1. ENE/EIE 325 Electromagnetic Fields and Waves
Lecture 6
Conductor, Semiconductor, Dielectric and Boundary Conditions

2. Introduction to electromagnetic material
The properties of electromagnetic material is
specified by , , .
Homogeneous material is the material that
possesses the same properties at every point
in the material.
Isotropic material is the material that its
properties are independent of direction.

3. Homogeneous and isotropic medium
Isotropic Homogeneous
Homogeneous and isotropic

4. Current and Current Density
Current density, J, measured in Amps/m2 , yields current
in Amps when it is integrated over a cross-sectional area.
The assumption would bethat the direction of J is normal
to the surface, and so we would write:
Current is a flux quantity and is defined as:

5. Current Density as a Vector Field
In reality, the direction of current flow may not be normal to the
surface in question, so we treat current density as a vector, and write
the incremental flux through the small surface in the usual way:
where S = n da
Then, the current through a large surface is found through the flux integral:

6. Relation of Current to Charge Velocity
Suppose that in time t, the charge moves through a distance
x = L = vx t
Consider a charge Q, occupying volume v, moving in the positive x
direction at velocity vx.
In terms of the volume charge density, we may write:
Then or
The motion of the charge represents a current given by:

7. Relation of Current Density to Charge Velocity
We now have
The current density is then:
So that in general:
Convection Current Density

8. Continuity of Current
Qi(t)
Suppose that charge Qi is escaping from a volume through
closed surface S, to form current density J. Then the total
current is:
where the minus sign is needed to produce positive outward
flux, while the interior charge is decreasing with time.
We now apply the divergence theorem:
so that or
The integrands of the last expression
must be equal, leading to the
Equation of Continuity

9. Energy Band Structure in Three Material Types
Conductors exhibit no energy gap between valence and conduction bands so electrons move freely.
Insulators show large energy gaps, requiring large amounts of energy to promote electrons into the conduction band. When this occurs, the dielectric breaks down.
Semiconductors have a relatively small energy gap, so modest amounts of energy (applied through heat, light, or an electric field) may promote electrons from valence to conduction bands.

10. Electron Flow in Conductors
Free electrons move under the influence of an electric field. The applied force on
an electron of charge Q = -e will be
When forced, the electron accelerates to an equilibrium velocity, known as the
drift velocity:
where e is the electron mobility, expressed in units of m2/V-s. The drift velocity is
used to find the current density through:
from which we identify the conductivity
for the case of electron flow :
The expression:
is Ohm’s Law in point form
S/m
In a semiconductor, we have hole current as well, and

11. Resistance
Consider the cylindrical conductor shown here, with voltage V applied across the ends.
Current flows down the length, and is assumed to be uniformly distributed over the
cross-section, S.
First, we can write the voltage and current in the cylinder in terms of field quantities:
Using Ohm’s Law:
We find the resistance of the cylinder: a b

12. General Expression for Resistanceb

13. Ex1 A semiconductor sample has a rectangular cross-section 1. 5 by 2
Ex1 A semiconductor sample has a rectangular cross-section 1.5 by 2.0 mm, and a length of 11.0mm. The material has electron and hole densities of 1.8×1018 and 3.0×1015 m−3, respectively. If μe = m2/V · s and μh = m2/V · s, find the resistance offered between the end faces of the sample.

14. Solution: Because
Substitution: 

15. Electrostatic Properties of Conductors
Charge can exist only on the surface as a surface charge density, s -- not in the interior.
Electric field cannot exist in the interior, nor can it possess a tangential component at
the surface (as will be shown next slide).
It follows from condition 2 that the surface of a conductor is an equipotential. s + E
solid conductor
Electric field at the surface
points in the normal
direction
E = 0 inside
Consider a conductor, on which excess charge has been placed

16. Tangential Electric Field Boundary Condition
conductor
dielectric n
Over the rectangular integration path, we use
To find: or
These become negligible as h approaches zero.
Therefore
More formally:

17. Boundary Condition for the Normal Component of D
dielectric
conductor s
Gauss’ Law is applied to the cylindrical surface shown below:
This reduces to:
as h approaches zero
Therefore
More formally:

18. Summary Tangential E is zero At the surface:
Normal D is equal to the surface
charge density

19. Method of Images
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!

20. Forms of Image Charges
Each charge in a given configuration will have its own image

21. Example of the Image Method
Ex2 In this case, we are to find the surface charge density on the conducting plane at the
point (2,5,0). A 30-nCline charge lies parallel to the y axis at z = 3.
The first step is to replace the conducting plane by a line charge of -30 nC at z = -3.

22. Example (continued) Then Referring to the figure, we find:
We now add the two fields to get:
Referring to the figure, we find:
Then

23. Example (concluded) We now have the electric field at point P: Now
To find the charge density, use
n = az
Therefore
where

24. Electric Dipole and Dipole MomentQ d
p = Qd ax
In dielectric, charges are held in position (bound), and ideally there are no free charges
that can move and form a current. Atoms and molecules may be polar (having
separated positive and negative charges), or may be polarized by the application of an
electric field.
Consider such a polarized atom or molecule, which possesses a dipole moment, p, defined
as the charge magnitude present, Q, times the positive and negative charge separation,
d. Dipole moment is a vector that points from the negative to the positive charge.

25. Model of a Dielectric
A dielectric can be modeled as an ensemble of bound charges in free space, associated
with the atoms and molecules that make up the material. Some of these may have
intrinsic dipole moments, others not. In some materials (such as liquids), dipole moments
are in random directions.

26. Polarization Field The number of dipoles is expressed as a density, n
dipoles per unit volume. v 
The Polarization Field of the
medium is defined as:
[dipole moment/vol] or
[C/m2]

27. Polarization Field (with Electric Field Applied)
Introducing an electric field may increase the charge separation in each dipole, and
possibly re-orient dipoles so that there is some aggregate alignment, as shown here.
The effect is small, and is greatly exaggerated here!
The effect is to increase P.
= np
if all dipoles are identical

28. Migration of Bound Charge
Consider an electric field applied at an angle  to a surface normal as
shown. The resulting separation of bound charges (or re-orientation)
leads to positive bound charge crossing
upward through surface of
area S, while negative bound
charge crosses downward through
the surface.

29. Migration of Bound Charge
Dipole centers (red dots) that lie within the range (1/2) d cos above
or below the surface will transfer charge across the surface.

30. Bound Charge Motion as a Polarization Flux
The total bound charge that crosses the surface is given by:
volume P

31. Polarization Flux Through a Closed Surface
The accumulation of positive bound charge within a closed
surface means that the polarization vector must be pointing
inward. Therefore: S P + - qb

32. Bound and Free Charge
Now consider the charge within the closed surface consisting of bound charges, qb , and free charges, q.
The total charge will be the sum of all bound and free charges. We write Gauss’ Law in terms of the total charge, QT as: S P + qb E q QT
where
QT = Qb + Q
free charge
bound charge

33. Gauss Law for Free Charge
QT = Qb + Q
We now have:
and
where
combining these, we write:
we thus identify:
which we use in the familiar form
of Gauss’ Law:

34. Charge Densities
Taking the previous results and using the divergence theorem,
we find the point form expressions:
Bound Charge:
Total Charge:
Free Charge:

35. Electric Susceptibility and the Dielectric Constant
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

36. Isotropic vs. Anisotropic Media
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:

37. Boundary Condition for Tangential Electric Field
We use the fact that E is conservative:
So therefore:
Leading to:
More formally:

38. Boundary Condition for Normal Electric Flux Density
Region 1 1
Region 2 2
We apply Gauss’ Law to the cylindrical volume shown here,
in which cylinder height is allowed to approach zero, and there
is charge density s on the surface: s

39. Boundary Condition for Normal Electric Flux Density
The electric flux enters and exits only through the bottom
and top surfaces, respectively.
From which:
and if the charge density is
zero:
More formally:

40. Example of the Use of Dielectric Boundary Conditions
We wish to find the relation between the angles 1 and 2,
assuming no charge density on the surface.
The normal components of D will be continuous across the boundary,
so that:
Then, with tangential E continuous across
the boundary, it follows that:
or…
Now, taking the ratio of the two
underlined equations, we finally obtain:

41. Ex3 We locate a slab of Teflon in the region 0 ≤ x ≤ a, and assume free space where x < 0 and x > a. Find the fields within the Teflon (r = 2.1), given the uniform external field Eout = E0ax in free space.