5

Electric field in dielectric media

5.1 Continuous media

In previous sections we were studying electrostatic field of given charge configurations or electrostatic field around some ideal conductors. All electrons that belong to conduc- tivity bands in conductors possess mobility i.e. they can move inside whole conductor and dislocate on macroscopic distances. We shall call them free charges and denote by qf . There are no free charges in insulators. All their electric charges are bounded in- side atoms or molecules. Clearly, mobility of such charges is highly restricted. Bounded charges can move only inside molecules. Such materials cannot conduct electric current, however, they respond on applied external electric field by appearance of macroscopic dipole moment. Materials with these properties are called dielectrics. Dielectrics are electrically neutral materials, however, occasionally they can contain immersed electric charges. A fundamental property of dielectrics is theirs macroscopic polarity. It leads to ap- pearance of macroscopic dipole moment in response on applied external electric field. Macroscopic polarity has origin in polarization of molecules or atoms due to interac- tion of their charges with electric field. In presence of electric field electric clouds in atoms deform, what breaks the symmetry of charge configurations and results in in- duced electric dipole moment p, see Fig.5.1. In such a case, a dipole electric moment is induced on microscopic and macroscopic level. Another group of dielectrics are polar dielectrics. In such materials there exist microscopic electric dipole moment associated which each molecule (also without external electric field). Such moments do not lead to a macroscopic electric dipole moment because they cancel out for macroscopic num- ber of molecules. In external electric field a microscopic dipole moment p experiences

141 5. ELECTRIC FIELD IN DIELECTRIC MEDIA P. Klimas torque " " ⌧ =( nˆ) (qE)+( nˆ) ( qE)="nˆ E = p E. (5.1.1) 2 ⇥ 2 ⇥ ⇥ ⇥ It leads to appearance of macroscopic electric dipole moment, see Fig.5.2. This de-

(a) (b)

Figure 5.1 scription is highly simplified because in real situations electric dipoles experience si- multaneously the external electric field and the electric field of its closest neighbors. Microscopic description in such a case is useless so it is substituted by effective macro- scopic description. Note, that on microscopic level the electrostatic field cannot be introduced (thermal motion of atoms and molecules).

(a) (b)

Figure 5.2

A macroscopic description relays on replacing a microscopic electric field by a medium field in regions containing sufficiently big number of atoms. In such approach a quickly oscillating (in space and time) electromagnetic field is replaced by slowly

142 5.1 Continuous media varying effective field. Macroscopic electrostatic field appears when time dependence disappears in averaging process. A similar approach is performed in description of magnetic media. Unfortunately, description based on explicit averaging process is possible only for some simple models. In alternative approach we assume that a peace of matter can be represented by continuous medium whose properties are given by some continuous functions. The averaging process is treated as a formal procedure. It gives interpreta- tion of continuous functions but not their explicit form.

5.1.1 Microscopic and macroscopic fields

Let F (t, x) be a function which represent any microscopic quantity. We choose a mea- promediacao formal, sure function f(x) in region ⌦. The measure function is normalized in the following funacao de ensaio sense 3 d x0f(x0)=1. (5.1.2) Z⌦ For instance, this function can be chosen as

3 for r R f(x):= 4⇡R3 . (5.1.3) 0forr>R ⇢ The averaging procedure is defined by the integral

3 F (t, x) := d x0f(x0)F (t, x x0), (5.1.4) h i Z⌦ where adequate choice of the region ⌦ is very important. Roughly speaking, in order to obtain a slowly varying1 macroscopic electromagnetic field, the region ⌦ should contain approximately 106 molecules. In such a case, microscopic fields of high frequencies result in macroscopic slow-varying fields. The most important property of averaging (5.1.4) is its linearity:

@ 3 @F @F F (t, x) = d x0f(x0) (t, x x0)= (t, x), (5.1.5) @t h i @t @t Z⌦ ⌧ @ 3 @F @F F (t, x) = d x0f(x0) (t, x x0)= (t, x). (5.1.6) @xi h i @xi @xi Z⌦ ⌧ Clearly, the averaging process commutes with operation of differentiation. On the microscopic level (where the electromagnetic field can be already treated as a classical

1With timescale characteristic for macroscopic processes.

143 5. ELECTRIC FIELD IN DIELECTRIC MEDIA P. Klimas

Figure 5.3: The averaging process.

field) one cannot expect that electric fields is a solution of equation E =0. r⇥ m Instead, one has to consider a microscopic field which obeys equations 1 @B E + m =0, E =4⇡⇢ , r⇥ m c @t r· m m where ”m” stands for word microscopic. We shall denote a macroscopic average fields as E E , B B . (5.1.7) ⌘h mi ⌘h mi A macroscopic configuration is said to be electrostatic when contributions from term @tBm vanish in averaging process 1 @B E + m =0 E =0, (5.1.8) hr ⇥ mi c @t )r⇥ ⌧ 0 E =4⇡⇢ E =4⇡ ⇢ . (5.1.9) r· m m| {z } )r· h mi

5.1.2 Macroscopic Gauss’ law

A microscopic electric charge density ⇢m consists of microscopic charge density of free charges ⇢f and microscopic charge density of bounded charges ⇢b. The average values of these densities are given by ⇢ = ⇢ + ⇢ where ⇢ ⇢ . It follows that h mi h bi ⌘h f i macroscopic Gauss’ law takes the form

E =4⇡(⇢ + ⇢ ). (5.1.10) r· h bi Since there is no explicit expression for ⇢ then one has to express contribution from h bi this density in alternative way. It can be done in terms of new quantity called polariza- tion vector.

144 5.1 Continuous media

5.1.3 Polarization vector We shall consider a dielectric body that occupy a region V . In absence of immersed electric charges (⇢ 0) all molecules are electrically neutral so the whole body is ⌘ neutral as well d3x ⇢ =0. (5.1.11) h bi ZV This condition must hold for any shape of dielectric body. It means that function ⇢b h i in (5.1.11) is special in the sense that it is a divergence of a vector field. We shall denote this field by P . Such a vector field must be identically zero P 0 outside of ⌘ V (in region where there are no molecules) and it gives ( P )= ⇢ inside V . It r· h bi follows that one can extend a volume integral on a region ⌦ V . The border @⌦ can be infinitesimally close to @V , however, it neither touch nor cross @V . The additional volume does not change a value of the integral because ⇢ 0 in ⌦ (V @V ). h bi⌘ \ [ Applying the Gauss’ theorem we get

d3x ⇢ = d3x ⇢ = d3x P h bi h bi r· ZV Z⌦ I⌦ = da P =0, · Z@⌦ where P 0. |@⌦ ⌘ We still need to give physical interpretation for this quantity. First, we apply

Figure 5.4: Dielectric body V and the region ⌦.

Gauss’ law to a small box containing a piece of surface of the dielectric body, see Fig.5.5. Under assumption that area of the base of the box a is very small one can replace the surface integral by a product

d3x ⇢ = P nˆ da P nˆ P nˆ a = P a. b S2 S1 n V h i @V · ⇡2 · | · | 3 Z I 0 4 5 | {z }

145 5. ELECTRIC FIELD IN DIELECTRIC MEDIA P. Klimas

(a) (b)

Figure 5.5: Application of Gauss’ law to electrically polarized body.

The flux thorough the border of the box can be neglected for h pa. The value ⌧ of the integral is equal to the electric charge that encounters at the border @V . It follows that Pn has interpretation of surface electric charge density. Since there are no free charges in the problem we shall call it a polarized surface charge density and denote by p. It shows that a normal component of the polarization vector at @V is equal to surface polarized charge density:

= P P nˆ . (5.1.12) p n ⌘ · |@V

Second, a physical interpretation of P (x) can be given not only for x @V but also for 2 x V . In order to give such interpretation we consider a total electric dipole moment 2 p of the dielectric body. One can extend the integration volume on the region ⌦ V , where P 0, without changing the value of the integral |@⌦ ⌘

p := d3x x ⇢ = d3x x ⇢ = d3x x( P ) h bi h bi r· ZV Z⌦ Z⌦ 3 i 3 i = d x@i(xP )+ d x@ix P ⌦ ⌦ Z Z eˆi = da Px+ d3xP =|{z} d3xP . · I@⌦ Z⌦ ZV 0 | {z } It allows to conclude that the polarization vector has interpretation of volume density of electric dipole moment.

146 5.1 Continuous media

5.1.4 Electric induction vector Gauss’ law in absence of free charges reads E = 4⇡ P what admits definition r· r· of auxiliary field D:

D =0 D := E +4⇡P . (5.1.13) r· The field D is called electric induction or dislocation vector.

(a)

Figure 5.6: Flow of electrostatic induction. By construction D da3 =0. ·

We shall consider a fragment of flux tube limited by closed surface S that consists on lines of D and two surfaces orthogonal to these lines. This region is shown in Fig.5.6. Integrating Gauss’ law (5.1.13) on compact region V (where @V = S) and applying Gauss’ theorem one gets

D da =0 da D = da D, (5.1.14) · ) · · IS ZS1 ZS2 where da = da on S and da = da on S . It expresses conservation of flux of 1 1 2 2 D. Note that analogous conservation law characterizes incompressible flow in hydro- dynamics. When free electric charges are present then Gauss’ law must also contain a macroscopic density of free charges

D =4⇡⇢. (5.1.15) r·

147 5. ELECTRIC FIELD IN DIELECTRIC MEDIA P. Klimas

5.1.5 Dielectric permittivity Macroscopic electrostatic equations E =0and D =0have solution only if r⇥ r· there is given constitutive relation which express electric induction D (or polarization vector P ) in a function of electric field E. A generic form of such relation is given by a set of nonlinear functions of electric field. In linear media the relation takes the form

i j P = ijE , (5.1.16) where matrix ij represent components of susceptibility tensor. These components are constant in uniform media. For isotropic media the tensor is diagonal and given by ij = eij, where e is called susceptibility constant. Electrostatic induction D in linear media can be written in the form

i i i j j D = E +4⇡P =(ij +4⇡ij)E = "ijE , (5.1.17) where "ij := ij +4⇡ij stands for permittivity tensor. In particular, in isotropic homogeneous media "ij = "ij, where " stands for dielectric constant

" := 1 + 4⇡e. (5.1.18)

Since is always non-negative function, then " 1. In empty space =0, " =1 e e and therefore D = E.

Note, that relation between fields E and D in empty space satisfy relation D = "0E in SI unit system. It shows that international unit system is not really "economic". Moreover, fields E and D in the SI system have different physical dimensions.

5.1.6 Poisson’s equation Electrostatic equation E =0in continuous media possesses solution on the form r⇥ E = '. Substituting this expression into Gauss’ law we get r (" ')= 4⇡⇢, r· r which is also true in non-homogeneous media. In homogeneous media this equation can be written in the form ⇢ 2' = 4⇡ . (5.1.19) r " Since " 1, then one can conclude that in dielectric media there exist screening effect i.e. the electrostatic potential takes the form exactly like in empty space except that the charge density ⇢ is replaced by ⇢eff <⇢.

148 5.1 Continuous media

5.1.7 Example: Dielectric ball with a point-like immersed charge We shall consider a simple example. A point-like electric charge q encounters at the center of dielectric ball with radius R, see Fig.5.7(a). The problem possesses spherical symmetry so gaussian surfaces can be chosen as spheres Sr with radius r and center at the center of the ball. Gauss’ law takes the form

D da =4⇡q, · ISr where amplitude of electrostatic induction vector depends only on radial coordinate

D(r)="(r)E(r)er where "(r)="✓(R r)+✓(r R) i.e. "(r)=" = const for rR.

(a) (b)

Figure 5.7: (a) Dielectric ball with a point-like immersed charge. (b) Polarized charge densities.

The amplitude of electric field reads

q 1 for rRthe field is equal to field of charge q.

149 5. ELECTRIC FIELD IN DIELECTRIC MEDIA P. Klimas

First of all, we check if there is any volume charge density responsible for this effect. A polarization vector inside the ball r

1 1 q P = E = (" 1)E = (1 1 ) e . (5.1.21) e 4⇡ 4⇡ " r2 r

A volume charge density of bounded charges reads ⇢ = P , where h bi r · 1 e P = (1 1 )q r r· 4⇡ " r· r2 1 ⇣1 ⌘ 1 = (1 1 )q @ r2 sin ✓ +0+0 =0. 4⇡ " r2 sin ✓ r r2  ✓ ◆ We conclude that there are no volume charge density associated with bounded charges ⇢ =0. However, there can still exist surface charge density (polarized charge). h bi p Indeed, considering a normal component of the polarization vector at the surface r = R we find 1 q = e P = (1 1 ) . (5.1.22) p r · |r=R 4⇡ " R2

The total polarized electric charge at SR has value

2 1 q = d⌦R =(1 )q = q q0, (5.1.23) p p " ISR where q0 is the electric charge inside any sphere with radius r

q0 =(q0 q)+q = q + q. p

In other words, in vicinity of the charge q there exist a polarized charge q . In such p idealized situation the polarized charge can be interpreted as point-like object. However, if one substitute the ball by a shell with radii r0 and R then on the surface Sr0 there would appear a polarized charge q distributed with surface density p

qp 1 1 1 p(r = r0)= 2 = (1 " ) 2 . (5.1.24) 4⇡r0 4⇡ r0

Such situation is sketched in Fig.5.7(b).

150 5.1 Continuous media

5.1.8 Boundary conditions for dielectrics We consider two isotropic dielectric media that are characterized by dielectric constants "1 and "2 and separated by surface of interface S. Let nˆ be a vector normal to the surface S oriented from medium 1 to medium 2. We shall consider Gauss’ law in the region that corresponds with a small cylinder whose basis are locally parallel to surface S d3x D =4⇡ d3x⇢. (5.1.25) r· ZV ZV This situation is sketched in Fig.5.8(a). The region of integration is such that a small part of surface S encounters always between bases of the cylinder. Considering that height of the cylinder h is very small, h a, we find that expression d3x⇢ ⌧ V does not vanish in the limit h 0 only if there exist a surface charge density of free ! R charges := lim ⇢h. (5.1.26) ⇢ !1h 0 !

(a) (b)

Figure 5.8: Determination of boundary conditions in electrostatics.

Gauss’ law applied to volume V reads

daD nˆ =4⇡ da , (5.1.27) · I@V ZS where S is a small piece of surface S which encounters inside the cylinder. We as- sume that area of S is sufficiently small to have approximation const on S. ⇡ Consequently, da a. In the limit h 0 the only contribution to the flux S ⇡ ! comes from bases of the cylinder R da D nˆ + da D ( nˆ) (D D ) nˆa (5.1.28) · · ⇡ 2 1 · ZS2 ZS1

151 5. ELECTRIC FIELD IN DIELECTRIC MEDIA P. Klimas what gives (D D ) nˆ =4⇡. (5.1.29) 2 1 · It means that discontinuity of normal component of electrostatic induction D is pro- portional to surface charge density of electric free charges. A normal component of in- duction vector D is continuous in absence of free charges. Obviously, the normal com- ponent of electric field E cannot be continuous because " E = " E and " = " . 2 2n 1 1n 1 6 2 The electrostatic potential must then satisfy relation

@' @' " 2 = " 1 . (5.1.30) 2 @n 1 @n

Integrating equation E =0over small rectangle with two edges parallel to the r⇥ surface S we obtain relation between tangent components of electric field E on both sides of the surface of interface. We assume that length l of the edges is sufficiently small. The remaining two edges have length h l. Applying Stokes theorem we ⌧ get da nˆ ( E)=0 dl E =0, (5.1.31) 0 · r⇥ ) · ZS0 IC where vector nˆ0 is perpendicular to surface S0 representing the rectangle and it is tan- gent to the surface of interface S. Let tˆ:= nˆ nˆ be another vector tangent to surface ⇥ 0 S. In the limit h 0 the integral can be approximated as follows !

lim dl E = ( tˆ E)dl + (tˆ E)dl ( E2 + E1) tˆl. (5.1.32) h 0 · · · ⇡ · ! IC ZC2 ZC1 It follows that tangent component of the electric field is continuous across the surface of interface

E2t = E1t. (5.1.33)

Consequently, the electrostatic potential satisfies condition

tˆ ' = tˆ ' . (5.1.34) ·r 2 ·r 1 In numerous cases it is more convenient substitute this condition by condition of con- tinuity of electric potential what leads to two conditions. In absence of free electric charges they read @' @' ' = ' ,"2 = " 1 . (5.1.35) 2 1 2 @n 1 @n

152 5.1 Continuous media

5.1.9 Example: Dielectric ball in external electrostatic field We consider a problem of dielectric ball with radius a in external uniform electrostatic field E0eˆ3. We choose spherical coordinates (r, ✓,). The problem has axial symmetry so electrostatic potential cannot depend on variable . We denote by 1 a ball bulk and by 2 rest of space. Dielectric constant takes value "(r)=" ">1 for r

153 5. ELECTRIC FIELD IN DIELECTRIC MEDIA P. Klimas

This condition can be replaced by condition of continuity of the potential

'1(r = a, ✓)='2(r = a, ✓). (5.1.39)

Electrostatic potentials ' are given by solutions of Laplace’s equation 2' =0 1/2 r 1/2 and they read

1 l l 1 '1(r, ✓)= (Alr + Blr )Pl(cos ✓), (5.1.40) Xl=0 1 l l 1 '2(r, ✓)= (A˜lr + B˜lr )Pl(cos ✓). (5.1.41) Xl=0 In the first stage we impose conditions at r =0and at r = . Solution (5.1.40) must 1 (l+1) be regular at r =0so one has to set Bl =0for l =0, 1,.... For r terms r !1 in (5.1.41) can be neglected. The solution is then

1 l 1 '2(r, ✓)=A˜0 + A˜1 r cos ✓ + A˜lr Pl(cos ✓)+O(r ). (5.1.42) 3 x Xl=2 It follows that condition (5.1.36)| {z leads} to expression

1 l A˜1 A˜l lim @3(r Pl(cos ✓)) = E0 r l=2 !1 = const X 6 which holds for any ✓ if A˜ = E and A˜| =0{zfor l =2} , 3,.... Solutions ' and ' 1 0 l 1 2 which have desired behaviour at r =0and r = read 1 1 l '1(r, ✓)= Alr Pl(cos ✓), (5.1.43) Xl=0 ˜ 1 B0 2 l 1 ' (r, ✓)=A˜ + + E r + B˜ r P (cos ✓)+ B˜ r P (cos ✓). (5.1.44) 2 0 r 0 1 1 l l ⇣ ⌘ Xl=2 Now we impose boundary condition (5.1.37)

1 l 1 "A1P1(cos ✓)+ "Alla Pl(cos ✓) Xl=2 ˜ 1 B0 3 l 2 = +( E 2B˜ a )P (cos ✓) B˜ (l +1)a P (cos ✓) a2 0 1 1 l l Xl=2

154 5.1 Continuous media which gives

˜ 1 B0 2 l 1 l +1 + "A + B˜ + E P (cos ✓)+ "la A + B˜ P (cos ✓)=0. a2 1 a3 1 0 1 l al+2 l l ✓ ◆ Xl=2 ✓ ◆ 0 0 0 |{z} | {z } | {z } It follows immediately from the last equation that B˜0 =0. Similarly, imposing condi- tion (5.1.39) one gets

˜ 1 ˜ ˜ B1 l Bl A0 A0 + aA1 2 + aE0 P1(cos ✓)+ a Al l+1 Pl(cos ✓)=0. a ! a ! 0 Xl=2 0 0 | {z } Notice that continuity| of{z tangent components} of electric| field{z @✓'1}= @✓'2 results in equation

˜ 1 ˜ B1 l Bl aA1 2 + aE0 @✓P1(cos ✓)+ a Al l+1 @✓Pl(cos ✓)=0. a ! a ! Xl=2 0 0

The set| of equations{z } | {z }

"A + 2 B˜ = E , 1 a3 1 0 (5.1.45) A 1 B˜ = E ⇢ 1 a3 1 0 has solution 3 " 1 A = E , B˜ = a3E , (5.1.46) 1 " +2 0 1 " +2 0 whereas the set for l 2 "lA + l+1 B˜ =0, l a2l+1 l (5.1.47) A 1 B˜ =0 ⇢ l a2l+1 l has solution

Al =0, B˜l =0,l=2, 3,.... (5.1.48)

The constant A˜0 = A0 is free. The electrostatic potentials in region 1 and 2 read

x3 3 ' (r, ✓)=A E r cos ✓ (5.1.49) 1 0 " +2 0 z }| {

155 5. ELECTRIC FIELD IN DIELECTRIC MEDIA P. Klimas

Figure 5.10: Electric field and potential of dielectric ball in external electric field. Section in plane x2 =0. and r " 1 a 2 ' (r, ✓)=A + E a + cos ✓. (5.1.50) 2 0 0 a " +2 r  ⇣ ⌘ Electric field inside the sphere is uniform according to @' 3E E = ' = 1 eˆ = 0 eˆ . (5.1.51) 1 r 1 @x3 3 " +2 3

Polarization vector P can be obtained from expression "E1 = E1 +4⇡P and it reads " 1 3 " 1 P = E = E eˆ . (5.1.52) 4⇡ 1 4⇡ " +2 0 3 It allows to obtain a total dipole moment of the ball " 1 p = d3x P = a3 E eˆ . (5.1.53) " +2 0 3 ZV ✓ ◆ p | {z }

156 5.1 Continuous media

A surface density of polarized reads 3 " 1 = P eˆ e = E cos ✓ (5.1.54) p 3 · r 4⇡ " +2 0 cos ✓

2 and it leads to total polarized charge| {zqp =} d⌦ a p =0. Electric field outside the sphere is a superposition of external electrostatic uniform H field and a field of the polarized ball

@'2 1 @'2 E = e eˆ 2 @r rˆ r @✓ ✓ 1 " 1 a2 a r " 1 a2 =E a +2 cos ✓ e + E ( sin ✓)eˆ 0 a " +2r3 rˆ 0 r a " +2r2 ✓   " 1 a 3 =E (cos ✓ e sin ✓eˆ) + E (2 cos ✓e +sin✓eˆ) 0 rˆ ✓ 0 " +2 r rˆ ✓ eˆ3 ⇣ ⌘ 3(eˆ3 erˆ)erˆ eˆ3 p · r3 | 3(p{zrˆ)rˆ p} | {z } =E eˆ + · . | {z } (5.1.55) 0 3 r3 This result shows that electric field outside a dielectric ball, polarized by uniform elec- tric field, is equivalent to electric field of a point-like electric dipole at the center of the ball (plus the external uniform field). Note, that the case of conductive sphere can be obtained taking limit " . !1

157