Chapter 4. Electrostatics of Macroscopic Media 4.1 Multipole Expansion Approximate potentials at large distances x 3 x x' d x' x' (x') x a Fig 4.1 We consider the potential in the far-field region (see Fig. 4.1 where | | ) due to a localized charge distribution ( for ). If the total charge is q, it is a good approximation to treat the charge distribution as a point charge, so . Even if q is zero, the potential does not vanish, but it decays much faster than . We will discuss more details about how the potential behaves in the far-field region. Electric dipole r+ +q x r- Fig 4.2. An electric dipole consists of two d equal and opposite charges +q and –q separated by a displacement d. -q We begin with a simple, yet exceedingly important case of charge distribution. Two equal and opposite charges separated by a small distance form an electric dipole. Suppose that +q and –q are separated by a displacement vector d as shown in Figure 4.2, then the potential at x is ( ) [( ) ( ) ] (4.1) 1 In the far-field region for | | , (4.2) [( ) ( )] This reduces to the coordinate independent expression (4.3) where is the electric dipole moment. For the dipole p along the z-axis, the electric fields take the form (4.4) { From this, we can obtain the coordinate independent expression (4.5) where is a unit vector. Fig 4.3. Field of an electric dipole 2 Multipole expansion We can expand the potential due to the charge distribution (1.12) ∫ | | using Eq. 3.68 (3.68) ∑ ∑ | | In the far-field region, . Then we find (4.6) ∑ [∫ ] We can rewrite the equation (4.7) ∑ where the coefficients (4.8) ∫ are called multipole moments. This is the multipole expansion of in powers of . The first term ( ) is the monopole contribution ( ); the second ( ) is the dipole ( ); the third is quadrupole; and so on. Monopole moment or total charge q ( √ : (4.9) ∫ Electric dipole moment p ( linear combinations of ): (4.10) ∫ Quadrupole moment tensor ( linear combinations of ): (4.11) ∫( ) 3 The expansion of in rectangular coordinates (4.12) [ ∑ ] Energy of a charge distribution in an external field If a localized charge distribution is placed in an external potential , the electrostatic energy of the system is (4.13) ∫ If is slowly varying over the region of , we can expand it in a Taylor series ∑ ∑ ( ) (4.14) ∑ ( ) Then, the energy takes the form (4.15) ∑ 4.2 Polarization and Electric Displacement in Macroscopic Media Dielectrics Properties of an ideal dielectric material It has no free charges. Instead, all charges are attached to specific atoms or molecules. Electric fields can induce only small displacements from their equilibrium positions. In a macroscopic scale, the effects of the electric fields can be visualized as a displacement of the entire positive charge in the dielectric relative to the negative charge. The dielectric is said to be polarized. Electric Polarization If an electric field is applied to a medium composed of many atoms and molecules, each atom or molecule forms a dipole pi due to the field induced displacements of the bound charges (see Fig. 4.4). Typically, this induced dipole moment is approximately proportional to the field: (4.16) 4 where is called atomic polarizability. These little dipoles are aligned along the direction of the field, and the material becomes polarized. An electric polarization P is defined as dipole moment per unit volume: (4.17) ∑ is a volume element which contains many atoms, yet it is infinitesimally small in the macroscopic scale. N is the number of atoms per unit volume and is the average dipole moment of the atoms. E pi + + + + 1 + + P pi + dV i Fig 4.4. An external electric field + + + induces electric polarization in a dielectric medium. dV Bound charges The dipole moment of is , so the total electric potential (see Eq. 4.3) is (4.18) ∫ | | We can rewrite this equation as (4.19) ∫ | | Integrating by parts gives (4.20) {∫ [ ] ∫ } | | | | Using the divergence theorem (4.21) {∫ ∫ } | | | | 5 where is a surface element and n is the normal unit vector. Here we define surface and volume charge densities: (4.22) and (4.23) Then, the potential due to the bound charges becomes (4.24) ∫ ∫ | | | | + + + b Fig 4.5. Origin of bound charge density. + b Electric displacement When a material system includes free charges as well as bound charges , the total charge density can be written: (4.25) And Gauss’s law reads (4.26) With the definition of the electric displacement D, (4.27) Equation 4.26 becomes (4.28) When an averaging is made of the homogeneous equation, , the same equation (4.29) holds for the macroscopic, electric field E. This means that the electric field is still derivable from a potential in electrostatics. Equations 4.28 and 4.29 are the two electrostatic equations in the macroscopic scale. 6 Electric susceptibility, permittivity, and dielectric constant For many substances (we suppose that the media are isotropic), the polarization is proportional to the field, provided E is not too strong: (4.30) The constant is called the electric susceptibility of the medium. The displacement D is therefore proportional to E, (4.31) where is electric permittivity and is called the dielectric constant or relative electric permittivity. Boundary conditions on the field vectors n 21 D2 E2 S l Fig 4.6. Boundary conditions on the 2 field vectors at the interface between S 1 L two media may be obtained by applying Gauss’s law to surface S and D E 1 1 integrating around the path L. Consider two media, 1 and 2, in contact as shown in Fig. 4.6. We shall assume that there is a surface charge density . Applying the Gauss’s law to the small pill box S, we obtain (4.32) This leads to (4.33) i.e., (4.34) Thus the discontinuity in the normal component of D is given by the surface density of free charge on the interface. The line integral of around the path L must be zero: (4.35) This gives (4.36) i.e, (4.37) Thus the tangential component of the electric field is continuous across an interface. 7 4.3 Boundary-Value Problems with Dielectrics If the dielectrics of interest are linear, isotropic, and homogeneous, (Eq. 4.31), where is a constant characteristic of the material, and we may write (4.38) Since still holds, the electric field is derivable from a scalar potential , i.e., , so that (4.39) Thus the potential in the dielectric satisfies the Poisson’s equation; the only difference between this equation and the corresponding equation for the potential in vacuum is that replaces (vacuum permittivity). In most cases of interest dielectrics contains no charge, i.e., . In those circumstances, the potential satisfies Laplaces equation throughout the body of dielectric: (4.40) An electrostatic problem involving linear, isotropic, and homogeneous dielectrics reduces, therefore, to finding solutions of Laplace’s equation in each medium and joining the solutions in the various media by means of the boundary conditions. We treat a few examples of the various techniques applied to dielectric media. Point charge near a plane interface of dielectric media x 2 1 q z d Fig 4.7. We consider a point charge q embedded in a semi-infinite dielectric a distance d away from a plane interface ( ) that separates the first medium from another semi-infinite dielectric as shown in Fig. 4.7. From Eqs. 3.34 and 3.37, we obtain the boundary conditions: | | (4.41) { | | | | 8 We apply the method of images to find the potential satisfying these boundary conditions (see Fig. 4.8). For the potential in the region , we locate an image charge q’ at . Then the potential at a point described by cylindrical coordinates is (4.42) ( ) where √ and √ (4.43) For the potential in the region , we locate an image charge q’’ at . Then the potential at a point is (4.44) (a) In the region z>0 (b) In the region z<0 1 1 2 2 P P R R2 1 R1 q’ q q’’ z z d d d d ,q' ,q" 2 1 1 2 Fig 4.8. (a) The potential for is due to q and an image charge q’ at . (b) The potential for is due to an image charge q’’ at . The first two boundary conditions in Eq. 4.41 are for the tangential components of the electric field: ( )| | [ ] (4.45) The third boundary condition in Eq. 4.41 is for the normal component of the displacement: 9 ( )| | (4.46) From Eqs. 4.45 and 4.46, we obtain the image charges q’ and q”: (4.47) ( ) ( ) { Figure 4.8 shows the lines of D for two cases and for . 2 1 2 1 Fig 4.8. Lines of electric displacement The surface charge density is given by (Eq. 4.22). Therefore, the polarization-surface- charge density on the interface is (4.48) Since , ( )| | (4.49) In the limit ( behaves like a conductor) and , Eq. 4.49 becomes equivalent to Eq. 2.2 for a point charge in front of a conducting surface. 10 Dielectric sphere in a uniform electric field A dielectric sphere of radius a and permittivity is placed in a region of space containing an initially uniform electric field as shown in Fig. 4.9. The origin of our coordinate system is taken at the center of the sphere, and the electric field is aligned along the z-axis. We should like to determine how the electric fields are modified by the dielectric sphere.