Electromagnetic Properties of
Materials
All
materials can be
classified according to
their electrical properties
into three
types
conductors, semiconductors, and
insulators (or dielectrics). In terms of
the crude atomic
model
of an atom
consisting of a
positively charged nucleus
with orbiting electrons,
the electrons in the
outermost shells of
the atoms of
conductors are very
loosely held and migrate
easily from one atom to another. Most
metals belong to this group
. The electrons in the atoms of insulators or dielectrics,
however, are confined to their orbits; they cannot be
liberated in normal circumstances,
even by the application of an external electric field. The
electrical properties
of semiconductors fall
between those of
conductors and insulators
in that they possess a relatively
small number of freely movable charges.
In
terms of the
band theory of
solids we find
that there are
allowed energy bands
for electrons, each band
consisting of many
closely spaced, discrete
energy states.
Between
these energy bands there may be
forbidden regions or gaps where no electrons of the solid's
atom can reside. Conductors have an upper energy band
partially filled with electrons or an
upper pair of overlapping bands that
are partially filled so that the electrons in these bands can move from one to
another with only a small change in energy.
Insulators or dielectrics
are materials with a completely
filled upper band, so conduction could not normally occur
because of the existence of a large
energy gap to the next higher band. If
the energy gap of
the forbidden region is relatively
small, small amounts of external energy may be sufficient
to
excite the electrons
in the filled
upper band to
jump into the
next band, causing
conduction. Such materials are semiconductors
Conductors in Static Field
Assume for
the present that
some positive (or
negative) charges are
introduced in the
interior of a conductor. An electric field will be set up in the
conductor, the field exerting a
force on the charges and making them
move away from one another. This
movement will
continue until
all the charges
reach the conductor
surface and redistribute
themselves in such a
way that both
the charge and
the field inside
vanish. Hence, inside
a conductor
(under
static conditions), the
volume charge density
in Cm
−3
rr
= 0. When
there is no
charge in the interior of a conductor
(r=0), E must be zero.
The
charge distribution on
the surface of
a conductor depends
on the shape
of the
surface. Obviously,
the charges would
not be in
a state of equilibrium if
there were a
tangential component
of the electric
field intensity that
produces a tangential
force and
moves the charges. Therefore, under static conditions the E
field on a conductor surface
is
everywhere normal to
the surface. In
other words, the
surface of a
conductor is an
equipotential surface under static
conditions. As a matter of fact, since E
= 0 everywhere
inside a conductor, the whole
conductor has the same electrostatic potential.
A finite time is
required for
the charges to
redistribute on a
conductor surface and
reach the equilibrium
state. This time depends on the conductivity of the
material. For a good conductor such as
copper this time is of the order of
10
−19
(s), a very brief transient.
Conductors Carrying Steady
Electric Currents
Conduction currents
in conductors and
semiconductors are caused
by drift motion
of
conduction electrons
and/or holes. In
their normal state
the atoms consist
of positively
charged nuclei
surrounded by electrons
in a shell-like
arrangement. The electrons
in the
inner shells are tightly bound to the
nuclei and are not free to move away.
The electrons in may wander from
one atom to
another in a
random manner. The
atoms, on the
average,
remain electrically neutral, and
there is no net drift motion of electrons. When an external
electric field is applied on a
conductor, an organized motion of the conduction electrons will
result, producing an electric
current. The average drift velocity of
the electrons is very low
(on the order
of 10
-4
or 10
-3
m/s) even for very good conductors
because they collide with the
atoms in the course of their motion,
dissipating part of their kinetic energy as heat. Even
with
the drift motion
of conduction electrons,
a conductor remains
electrically neutral.
Electric forces prevent excess
electrons from accumulating at any point in a conductor.
Consider the
steady motion of one kind
of charge carriers,
each of charge
q (which is
negative for electrons), across an
element of surface Δs with a velocity u.
If N is the number
of charge carriers per unit volume,
then in time Δt each charge carrier moves a distance u Δt,
and the amount of charge carrier
passing through the surface Δs is
Δ Δ Δ Q Nq s
t
n = • u
a (C)
Since current is the time rate of
change of charge, we have
Δ
Δ
Δ
Δ Δ I
Q
t
Nq s
n = = • =
• u a J
s (A)
where
J u = Nq (A/m
2
) is
the volume current
density, or simply
current density and
Δ s=a n Δs.
It
can be justified
analytically that for
most conducting materials
the average drift
velocity is directly proportional to
the applied external electric field strength.
For metalic
conductors we write
u E = −m
e
(m/s)
where m e is the electron mobility measured in (m
2
/Vs). The electron mobility for copper is
3.2×10
-3
(m
2
/Vs). It
is 1.4×10
-4
(m
2
/Vs) for
aluminum and 5.2×10
-3
(m
2
/Vs) for
silver.
Therefore, we obtain the point form
of Ohm's law:
J E E = − = r m s
e
e
(A/m
2
)
where
r e =−Ne is the
charge density of
the drifting electrons,
and s=−r e m e a macroscopic
constitutive parameter of the medium
known as conductivity. The SI unit for
conductivity
is ampere per volt-meter (A/Vm) or
siemens per meter (S/m). The reciprocal of conductivity
is known as resistivity in ohm-meters
(Ωm).
In the physical world we have an
abundance of "good conductors" such as silver, copper,
gold, and aluminum, whose
conductivities are of the order of 10
7
(S/m).
There are super-
conducting materials whose
conductivities are essentially infinite (in excess of 10
20
S/m) at
the outermost shells of a conductor
atom do not completely fill the shells; they are valence or
conduction electrons and are only
very loosely bound to the nuclei. These
latter electrons cryogenic
temperatures. They are
called superconductors. Because
of the requirement
of
extremely low
temperatures, they have
not found much
practical use. However,
this
situation is
expected to change
in the near
future, since scientists
have recently found
temperatures (20-30 degrees above 77
K boiling point of nitrogen, raising the possibility of
using
inexpensive liquid nitrogen
as coolant). At
the present time
the brittleness of the
ceramic materials
and limitations on
usable current density
and magnetic field
strength
remain obstacles
to industrial applications. Room-temperature superconductivity is
still a
dream.
For
semiconductors, conductivity depends
on the concentration
and mobility of
both
electrons and holes:
s r m
r m = − + e e h
h
where the subscript h denotes hole.
Resistance Calculation
Consider a piece of homogeneous
material of conductivity s, length l, and uniform cross
section A,
as shown below.
Within the conductor,
J=sE, where both
J and E
are in the
direction of current flow. The potential difference or voltage between
terminals 1 and 2 is
V El
12 =
or
E V l
= 12
and the total current is
I JA EA
A
= •
= = ∫ J dA s
=
sA
l
V
12
or I
V
R
= 12
where R
l
A
=
s
is
the resistance between
two terminals. The
unit for resistance
is Ohms (Ω).
The
reciprocal of resistance is defined
as conductance or G=1/R.
The unit for
conductance is
siemens (S)
or (Ω -1
). This
equation can be
applied directly to
uniform cross sectioned
bodies operating at low frequencies..
What is a dielectric?
The dielectric constant of a material
measures how the material responds
to an applied external electric field.
If the atoms in the material have a
dipole moment they will tend to
orient
themselves in the applied field so the
net field in the material is reduced.
Internal electric
field reduced from
the
vacuum value by
the dielectric
constant ǫ r .
Dielectric are non-metallic
substances (gas, liquid, or solid).
Many practical applications like
storing energy in capacitors,
piezoelectric for making
measurements, accumulating charge in an
accelerator