Maxwells equations for a dielectric Four equations for 6 unknowns Cant be solved We need a relation between induction D and field E External field is usually small compared to the internal molecular fields ID: 675567
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Slide1
The Permittivity
LL 8 Section 7Slide2
Maxwell’s equations for a dielectric
Four equations for 6 unknowns
Can’t be solved
We need a relation between induction D and field E.Slide3
External field is usually small compared to the internal molecular fields
Polarization is due to external fieldSlide4
Isotropic dielectric
D
||
ED= e E
e
= permittivity
Depends on substance
Depends on thermodynamic stateSlide5
Polarization is also proportional to the field
For a rarified medium (gas),
k
is proportional to density
e
> 1
k
> 0
Polarization coefficient
or
Dielectric susceptibilitySlide6
Boundary conditions at surface between two dielectrics
The normal component of the field is discontinuous
Linear isotropic dielectricSlide7
Spatially uniform neutral dielectrics have no volume charge density
Homogenous dielectrics have spatially uniform permittivitySlide8
Volume charge density can occur for inhomogeneous dielectric
0 for neutral dielectricSlide9
What equation is satisfied by the potential in a neutral dielectric?
If dielectric is homogeneous, then
Laplace’s equation holds
For inhomogeneous dielectric, Laplace’s equation doesn’t hold.Slide10
Boundary conditions on the potential
Potential is continuous.
(They could differ by a constant, but they wouldn’t they would also be different at infinity.)
Slope of
f
in normal direction is discontinuous
f
1
= f
2
Slide11
Piecewise homogeneous dielectric
(e.g. multilayer optical films)
in each homogeneous region.
Permittivity appears only in the boundary conditions
Boundary conditions involve only the ratios
Same solution as Slide12
Conductor surrounded by homogeneous isotropic dielectric
0 everywhere
Constant on the conductor surfaceSlide13
vacuum
If
e
D
= e
V
, then
f
D
=
f
V
/
e.If
fD = fV, then eD
= e eV Slide14
For a given charge on the conductor (e
V
=
eD)
and
E
are reduced by the factor
e
compared to their values in vacuum.
The charge on the conductor is screened by the surface polarization charge on the dielectric.
Charged conductor in vacuum
Charged conductor surrounded by dielectricSlide15
If the potential of the conductor is maintained as it is placed inside dielectric
f
V
=
f
D
e
D
=
e
e
V
Field is unchanged but the charge is increased by the factor e.The extra charge is put there by the battery.
+
v
+++
v
eSlide16
An uncharged conductor behaves like a dielectric with infinite permittivity
Normal component of electric induction is continuous
But
Requires Slide17
Landau & Gaussian Units
“Permittivity”
or
“dielectric constant”
(
unitless
)
Conventions for linear dielectrics
Other books & S.I. units
D
=
e
E
Polarization coefficient
or
Susceptibility
(
unitless
)
Relative permittivity
or
Dielectric constant
(
unitless
)
Permittivity
(not
unitless
)
Susceptibility (
unitless
)