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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)
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