)|

|

[

]

The third boundary condition in Eq. 4.41 is for the normal component of the displacement:

zq

d

1 1

d

q’

P

zq’’

d

2 2

d

PR1R2 R1

(a) In the region z>0 (b) In the region z<0

',12 q ",21 q

(4.42)

(4.43)

(4.44)

(4.45)

10

(

)|

|

From Eqs. 4.45 and 4.46, we obtain the image charges q’ and q”:

{

(

)

(

)

Figure 4.8 shows the lines of D for two cases and

for .

The surface charge density is given by (Eq. 4.22). Therefore, the polarization-surface-

charge density on the interface is

Since ,

(

)|

|

In the limit ( behaves like a conductor) and , Eq. 4.49 becomes equivalent to

Eq. 2.2 for a point charge in front of a conducting surface.

12 12

(4.46)

(4.47)

Fig 4.8. Lines of electric

displacement

(4.48)

(4.49)

11

Dielectric sphere in a uniform electric field

A dielectric sphere of radius a and permittivity is placed in a region of space containing an

initially uniform electric field as shown in Fig. 4.9. The origin of our coordinate

system is taken at the center of the sphere, and the electric field is aligned along the z-axis. We

should like to determine how the electric fields are modified by the dielectric sphere.

Inside and outside potential

From the azimuthal symmetry of the geometry we can take the solution to be of the form:

(i) Outside:

∑[

]

∑

(4.50)

At large distances from the sphere, i.e., for the region , the potential is given by

Accordingly, we can immediately set all except for equal to zero.

(ii) Inside:

∑

Since is finite at , terms must vanish.

Boundary conditions at

(i) Tangential E:

|

|

(4.53)

or (4.54)

(ii) Normal D:

|

|

(4.55)

a

P

z

r

0E 0E

Fig 4.9.

(4.52)

(4.51)

Fig 3.2.

12

Applying boundary condition (i) (Eq. 4.54) tells us that

∑

∑

We deduce from this that

{

We apply boundary condition (ii) results in

∑

∑

We deduce from this that

{

The equations 4.57 and 4.60 can be satisfied only if

{ (

)

(

)

where is the dielectric constant (or relative electric permittivity). From Eqs. 4.58 and

4.61, we can deduce that for all . The potential is therefore

(

)

(

)

Electric field and polarization

Equation 4.64 tells us that the field inside the sphere is a constant in the z direction:

(

)

(4.56)

(4.57)

(4.58)

(4.59)

(4.60)

(4.61)

(4.62)

(4.63)

(4.64)

(4.65)

(4.66)

13

For (no dielectric), this reduces as expected to . The field outside the dielectric is

clearly composed of the original constant field and a field which has a characteristic dipole

distribution with dipole moment of

(

)

We compare this with that from integrating the polarization P over the sphere. Insider the

dielectric we have

(

)

Since P is constant, we obtain the total dipole moment

(

) (

) which is equal to Eq. 4.67.

Surface charge density

Fig. 4.10

The uniform external electric field induces the constant polarization inside a dielectric sphere

(Eq. 4.68), and the induced polarization gives rise to surface charge which produces opposing

electric field if , as illustrated in Fig. 4.10. The surface charge density (Eq. 4.22) is

(

) (

)

Spherical cavity in a dielectric medium

Fig. 4.11

Figure 4.11 sketches the problem of a spherical cavity of radius a in a dielectric medium ( ) with

an external field . We can obtain the solution of this problem by switching and in

0E0EP

(a) polarization (b) Electric field due to surface charge

a

0E

z

0

(4.67)

(4.68)

(4.69)

14

the solution of the previous problem (i.e., ). For example, the field

inside the cavity is constant in the z direction:

(

)

The field outside the dielectric is composed of the original constant field and a field of the

dipole moment

(

)

which is oriented oppositely to the applied field if .

4.4 Microscopic Theory of Dielectrics

We now examine the molecular nature of the dielectric, and see how the electric field

responsible for polarizing the molecule is related to the macroscopic electric field. Our

discussion is in terms of simple classical models of the molecular properties, although a proper

treatment necessarily would involve quantum mechanical consideration. On the basis of a

simple molecular model it is possible to understand the linear behavior that is characteristic of

a large class of dielectric materials.

Molecular polarizability and electric susceptibility

Molecular field and macroscopic field

The electric susceptibility is defined through the relation (Eq. 4.30), where is the

macroscopic electric field. The electric field responsible for polarizing a molecule of the

dielectric is called the molecular field . is different from because the polarization of

other molecules gives rise to an internal field , so that we can write .

Internal field

In order to find out , we consider an imaginary sphere which contains neighboring molecules.

It is much larger than the molecules, yet infinitesimally small in the macroscopic scale. The

geometry is shown in Fig. 4.12. Then we can decompose into two terms: ,

where is the field due to the neighboring molecules close to the given molecule and is

E

+

++

+

+

+

+

+

+

pmol

mE

b

(4.70)

(4.71)

Fig 4.12. The dielectric outside

the cavity is replaced by a system

of polarization charges .

15

the contribution from all the other molecules. arises from surface charge density on

the cavity surface. Using spherical coordinates, we obtain

∫

∫ ∫

The x and y components vanish because they include the integrals of ∫

and

∫

, respectively. Therefore,

∫ ∫

Now we consider the term, . If the many molecules are randomly distributed in position,

then . This is the case if the dielectric is a gas or a liquid. If the dipoles in the cavity

are located at the regular atomic positions of a cubic crystal, then again (you may

refer to the proof in the textbook, pp. 160-161). We restrict further discussion to the rather

large classs of materials in which . Then,

Polarization and molecular polarizability

The polarization vector is defined as

where N is the number of molecules per unit volume and is the dipole moment of the

molecules. We define the molecular polarizability as

Combining Eqs. 4.73, 4.74, and 4.75, we obtain

(

)

Using (Eq. 4.30), we find

as the relation between susceptibility (the macroscopic parameter) and molecular polarizability

(the macroscopic parameter).

(4.72)

(4.74)

(4.75)

(4.73)

(4.76)

(4.77)

16

Using , we find

(

)

This is called the Clausius-Mossotti equation.

Models for the molecular polarizability

The molecules of a dielectric may be classified as polar or nonpolar. A polar molecule such as

H2O and CO has a permanent dipole moment, even in the absence of a polarizing field Em. In

nonpolar molecules, the “centers of gravity” of the positive and negative charge distributions

normally coincide. Symmetrical molecules such as O2, monoatomic molecules such as He, and

monoatomic solids such as Si fall into this category. We will discuss simple models for these

polar and nonpolar molecules.

Induced dipoles: simple harmonic oscillator model

The application of an electric field causes a relative displacement of the positive and negative

charges in nonpolar molecules, and the molecular dipoles so created are called induced dipoles.

To estimate the induced dipole moments we consider a simple harmonic oscillator model of

bound charges (electrons and ions). Each charge e is bound under the action of a restoring force

by an applied electric field

where m is the mass of the charge, and is the frequency of oscillation about equilibrium.

Consequently the induced dipole moment is

Therefore the polarizability is

For a bound electron, a typical oscillation frequency is in the optical range, i.e., Hz.

Then the electronic contribution is m3. For gases at NTP, m-3, so that

their susceptibilities, (see Eq. 4.77), are of the order of at best. For example, the

experimental value of dielectric constant for air is 1.00054. For solids or liquid

dielectrics, m-3, therefore the susceptibility can be of the order of unity.

(4.78)

(4.79)

(4.80)

(4.81)

17

Polar molecules: Langevin-Debye formula

In the absence of an electric field a macroscopic piece of polar dielectric is not polarized, since

thermal agitation keeps the molecules randomly oriented. If the polar dielectric is subjected to

an electric field, the individual dipoles experience torques which tend to align them with the

field. The average effective dipole moment per molecule may be calculated by means of a

principle from statistical mechanics. At temperature T the probability of finding a particular

molecular energy or Hamiltonian H is proportional to

For a polar molecule in the presence of an electric field , the Hamiltonian includes the

potential energy (see Eq. 4.16),

Where is a permanent dipole moment. Then the Hamiltonian is given by

where is a function of only the “internal” coordinates of the molecule (e.g., kinetic energy)

so that it is independent of the applied field. Using the Boltzmann factor Eq. 4.82 we can write

the average dipole moment as:

⟨ ⟩ ∫

∫

[ (

)

]

Here the components of ⟨ ⟩ not parallel to vanish. In general, the dipole potential energy

is much smaller than the thermal energy except at very low temperature. Then

⟨ ⟩

Therefore the polarizability of the polar molecule is

In general, induced dipole effects are also present in polar molecules, yet they are independent

of temperature. Then, the total molecular polarizability is

(4.84)

(4.83)

(4.85)

(4.86)

(4.87)

(4.88)

(4.82)

18

4.5 Electrostatic Energy in Dielectric Media and Forces on Dielectrics

Energy in dielectric systems

We discuss the electrostatic energy of an arbitrary distribution of charge in dielectric media

characterized by the macroscopic charge density . The work done to make a small change

in is

∫

Where is the potential due to the charge density already present. Since ,

, where is the resulting change in , so

∫

Now and hence (integrating by parts)

∫ ∫

The divergence theorem turns the first term into a surface integral, which vanishes if is

localized and we integrate over all of space. Therefore, the work done is equal to

∫

So far, this applies to any material. Now, if the medium is a linear dielectric, then so

Thus

(

∫ )

The total work done, then, as we build the free charge up from zero to the final configuration, is

∫

Parallel-plate capacitor filled with a dielectric medium

V d

+Q

-Q

A

(4.89)

(4.90)

(4.91)

(4.92)

(4.93)

(4.94)

(4.95)

Fig 4.13.

19

We shall find the electrostatic energy stored in a parallel-plate capacitor. Its geometry is shown

in Fig. 4.13: two conducting plates of area A (charged with +Q and -Q) is separated by d (we

assume that d is very small compared with the dimensions of the plates), and the gap is filled

with dielectric ( ).

(i) Capacitance

The electric field between the plates is

The potential difference . Therefore,

(ii) Electrostatic energy

Using Eq. 1.40, we obtain the electrostatic energy stored in the capacitor.

This is consistent with Eq. 4.95:

∫

Forces on dielectrics

We have just developed a procedure for calculating the electrostatic energy of a charge system

including dielectric media. We now discuss how the force on one of the objects in the charge

system may be calculated from this electrostatic energy. We assume all the charge resides on

the surfaces on the conductors.

Constant total charge

Let us suppose we are dealing with an isolated system composed of a number of parts

(conductors, point charges, dielectrics) and allow one of these parts to make a small

displacement under the influence of the electrical forces acting upon it. The work

performed by the electrical force on the system is

Because the system is isolated, this work is done at the expense of the electrostatic energy ;

in other words, the change in the electrostatic energy is . Therefore,

(

)

where the subscript Q has been added to denote that the system is isolated, and hence its total

charge remains constant during the displacement .

(4.99)

(4.100)

(4.96)

(4.97)

(4.98)

20

Fixed potential

We assume that all the conductors of the system are maintained at fixed potentials, , by

means of external sources of energy (e.g., by means of batteries). Then, the work performed

where is the work supplied by the batteries. The electrostatic energy W of the system (see

Eq. 1.36) is given as

∑

Since s are constant,

∑

Furthermore, the work supplied by the batteries is the work required to move each of the

charge increments from zero potential to the potential of the appropriate conductor,

therefore,

∑

Consequently, , and hence

(

)

Here the subscript V is used to denote that all potentials are maintained constant.

Dielectric slab within a parallel-plate capacitor

As an example of the energy method, we consider a parallel-plate capacitor in which a dielectric

slab ( ) is partially inserted. The dimensions of each plate are length and width . The

separation between them is . The geometry is illustrated in Fig. 4.14. We shall calculate the

force tending to pull the dielectric slab back into place. We consider two cases of (i) a constant

potential difference V and (ii) a constant total charge Q.

l

V

x

d

+Q

-Q

Fig 4.14. Dielectric slab partially

withdrawn from the gap between

two charged plates.

(4.101)

(4.102)

(4.103)

(4.104)

(4.105)

21

(i) Constant potential difference V

Since the electric field is the same everywhere between the plates, we find

∫

(

)

(

)

The force may be calculated from Eq. 4.106:

(ii) Constant total charge Q

The energy stored in the capacitor (see Eq. 1.42) is

and the capacitance in this case is

[ ]

We apply Eq. 4.101 to obtain the force:

Since

, we find

Eq. 4.111 has the same expression with Eq. 4.107, but the force of constant charge (Eq. 4.111)

is a function of (C varies with x) while the force of constant potential (Eq. 4.107) is

independent of x.

(4.107)

(4.108)

(4.109)

(4.110)

(4.111)

(4.106)