3.2.2 Boundary conditions Boundary conditions for the electrostatic field. The boundary conditions found for the dynamic electric field remain valid in the electrostatic case. Thus n12×(E1-E2) (3.32) Here f12 points into region 1 from region 2. Because the static curl and divergence equations are independent, so are the boundary conditions(3. 32)and(3. 33) For a linear and isotropic dielectric where D=EE, equation(3. 33)becomes (∈E1-∈2E2)=p (334) Alternatively, using D=EoE+P we can write (3. 33)as (Ps pps +p (335) n. P is the polarization surface charge with f pointing outward from the material body We can also write the boundary conditions in terms of the electrostatic potential. with E=-Vq, equation(3. 32) becomes d1(r)=Φ2(r) for all points r on the surface. Actually p and p2 may differ by a constant; because this constant is eliminated when the gradient is taken to find E, it is generally ignored We can write (3. 35)as dd1a中2 =-Ps-pPsl-PPs2 where the normal derivative is taken in the f12 direction. For a linear, isotropic dielectric (337) Again, we note that (3.36)and (3. 37) are independent Boundary conditions for steady electric current. The boundary condition on the normal component of current found in 8 2.8.2 remains valid in the steady current case. Assume that the boundary exists between two linear, isotropic conducting regions having onstitutive parameters(E1, 01) and(E2, 2), respectively. By(2.198)we have n12·(J1-J2)=-Vx·J (338) where f12 points into region 1 from region 2. A surface current will not appear on the boundary between two regions having finite conductivity, although a surface charge may cumulate there during the transient period when the currents are established 31]. If is influenced to move from the surface, it will move into the adjacent regions, ②2001 by CRC Press LLC3.2.2 Boundary conditions Boundary conditions for the electrostatic field. The boundary conditions found for the dynamic electric field remain valid in the electrostatic case. Thus nˆ 12 × (E1 − E2) = 0 (3.32) and nˆ 12 · (D1 − D2) = ρs. (3.33) Here nˆ 12 points into region 1 from region 2. Because the static curl and divergence equations are independent, so are the boundary conditions (3.32)and (3.33). For a linear and isotropic dielectric where D = E, equation (3.33)becomes nˆ 12 · ( 1E1 − 2E2) = ρs. (3.34) Alternatively, using D = 0E + P we can write (3.33)as nˆ 12 · (E1 − E2) = 1 0 (ρs + ρPs1 + ρPs2) (3.35) where ρPs = nˆ · P is the polarization surface charge with nˆ pointing outward from the material body. We can also write the boundary conditions in terms of the electrostatic potential. With E = −∇, equation (3.32)becomes 1(r) = 2(r) (3.36) for all points r on the surface. Actually 1 and 2 may differ by a constant; because this constant is eliminated when the gradient is taken to find E, it is generally ignored. We can write (3.35)as 0 ∂1 ∂n − ∂2 ∂n  = −ρs − ρPs1 − ρPs2 where the normal derivative is taken in the nˆ 12 direction. For a linear, isotropic dielectric (3.33)becomes 1 ∂1 ∂n − 2 ∂2 ∂n = −ρs. (3.37) Again, we note that (3.36)and (3.37)are independent. Boundary conditions for steady electric current. The boundary condition on the normal component of current found in § 2.8.2 remains valid in the steady current case. Assume that the boundary exists between two linear, isotropic conducting regions having constitutive parameters ( 1,σ1) and ( 2,σ2), respectively. By (2.198) we have nˆ 12 · (J1 − J2) = −∇s · Js (3.38) where nˆ 12 points into region 1 from region 2. A surface current will not appear on the boundary between two regions having finite conductivity, although a surface charge may accumulate there during the transient period when the currents are established [31]. If charge is influenced to move from the surface, it will move into the adjacent regions