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Home , Dielectric, Dipole, Electric dipole moment, Electric susceptibility, Electrostatics, Permittivity

Chapter 4. of Macroscopic Media

4.1

Approximate potentials at large distances x 3 x  x' d x' x' (x') x

a Fig 4.1 We consider the potential in the far- 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

r+ +q x  r- Fig 4.2. An electric dipole consists of two d equal and opposite charges +q and –q separated by a d. -q

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

( ) [( ) ( ) ]

(4.1)

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In the far-field region for | | ,

(4.2) [( ) ( )]

This reduces to the coordinate independent expression

(4.3)

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

(4.4)

{

From this, we can obtain the coordinate independent expression

(4.5)

where is a unit vector.

Fig 4.3. Field of an electric dipole

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Multipole expansion We can expand the potential due to the charge distribution

(1.12) ∫ | | using Eq. 3.68

(3.68)

∑ ∑ | |

In the far-field region, . Then we find

(4.6) ∑ [∫ ]

We can rewrite the equation

(4.7) ∑

where the coefficients

(4.8) ∫ are called multipole moments. This is the multipole expansion of in powers of . The first term ( ) is the monopole contribution ( ); the ( ) is the dipole ( ); the third is ; and so on.

 Monopole moment or total charge q ( √ :

∫ (4.9)

p ( linear combinations of ):

∫ (4.10)

 Quadrupole moment ( linear combinations of ):

(4.11) ∫( )

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The expansion of in rectangular coordinates

(4.12)

[ ∑ ]

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

(4.13) ∫

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

∑

∑ ( )

∑ ( ) (4.14)

Then, the energy takes the form

(4.15) ∑

4.2 and Electric Displacement in Macroscopic Media

Dielectrics Properties of an ideal material  It has no free charges. Instead, all charges are attached to specific or .  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 is applied to a medium composed of many atoms and molecules, each or forms a dipole 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:

(4.16)

4 where is called atomic . These little 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: (4.17) ∑

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.

E

pi + + + + 1 + + P  pi + dV i Fig 4.4. An external electric field + + + induces electric polarization in a dielectric medium. dV

Bound charges The dipole moment of is , so the total (see Eq. 4.3) is

(4.18) ∫ | | We can rewrite this equation as

(4.19) ∫ | | Integrating by parts gives

(4.20) {∫ [ ] ∫ } | | | | Using the theorem

(4.21) {∫ ∫ } | | | |

5 where is a surface element and n is the normal unit vector. Here we define surface and volume charge densities:

(4.22) and (4.23)

Then, the potential due to the bound charges becomes

(4.24) ∫ ∫

| | | |

+ + + b Fig 4.5. Origin of bound .

+  b

Electric displacement

When a material system includes free charges as well as bound charges , the total charge density can be written:

(4.25) And Gauss’s law reads (4.26)

With the definition of the electric displacement D, (4.27)

Equation 4.26 becomes (4.28)

When an averaging is made of the homogeneous equation, , the same equation (4.29) 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.

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Electric susceptibility, , 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:

(4.30)

The constant is called the of the medium. The displacement D is therefore proportional to E,

(4.31) where is electric permittivity and is called the dielectric constant or relative electric permittivity. Boundary conditions on the field vectors n 21 D2 E2 S l Fig 4.6. Boundary conditions on the 2 field vectors at the interface between S 1 L two media may be obtained by applying Gauss’s law to surface S and D E 1 1 integrating around the path L.

Consider two media, 1 and 2, in contact as shown in Fig. 4.6. We shall assume that there is a density . Applying the Gauss’s law to the small pill box S, we obtain

(4.32) This leads to

(4.33) i.e., (4.34) Thus the discontinuity in the normal component of D is given by the surface density of free charge on the interface. The of around the path L must be zero: (4.35)

This gives

(4.36) i.e, (4.37) Thus the tangential component of the electric field is continuous across an interface.

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4.3 Boundary-Value Problems with

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

(4.38)

Since still holds, the electric field is derivable from a , i.e., , so that

(4.39)

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 is that replaces (). In most cases of interest dielectrics contains no charge, i.e., . In those circumstances, the potential satisfies Laplaces equation throughout the body of dielectric: (4.40) 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

x

 2 1 q z

d Fig 4.7.

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:

| | (4.41) { | | | | 8

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

(4.42) ( ) where

√ and √ (4.43) For the potential in the region , we locate an image charge q’’ at . Then the potential at a point is

(4.44)

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

1 1  2  2 P P R R2 1 R1 q’ q q’’ z z

d d d d   ,q'   ,q" 2 1 1 2 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:

( )| |

[ ]

(4.45)

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

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( )| |

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

( ) (4.47)

( ) {

Figure 4.8 shows the lines of D for two cases and for .

 2  1  2  1

Fig 4.8. Lines of electric displacement

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

(4.48)

Since ,

( )| |

(4.49)

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.

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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.

r P

E 0 a  z  Fig 4.9.

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 (4.51)

Accordingly, we can immediately set all except for equal to zero. (ii) Inside:

(4.52) ∑

Fig 3.2. Since is finite at , terms must vanish. Boundary conditions at

(i) Tangential E: | | (4.53)

or (4.54)

(ii) Normal D: | | (4.55)

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

(4.56)

∑ ∑

We deduce from this that

(4.57)

{ (4.58)

We apply boundary condition (ii) results in

(4.59) ∑ ∑

We deduce from this that

(4.60)

{ (4.61)

The equations 4.57 and 4.60 can be satisfied only if

( ) (4.62)

{

( ) (4.63)

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

( ) (4.64)

(4.65) ( )

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

(4.66) ( )

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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

(4.67) ( )

We compare this with that from integrating the polarization P over the sphere. Insider the dielectric we have (4.68) ( )

Since P is constant, we obtain the total dipole moment

( ) ( ) which is equal to Eq. 4.67.

Surface charge density

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

E 0 P E0

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

(4.69) ( ) ( )

Spherical cavity in a dielectric medium

E0 a z

 0  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

13 the solution of the previous problem (i.e., ). For example, the field inside the cavity is constant in the z direction:

( ) (4.70)

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

(4.71) ( )

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 .

+  b E + pmol + + + + Fig 4.12. The dielectric outside + Em +  the cavity is replaced by a system

+ of polarization charges .

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

14 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,

(4.72) ∫ ∫

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

(4.73)

Polarization and molecular polarizability The polarization vector is defined as

(4.74) where N is the number of molecules per unit volume and is the dipole moment of the molecules. We define the molecular polarizability as

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

( ) (4.76)

Using (Eq. 4.30), we find

(4.77)

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

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Using , we find

(4.78) ( )

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 ” of the positive and negative charge distributions normally coincide. Symmetrical molecules such as O2, monoatomic molecules such as He, and monoatomic 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 ( and ). Each charge e is bound under the action of a restoring by an applied electric field

(4.79) where m is the mass of the charge, and is the of about equilibrium. Consequently the induced dipole moment is

(4.80)

Therefore the polarizability is

(4.81)

For a bound , a typical oscillation frequency is in the optical range, i.e., Hz. 3 -3 Then the electronic contribution is m . For gases at NTP, m , 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.

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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 which tend to align them with the field. The average effective dipole moment per molecule may be calculated by means of a principle from . At temperature T the probability of finding a particular molecular energy or Hamiltonian H is proportional to

(4.82)

For a polar molecule in the presence of an electric field , the Hamiltonian includes the (see Eq. 4.16),

(4.83)

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

(4.84) 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:

(4.85) ∫ 〈 〉 [ ( ) ]

∫

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

(4.86) 〈 〉

Therefore the polarizability of the polar molecule is

(4.87)

In general, induced dipole effects are also present in polar molecules, yet they are independent of temperature. Then, the total molecular polarizability is

(4.88)

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4.5 Electrostatic Energy in Dielectric Media and 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 done to make a small change in is

∫ (4.89)

Where is the potential due to the charge density already present. Since , , where is the resulting change in , so

∫ (4.90)

Now and hence (integrating by parts)

∫ ∫ (4.91)

The 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 ∫ (4.92)

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

(4.93)

Thus

( ∫ ) (4.94)

The total work done, then, as we build the free charge up from zero to the final configuration, is (4.95) ∫

Parallel-plate filled with a dielectric medium

A +Q

 d V Fig 4.13.

-Q

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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) The electric field between the plates is

(4.96)

The potential difference . Therefore,

(4.97)

(ii) Electrostatic energy Using Eq. 1.40, we obtain the electrostatic energy stored in the capacitor.

(4.98)

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 (4.99)

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,

( ) (4.100) where the subscript Q has been added to denote that the system is isolated, and hence its total charge remains constant during the displacement .

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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

(4.101) where is the work supplied by the batteries. The electrostatic energy W of the system (see Eq. 1.36) is given as (4.102) ∑

Since s are constant, (4.103) ∑

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,

∑ (4.104)

Consequently, , and hence (4.105) ( )

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. x +Q

 d V Fig 4.14. Dielectric slab partially withdrawn from the gap between -Q two charged plates. l

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(i) Constant potential difference V Since the electric field is the same everywhere between the plates, we find

(4.106) ∫ ( ) ( )

The force may be calculated from Eq. 4.106:

(4.107)

(ii) Constant total charge Q The energy stored in the capacitor (see Eq. 1.42) is

(4.108) and the capacitance in this case is

(4.109) [ ]

We apply Eq. 4.101 to obtain the force:

(4.110)

Since , we find

(4.111)

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.

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