Phd and Mphil Thesis Classes

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 inside 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 appearance 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 induced 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 number 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 macroscopic 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 interpretation 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 polarization 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 p∆a. 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 σp. It shows that a normal component of the polarization vector at @V is equal to surface polarized charge density: σ = 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 eî = 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 .

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