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
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n
Qi(t)
^^ Why is current increasing ?
<<Some central repulsive force!
Free electrons are accelerated by an electric field. The applied force on an electron of charge Q = e is
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/Vs. 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)
S/m (electrons)
or
Over the rectangular integration path, we use
To find:
dielectric
n
These become negligible as h approaches zero.
Therefore
conductor
More formally:
Gauss’ Law is applied to the cylindrical surface shown below:
dielectric
This reduces to:
as h approaches zero
n
Therefore
s
conductor
More formally:
Tangential E is zero
At the surface:
Normal D is equal to the surface charge density
2
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The Theorem of Uniqueness states that if we are given a configuration of charges and
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!
Each charge in a given configuration will have its own image
Want to find surface charge density on conducting plane at the point (2,5,0). A 30nC line of charge lies parallel to the y axis at x=0, z = 3.
First step is to replace conducting plane with image line of charge 30 nC at z = 3.
Vectors from each line charge to observation point:
Electric Fields from each line charge
Add both fields to get: (x component cancels)

Electric Field at P is thus:
Displacement is thus
Charge density is
n
D
moment and position
amount of charge (Q) and offset (d) of charge
Introducing an electric field may increase the charge separation in each dipole, and possibly reorient dipoles so that
there is some aggregate alignment, as shown here. The effect is small, and is greatly exaggerated here!
E
The effect is to increase P.
n = charge/volume
p = polarization of individual dipole
P = polarization/volume
E
positive
(Note dot product sign, outward normal leaves opposite charge enclosed)
               
neutral
negative
+ + + + + + + + + + + + +
E
positive
               
neutral
negative
+ + + + + + + + + + + + +
Bound Charge:
Total Charge:
Free Charge:
Taking the previous results and using the divergence theorem, we find the point form expressions:
A stronger electric field results in a larger polarization in the medium. In a linear medium, the relation
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:
where
In an isotropic medium, the dielectric constant is invariant with direction of the applied electric field.
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:
imaginary permittivity peaks
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Requirements:
Debye for free & medium. ColeDavidson for low. (literature, biosystems)
Combined
9 variables fit over entire range, real & imaginary, 2stage fit, f = 8.2 ps
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Since E is conservative, we setup line integral straddling both dielectrics:
Left and right sides cancel, so
Leading to Continuity for tangential E
And Discontinuity for tangential D
E same, D higher in high permittivity material
n
Apply Gauss’ Law to the cylindrical volume straddling both dielectrics
s
Flux enters and exits only through top and bottom surfaces, zero on sides
Leading to Continuity for normal D (for ρS = 0)
And Discontinuity for normal E
D same. E lower in high permittivity material
high
low