Conductors and Dielectrics. Conductors Current, current density, drift velocity, continuity Energy bands in materials Mobility, micro/macro Ohm’s Law Boundary conditions on conductors Methods of Images Dielectrics Polarization, displacement, electric field
^^ Why is current increasing ?
But in reality the electrons are constantly bumping into things (like a terminal velocity) so they attain an equilibrium or drift velocity:
where eis the electron mobility, expressed in units of m2/V-s. The drift velocity is used in the current density through:
So Ohm’s Law in point form (material property)
With the conductivity given as:
S/m (electrons/holes)Ohm’s Law (microscopic form)
Over the rectangular integration path, we use
These become negligible as h approaches zero.
More formally:Boundary Condition for Tangential Electric Field E
This reduces to:
as h approaches zero
More formally:Boundary Condition for the Normal Displacement D
boundary conditions, there will exist only one potential and electric field solution.
In the electric dipole, the surface along the plane of symmetry is an equipotential with V = 0.
The same is true if a grounded conducting plane is located there.
So the boundary conditions and charges are identical in the upper half spaces of both configurations
(not in the lower half).
In effect, the positive point charge images across the conducting plane, allowing the conductor to be
replaced by the image. The field and potential distribution in the upper half space is now found much
more easily!Method of Images
First step is to replace conducting plane with image line of charge -30 nC at z = -3.Example of the Image Method
Electric Fields from each line charge
Add both fields to get: (x component cancels)Example of the Image Method (continued)
moment and position
amount of charge (Q) and offset (d) of charge
there is some aggregate alignment, as shown here. The effect is small, and is greatly exaggerated here!
The effect is to increase P.
n = charge/volume
p = polarization of individual dipole
P = polarization/volumePolarization as sum of dipole moments (per volume)
(Note dot product sign, outward normal leaves opposite charge enclosed)
- - - - - - - - - - - - - - - -
+ + + + + + + + + + + + +
between P and E is linear, and is given by:
where e is the electric susceptibility of the medium.
We may now write:
where the dielectric constant, or relative permittivity is defined as:
Leading to the overall permittivity of the medium:
whereElectric Susceptibility and the Dielectric Constant
This is not the case in an anisotropic medium (usually a crystal) in which the dielectric constant will vary
as the electric field is rotated in certain directions. In this case, the electric flux density vector components
must be evaluated separately through the dielectric tensor. The relation can be expressed in the form:Isotropic vs. Anisotropic Media
imaginary permittivity peaks
Debye for free & medium. Cole-Davidson for low. (literature, biosystems)
9 variables fit over entire range, real & imaginary, 2-stage fit, f = 8.2 ps
Left and right sides cancel, so
Leading to Continuity for tangential EBoundary Condition for Tangential Electric Field E
And Discontinuity for tangential D
E same, D higher in high permittivity material
Apply Gauss’ Law to the cylindrical volume straddling both dielectrics
Flux enters and exits only through top and bottom surfaces, zero on sidesBoundary Condition for Normal Displacement D
Leading to Continuity for normal D (for ρS = 0)
And Discontinuity for normal E
D same. E lower in high permittivity material