The Dielectric Constant and the Displacement Vector

where n 0 is the average number of atoms per cubic metre. This can be rewritten asZ ( )ε 0⃗E + n 0 ⃗p · ⃗dS = Q realUsing Eq. 5, we get our final equationZ (ε 0 + n 0 〈α〉 θ,ψ)⃗ E · ⃗dS = Q realWe define a new quantity, called the Displacement Vector:()⃗D = ε 0 + n 0 〈α〉 θ,ψ ⃗ EIn terms of ⃗D, the divergence theorem for Electrostatic Fields in a material become∇ ·⃗D = ρ (6)Defining ⃗D is not the great simplification. The great simplification is that ⃗D can beobtained from ⃗E using known, bulk properties of the material, namely n 0 and 〈α〉 θ,ψ .In textbooks, the presentation of this material is similar, but uses different jargon.For example, Jackson defines the Polarization Vector ⃗P defined by⃗P = n 0 〈⃗p〉which is nothing but our n 0 〈α〉 θ,ψ⃗E. In terms of ⃗P,⃗D = ε 0⃗E +⃗PHe obtains Eq. 6 via Eq. 4φ(⃗r) ==Z [1 ρreal4πε 0Z14πε 0−⃗P(⃗r ′ ) · ∇ 1 ]dV ′R 12 R 121[]ρR real − ∇ ′ ·⃗P(⃗r ′ ) dV ′12after some algebra. Hence,∇ · ⃗E = −∇ 2 φ = ρ real − ∇ ·⃗Pε 0which gets us to Eq. 6 again. The more mathematically minded student can look atthat derivation instead. In the end both approaches say the same thing.8