Problems with known polarization
    
    (pic) Uniformly polarized sphere

    Gauss' Law in Dielectric

        ε₀∇⋅𝐄 = ρ = ρfree + ρbound
                = ρfree - ∇⋅𝐏

        ε₀∇⋅𝐄 + ∇⋅𝐏 = ρfree

        ∇⋅(ε₀𝐄 + 𝐏) = ρfree/ε₀

        𝐃 ≡ ε₀𝐄 + 𝐏

    Bar Electret

        - Analogous to Magnetic bar

        Two limits and an intermediate case.

        ∇×𝐃 = ∇×(ε₀𝐄 + 𝐏) = ∇×𝐏

        𝐄 for bar electret, properties:

            (𝐄₂ - 𝐄₁)⋅𝐧̂₁→₂ = σ/ε₀ with σ = σᵦ + σfree

            Curl eq shows 𝐄"₂ - 𝐄"₁ = 0 (tangential components).

            (𝐃₂ - 𝐃₁)⋅𝐧̂₁→₂ = σfree

            (𝐃"₂ - 𝐃"₁) = 𝐏"₂ - 𝐏"₁

            𝐏 = ε₀χₑ𝐄

            • χₑ is the susceptibility of a linear dielectric material.

            𝐃 = ε₀𝐄 + 𝐏 = ε₀𝐄 + ε₀χₑ𝐄 = ε₀(1+χₑ) 𝐄

            • ε₀(1+χₑ) ≡ permittivity
            • 1+χₑ = dielectric constant

    Metal Sphere

        (pic) What is V(0)?

        (pic) ε becomes relevant in electric field


        𝐄 = ⎧ 𝐃/ε = Q𝐫̂/(4πεr²), R<r<b
            ⎨ 
            ⎩ 𝐃/ε = Q𝐫̂/(4πε₀r²), b<r 

                        R       b        ∞
        V(∞) - V(0) = -∫ 𝐄⋅d𝐥 - ∫ 𝐄⋅d𝐥 - ∫ 𝐄⋅d𝐥
                        0       R        b

                    =   0 - ...