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§4.2–3 Displacement

§4.2–3 Displacement. Christopher Crawford PHY 311 2014-03-07. Outline. Review – D= ε 0 E+P New Gauss ’ law – displacement field boundary conditions – obtained as usual

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§4.2–3 Displacement

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  1. §4.2–3 Displacement Christopher Crawford PHY 311 2014-03-07

  2. Outline • Review – D=ε0E+PNew Gauss’ law – displacement fieldboundary conditions – obtained as usual • Constitutive equation –ε = ε0εr = ε0(1+χe)Electric susceptibility – P vs E, compare: polarizabilityDielectric constant – amplification of free charge[relative] permittivity – D vs E • Examplesparallel plate capacitorpolarized sphere dielectric sphere in an external field

  3. New Gauss’ (flux) law: • MACROSCOPIC formulation • New field: D = ε0E + P (electric displacement) • Derived from E, P Gauss’ laws • Corresponding boundary condition Old (flow) law: • E field still responsible for force -> potential energy • V is still defined in terms of E • Boundary conditions: potential still continuous

  4. Polarizability vs. Susceptibility • Polarizability • Dipole moment of single atom in an electric field • Susceptibility • Polarization [density] of a material in an electric field • Relation between the two • Clausius-Mossotti relationship

  5. Dielectric material properties In general the polarization is an arbitrary function of: • Electric field, position(wavelength), time(frequency), temperature, … “Electrets” even have polarization independent of E However most materials satisfy the following properties which makes it much easier to calculate the fields: • Linear– χe independent of magnitude of E • Polarization proportional to electric field • Isotropic– χe independent of direction of E • Polarization in the same direction as electric field • Homogeneous– χe independent of position • Material doesn’t change from place to place

  6. Permittivity: constitutive equation • Link between D and E in Maxwell’s equations • Susceptibility • Relative permittivity (dielectric constant) • Permittivity of free space [vacuum] • Absolute permittivity • Relations between constants • Permittivity reflects the same material propertiesas susceptibility: linear, isotropic, homogeneous • In general it is a tensor (matrix) function ε(E,r,ω,…)

  7. Macroscopic potential formulation • Poisson’s equation  Laplace’s equation • Continuity boundary conditions

  8. Example: Parallel plates w/dielectric

  9. Example: Polarized dielectric

  10. Example: dielectric sphere in external E

  11. Example: dielectric sphere in external E

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