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58 lines
1.5 KiB
Plaintext
58 lines
1.5 KiB
Plaintext
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Problems with known polarization
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(pic) Uniformly polarized sphere
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Gauss' Law in Dielectric
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ε₀∇⋅𝐄 = ρ = ρfree + ρbound
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= ρfree - ∇⋅𝐏
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ε₀∇⋅𝐄 + ∇⋅𝐏 = ρfree
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∇⋅(ε₀𝐄 + 𝐏) = ρfree/ε₀
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𝐃 ≡ ε₀𝐄 + 𝐏
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Bar Electret
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- Analogous to Magnetic bar
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Two limits and an intermediate case.
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∇×𝐃 = ∇×(ε₀𝐄 + 𝐏) = ∇×𝐏
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𝐄 for bar electret, properties:
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(𝐄₂ - 𝐄₁)⋅𝐧̂₁→₂ = σ/ε₀ with σ = σᵦ + σfree
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Curl eq shows 𝐄"₂ - 𝐄"₁ = 0 (tangential components).
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(𝐃₂ - 𝐃₁)⋅𝐧̂₁→₂ = σfree
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(𝐃"₂ - 𝐃"₁) = 𝐏"₂ - 𝐏"₁
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𝐏 = ε₀χₑ𝐄
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• χₑ is the susceptibility of a linear dielectric material.
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𝐃 = ε₀𝐄 + 𝐏 = ε₀𝐄 + ε₀χₑ𝐄 = ε₀(1+χₑ) 𝐄
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• ε₀(1+χₑ) ≡ permittivity
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• 1+χₑ = dielectric constant
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Metal Sphere
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(pic) What is V(0)?
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(pic) ε becomes relevant in electric field
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𝐄 = ⎧ 𝐃/ε = Q𝐫̂/(4πεr²), R<r<b
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⎨
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⎩ 𝐃/ε = Q𝐫̂/(4πε₀r²), b<r
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R b ∞
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V(∞) - V(0) = -∫ 𝐄⋅d𝐥 - ∫ 𝐄⋅d𝐥 - ∫ 𝐄⋅d𝐥
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0 R b
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= 0 - ...
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