Polarization and the Polarization Vector

4.5 Materials in the Electric Field

4.5.3 Polarization and the Polarization Vector

The potential atR1¼a equals that at band the potential at the location of the point charge (R1¼0) is infinite as required. A plot of the electric field intensity and electric potential is shown inFigure4.24c. Note that the integration is done in segments because the expression for the electric field intensity in each segment is different.

(b) To calculate the charge densities on the inner surface, we use the fact thatE ¼0 inside the conductor. We draw a Gaussian surface atR¼a⁺(immediately inside the conductor so that the surface is included). For this Gaussian surface to produce zero field intensity, the total charge enclosed must be zero:

ð

s

E

^ds¼E4πR22 ¼Q ε0

þρsa4πa² ε0

¼0 ! ρsa¼ Q 4πa²

C m²

The surface charge density on the outer surface is calculated from the fact that the total charge in the system isQ[C].

Therefore, the total charge on the outer surface equals the total charge on the inner surface. This gives Q_b¼ Q_a¼Q ! ρsb¼ Q

4πb² C m²

FromFigure4.25cand from Coulomb’s law, it is clear that the dipole aligns itself with the external electric field. Now, consider a large number of molecules, all aligned in the direction of the externally applied electric field, as shown in Figure4.26. Two effects must now be taken into account. First, the total dipole moment that the material exhibits depends on the number of molecules and on the strength of individual molecular dipole moments. Second, each dipole moment, viewed as two point charges, generates an electric field pointing from the positive to the negative charge. This field opposes the external field, indicating that the field inside the dielectric is lower than the external field. A total dipole moment per unit volume can be defined by summing the dipole moments of individual molecules:

P¼ lim

Δv!0

1 Δv

X^N

i¼1

p_i C

m²

ð4:38Þ

This vector is called apolarization vectorand, based on the preceding discussion, is dependent on the external electric field intensity. While a useful definition, the polarization vector as given above is not really calculable in general. For one, the dipole moment of individual molecules is not known and is not easily measurable. In addition, not all molecules are polarized in an identical fashion. For example, we expect molecules near the surface to have different dipole moments than molecules inside the material. However, the net effect of the electric field due to polarization of the molecules is to reduce the electric field intensity inside the material since the external and internal electric fields oppose each other.

For an element of volumedv⁰, we can use the idea of the dipole and calculate the potential at a distanceR(seeExample 4.12)fromdv⁰as

dV¼ P

^{R^}

4πε0

R²dv⁰ ð4:39Þ

In this expression,Pdv⁰is the equivalent dipole moment of the volumedv⁰sincePwas defined as the dipole moment per unit volume inEq. (4.38). To obtain the potential due to the whole volume, we integrate over the volume of the charge distribution to obtain

V ¼ ð

v⁰

dV¼ ð

v⁰

P

^{R^}

4πε0

R²dv⁰ ½ V ð4:40Þ

It is well worth comparing this expression with that for the potential of a charge densityρvin any arbitrary volume. This was obtained inEq. (4.33)as

V¼ 1 4πε0

ð

v⁰

ρvdv⁰

j jR ½ V ð4:41Þ Comparing the two, we note thatEq. (4.40)can be written in the form ofEq. (4.41)if the equivalent charge density in the volumev⁰is written as

ρ_pv¼P

^{R^}

j jR

C m³

ð4:42Þ ++

++ ++ ++ +

−−

−−

−−

−−

−− Eexternal

− + − + − +

− + − + − +

+

− − + − +

− + − + − +

Figure 4.26 Polarization of charges in a dielectric by means of an external field

Thus, we conclude that due to the polarization vector, there is an equivalent polarization volume charge density throughout the volumev⁰. However, we would like to be able to separate the charge density into a surface charge density and a volume charge density. To do so, we rewriteEq. (4.40)as

V¼ 1 4πε0

ð

v⁰

P

_R^{R^}2dv⁰¼ 1 4πε0

ð

v⁰

P

^∇ _R^{1 dv0 ¼}_4πε¹

0

ð

v⁰

P

^∇0 _R^{1 dv0 ½ V ð4:43Þ}

The later step needs some explanation. We obtained inExample 2.10

∇ 1

R ¼ R^

R² ð4:44Þ

where (in Cartesian coordinates)R ¼((x–x⁰)²+ (y–y⁰)²+ (z–z⁰)²)^1/2. The integration is on the volumev⁰and therefore must be carried out in the primed coordinates. In the same example, we also found that performing the gradient with respect to the primed coordinates, we obtained

∇⁰ 1

R ¼ ∇ 1

R ¼R^

R² ð4:45Þ

The relations inEqs. (4.44)and(4.45)were used inEq. (4.43).

Now, we use the following vector identity[seeEq. (2.140)]:

∇

^ϕA¼ϕ∇

^AþA

^{∇ϕ ð4:46Þ}

whereϕis any scalar function andAany vector function. In this case, we can write the integrand inEq. (4.43)usingϕ¼1/R andA¼P:

P

^∇0 _R^{1 ¼∇0}

_R^1P_R^1∇0

^{P ð4:47Þ}

Substituting this intoEq. (4.43)gives V ¼ 1

4πε0

ð

v⁰

∇⁰

_R^1Pdv0_4πε¹

0

ð

v⁰

1

R∇⁰

^{P dv0 ½ V ð4:48Þ}

The first integral can be converted into a surface integral using the divergence theorem:

ð

v⁰

∇⁰

_R^1Pdv0¼þ

s⁰

1

RP

^{ds0 ¼þ}

s⁰

1

R^ðP

^{n^Þds0 ð4:49Þ}

and, finally, substituting this for the first integral inEq. (4.48), we get

V¼ 1 4πε0

þ

s⁰

1

R^ðP

^{n^Þds0þ}_4πε¹

0

ð

v⁰

∇⁰

^P

R dv⁰ ½ V ð4:50Þ

Comparing the two terms with the general expressions for the potential due to surface charge densities and volume charge densities inEqs. (4.32)and(4.33), we can rewrite the potential as that due to an equivalent polarization surface charge densityρps[C/m²] and an equivalent polarization volume charge densityρpv[C/m³] as

V¼ 1 4πε0

þ

s⁰

ρps

Rds⁰þ 1 4πε0

ð

v⁰

ρpv

Rdv⁰ ½ V ð4:51Þ

172 4 Gauss’s Law and the Electric Potential

where the surface and volume charge densities are ρps¼P

^{n^0} _m^C2

and ρpv¼ ∇⁰

^P _m^C3

ð4:52Þ

In general, we can drop the primed notation since this relation only applies in the volume in which polarization exists (and, therefore, there is no specific need to identify the volume), but it is left intact here to signify the fact that polarization can only take place inside the material.

When molecules of dielectrics align with the external field, there are two possible situations:

(1) The dielectric is uniform in its properties. All molecules have identical dipole moments, and because of this, all internal charges cancel each other with the exception of charges on the surface of the material, as shown inFigure4.27a. This can be explained simply by proximity of the positive and negative charges. The net effect is an induced (polarization) surface charge density due to the effect of the external electric field. The total field now is the combined external field and the internal field due to induced surface charges as shown.

(2) The dielectric is not uniform in its properties. Molecules at different locations are polarized differently. Although some of the charges cancel each other, some do not, and the net effect is a charge density on the surface and in the interior of the material, as shown inFigure4.27b. Again, the net effect is that of a combined electric field intensity due the external and internal fields.