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Section 14_Dielectric_Properties_of_Insulators

Section 14_Dielectric_Properties_of_Insulators - Physics...

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Physics 927 E.Y.Tsymbal 1 Section 14: Dielectric properties of insulators The central quantity in the physics of dielectrics is the polarization of the material P . The polarization P is defined as the dipole moment p per unit volume. The dipole moment of a system of charges is given by i i i q = p r (1) where r i is the position vector of charge q i . The value of the sum is independent of the choice of the origin of system, provided that the system in neutral. The simplest case of an electric dipole is a system consisting of a positive and negative charge, so that the dipole moment is equal to q a , where a is a vector connecting the two charges (from negative to positive). The electric field produces by the dipole moment at distances much larger than a is given by (in CGS units) 2 5 3( ) ( ) r r = p r r p E r (2) According to electrostatics the electric field E is related to a scalar potential as follows ϕ = −∇ E , (3) which gives the potential 3 ( ) r ϕ = p r r (4) When solving electrostatics problem in many cases it is more convenient to work with the potential rather than the field. A dielectric acquires a polarization due to an applied electric field. This polarization is the consequence of redistribution of charges inside the dielectric. From the macroscopic point of view the dielectric can be considered as a material with no net charges in the interior of the material and induced negative and positive charges on the left and right surfaces of the dielectric. The fact that the average charge inside the dielectric is zero can be understood if we take a macroscopic volume, it will contain equal amount of positive and negative charges and the net charge will be zero. On the other hand if we consider a volume including a boundary perpendicular to the direction of polarization, there is a net positive (negative) charge on the surface which is not compensated by charges inside the dielectric. Therefore, the polarization charge appears on the surface on the dielectric. Polarization of the dielectric produces a macroscopic electric field, which is determined by these surface charges. This can be seen from the following consideration. The electrostatic potential (4) produced by a dipole can be represented as 1 ( ) r ϕ = ⋅∇ r p . (5) Therefore for a volume distribution of the polarization we have:
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Physics 927 E.Y.Tsymbal 2 1 ( ) ( ) dV r r r ϕ ʹஒ = ⋅∇ ʹஒ r P , (6) where the integration is performed over the volume of the dielectric. Assuming for simplicity that the polarization P is constant throughout the medium and applying the Gauss theorem we obtain: ( ) dV dS dS r r r r r r σ ϕ = = = ʹஒ ʹஒ ʹஒ P n P r . (7) where the integration is performed over the surface of the dielectric and σ = n P is the fictitious surface charge density. Here n is the unit normal to the surface, directed outward from the polarized medium.
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