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

2. Outline
Review – D=ε0E+P New Gauss’ law – displacement field boundary conditions – obtained as usual
Constitutive equation – ε = ε0εr = ε0(1+χe) Electric susceptibility – P vs E, compare: polarizability Dielectric constant – amplification of free charge [relative] permittivity – D vs E
Examples parallel plate capacitor polarized sphere dielectric sphere in an external field

3. New Gauss’ (flux) law: Old (flow) 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
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 properties as 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