Problems
Application 6.2 Refraction of Field Lines at a Dielectric Interface
6.5 Simple Dielectric Matter 173
d
V 0
E = E0
Figure 6.9: A capacitor with fixed plate potentials. Plus and minus signs denote polarization charge at the surface of the dielectric. A few lines of the electric fieldE=Ezˆare indicated.
The sum of (6.65) and (6.64) is exactly the free charge densityσ0=0V /dthat produces the field E0when the dielectric is absent. In other words,E=E0when the dielectric is present because charge flows from the battery (or whatever maintains the plates at fixed potential) to the surface of the plates to exactly cancel the polarization charge on the adjacent dielectric surfaces. Faraday exploited (6.65) to determineκ for many materials. His method was to measure the change in the amount of charge drawn onto the metal plates of a (spherical) capacitor when a dielectric was interposed between them.
The SI unit of capacitance is called thefaradto honor this aspect of Faraday’s work.
Macroscopically, the charge densities (6.65) and (6.64) are spatially coincident. This is not so microscopically, and (6.65) implies that the local electric field immediately adjacent to both metal plates islargerin magnitude than the electric field produced by the vacuum capacitor. If so, there must also be regions inside the dielectric where the electric fields is smaller than (or oppositely directed from) the field of the vacuum capacitor. We will return to this point at the end of Section 6.6.2.
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174 DIELECTRIC MATTER: POLARIZATION AND ITS CONSEQUENCES
The case21is interesting because bothα1≈0 andα2≈π/2 satisfy the law of refraction. The first solution is consistent withE=(σ/0) ˆnfor the electric field just outside the surface of a perfect conductor. The second solution says that the lines ofDandEin the high-permittivity material are nearly parallel to the interface with the low-permittivity material. The actual solution adopted by Nature depends on the detailed geometry of the problem, including the positions of free charges away
from the interface.
6.5.5 Potential Theory for a Simple Dielectric
Gauss’ law for simple dielectric matter (6.39) combined withE= −∇ϕtells us that the electrostatic potential inside a simple dielectric with permittivity=0κ satisfies Poisson’s equation withρ(r) replaced byρf(r)/κ:
0∇2ϕ= −ρf/κ. (6.69)
As discussed in Section 3.3.2, the matching conditions (6.33) and (6.31) can be expressed entirely in potential language, namely,
ϕ1(rS)=ϕ2(rS) (6.70)
and
κ2
∂ϕ2
∂n2 −κ1
∂ϕ1
∂n2
S
= σf
0
. (6.71)
When there is no free charge anywhere, the Poisson-like equation (6.69) reduces everywhere to Laplace’s equation,
∇2ϕ=0. (6.72)
The matching condition (6.71) simplifies similarly to κ1
∂ϕ1
∂n
S
= κ2
∂ϕ2
∂n
S
. (6.73)
Chapters 7 and 8 discuss methods to solve the Laplace and Poisson equations, respectively, when simple dielectric matter is present.
Example 6.3 Figure 6.11 shows a spherical cavity scooped out of an infinite medium with dielectric constantκ. Capacitor plates at infinity produce a fixed and uniform external fieldE0=E0z. Findˆ the potential and field everywhere.
0
ˆ
E z
r′ ′r R
Figure 6.11: A sphere of radiusRscooped out of a medium with dielectric constantκ. The field lines shown are for the external fieldE0=E0zˆonly.
6.6 The Physics of the Dielectric Constant 175
Solution: The fieldE0 induces a uniform polarizationP0=P0zˆin the original infinite uniform medium. After the cavity is created, a polarization charge density σ(θ)=P0·nˆ = −P0cosθ develops on its surface.10 This charge distribution produces an electric field which supplements E0. This is the situation analyzed in Application 4.3, where we showed that a charge density σ(θ)∝cosθ confined to sphere produces a potential that varies asrcosθ inside the sphere and cosθ/r2outside the sphere. Consequently, the total potential has the form
ϕ(r, θ)=
⎧⎪
⎨
⎪⎩
Arcosθ r≤R,
−E0r+ B r2
cosθ r≥R.
The contribution−E0rcosθ toϕ(r, θ) is associated with theunscreenedE0. This is consistent with the discussion immediately above of the parallel-plate capacitor with fixed plate potential.
The matching conditions (6.70) and (6.73) applied atr=Rdetermine the coefficientsAandB:
A= − 3κ
2κ+1E0 and B= − κ−1
2κ+1E0R3.
The total fieldE= −∇ϕoutside the cavity is sum ofE0and a pure dipole field. The total field inside the cavity is uniform, points in the same direction asE0, and is larger in magnitude thanE0:
Ein= −A= 3κ
2κ+1E0> E0.
Physically, this occurs because the polarization charge density is positive (negative) on the left (right) hemisphere of the cavity in Figure 6.11:
σP(θ)=P·nˆ= −0(κ−1)Eout·r.ˆ
It is worth noting thatσP(θ)=P·nˆisnotthe same asσ(θ)=P0·nˆintroduced at the beginning of the problem because the final electric field (and hence the final polarization) in the medium is not uniform. This fact does not invalidate our solution becauseσP(θ) andσ(θ) are both proportional to cosθ, which is the only information we used to help determine the form ofϕ(r, θ) above.
6.6 The Physics of the Dielectric Constant
Table 6.1 lists the experimental dielectric constants for two gases (helium and nitrogen), two liquids (methane and water), and two solids (silicon dioxide and silicon). The electrostatics of the physical models which predict these numbers is interesting. All make essential use of thelocal electric field that exists inside a polarized dielectric.
6.6.1 The Electric Polarizability
Example 5.1 of Section 5.2 demonstrated that a uniform electric fieldE0induces an electric dipole momentp=4π 0R3E0in a conducting sphere of radiusR. More generally, we define thepolarizability αof any small dielectric body as the proportionality constant between a uniform fieldE0and the dipole momentpinduced in the body by that field:11
p=α0E0. (6.74)
10 There is a negative sign because ˆnis theoutwardunit normal to the dielectric surface.
11 Compare (6.74) with (6.35).
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176 DIELECTRIC MATTER: POLARIZATION AND ITS CONSEQUENCES
Table 6.1. Dielectric constants at 105Pa (1 atm) and 20oC.
Substance κ
He 1.000065
N2 1.00055
CH4 1.7
SiO2 4.5
Si 11.8
H2O 80
Equation (6.74) generalizes to microscopic bodies like atoms and molecules where the value of the induced dipole moment depends on the value of a microscopic electric field at the position of the atom or molecule:
p(r)=α0Emicro(r). (6.75)
Hundreds of research papers have been devoted to calculations and measurements of electric polariz- abilities.
6.6.2 The Clausius-Mossotti Formula
The Lorentz model for polarizable matter (Section 6.2.2) relates the macroscopic dielectric constantκ [see (6.36)] of a dielectric gas to the microscopic polarizabilityα[see (6.75)] of the constituent atoms or molecules. The main ingredient is the spatial integral over a suitable averaging volumeof the microscopic electric field on the right side of (6.75):12
E(R)= 1
d3rEmicro(r). (6.76)
We also need an alternative definition of the cell-averaged dipole momentpRfirst defined in (6.10), namely,
pR= 1
d3rp(r). (6.77)
Finally, becausen=1/ is the density of molecules in the gas, (6.10), (6.76), and (6.77) combine to generate an expression for the polarizationPwhich appears in the macroscopic equation (6.25):
D(R)=0E(R)+P(R)=0E(R)+nα0E(R). (6.78) Comparing (6.78) with (6.37) shows that the static dielectric constant is
κ=1+nα. (6.79)
Our derivation of (6.79) implicitly assumes that the dimensionless parameternαis much smaller than one. To see this, note first that the field in (6.75) which polarizes the molecule in a given cell cannot include the microscopic fieldEself(r) produced by the molecule itself. On the other hand, the macroscopic fieldE(R) in (6.78) is the average of the field in a given cell fromallsources, including
12 This is the Lorentz averaging procedure discussed in Section 2.3.1. We follow the notation of that section wherer denotes a microscopic spatial variable andRdenotes a macroscopic spatial variable.
6.6 The Physics of the Dielectric Constant 177
Eself(r). Therefore, it is necessary to subtract the Lorentz average of the self-field when we calculate the polarization:
P(R)=n α0
⎧⎨
⎩E(R)− 1
d3sEself(s)
⎫⎬
⎭=nα0Elocal(R). (6.80)
Thelocal fieldElocal(R) defined by (6.80) is a better approximation to the macroscopic field that polarizes each molecule. We have performed the integral in (6.80) already for the case of a spherical cell (Example 4.1). The result given there is a good approximation for non-spherical cells also, particularly if the cell is large compared to the size of the molecular charge distribution:
1
d3sEself(s)= − p
30= −P(R)
30 . (6.81)
Substituting (6.81) into (6.80) gives an algebraic equation forP(R). The solution, P(R)= nα0
1−nα/3E(R), (6.82)
together with the leftmost equation in (6.78) gives the desired result for the dielectric constant:
κ=1+ nα
1−nα/3. (6.83)
Only whennα1 does (6.83) reproduce (6.79). In that limit, the averaging volumeis large and the self-field makes a negligible contribution to the Lorentz average. Experimental tests of (6.83) usually employ an equivalent form known as theClausius-Mossotti formula:
κ−1 κ+2 = nα
3 . (6.84)
Non-polar gases and a few simple liquids like He, N2, and C6H6obey formula (6.84) rather well. The polarizability is either measured independently or estimated fromα=3V.13
We can also combine (6.80) with (6.82) to discover that (cf. Example 6.1) Elocal(R)= E(R)
1−nα/3 = κ+2
3 E(R). (6.85)
This formula confirms the remark made at the end of Section 6.5.4 that the local electric field typically exceeds the average macroscopic field inside a dielectric.Elocal(R) is due to all theothermolecules, so it is characteristic of the volumebetweengas molecules. This may be contrasted with the volume insideeach molecule where (the reader may wish to confirm) the dipole self-field is very large and directed oppositely toElocal(R).
6.6.3 Polar Liquids and Solids
Table 6.1 reports the dielectric constants for liquid methane and water. Experiments show that CH4
obeys the Clausius-Mossotti relation (6.84) while H2O does not. The physical reason for this is that methane is a non-polar molecule while water is a polar molecule with a permanent electric dipole momentp0(see Figure 4.2). Onsager (1936) realized that the presence ofp0dramatically affects the local field needed to calculateκ. He derived a generalization of the Clausius-Mossotti formula using, in part, the cavity calculation of Example 6.3 generalized to include a dipolep0at the center of the cavity.
13 See the remark preceding (6.74).