Page 133 - Engineering Electromagnetics, 8th Edition

CHAPTER 5 Conductors and Dielectrics 115

the electron cannot accept additional energy in small amounts, and the material is an
insulator. This band structure is indicated in Figure 5.2b. Note that if a relatively large
amount of energy can be transferred to the electron, it may be sufficiently excited to
jump the gap into the next band where conduction can occur easily. Here the insulator
breaks down.
An intermediate condition occurs when only a small “forbidden region” separates
the two bands, as illustrated by Figure 5.2c. Small amounts of energy in the form of
heat, light, or an electric field may raise the energy of the electrons at the top of the
filled band and provide the basis for conduction. These materials are insulators which
display many of the properties of conductors and are called semiconductors.
Let us first consider the conductor. Here the valence electrons, or conduction,
or free, electrons, move under the influence of an electric field. With a field E,an
electron having a charge Q =−e will experience a force
F =−eE
In free space, the electron would accelerate and continuously increase its velocity
(and energy); in the crystalline material, the progress of the electron is impeded
by continual collisions with the thermally excited crystalline lattice structure, and a
constant average velocity is soon attained. This velocity v d is termed the drift velocity,
and it is linearly related to the electric field intensity by the mobility of the electron
in the given material. We designate mobility by the symbol µ (mu), so that
v d =−µ e E (6)
where µ is the mobility of an electron and is positive by definition. Note that the
electron velocity is in a direction opposite to the direction of E. Equation (6) also
shows that mobility is measured in the units of square meters per volt-second; typical
3
values are 0.0012 for aluminum, 0.0032 for copper, and 0.0056 for silver.
For these good conductors, a drift velocity of a few centimeters per second is
sufficient to produce a noticeable temperature rise and can cause the wire to melt if
the heat cannot be quickly removed by thermal conduction or radiation.
Substituting (6) into Eq. (3) of Section 5.1, we obtain

J =−ρ e µ e E (7)

where ρ e is the free-electron charge density, a negative value. The total charge density
ρ ν is zero because equal positive and negative charges are present in the neutral
material. The negative value of ρ e and the minus sign lead to a current density J that
is in the same direction as the electric field intensity E.
The relationship between J and E for a metallic conductor, however, is also
specified by the conductivity σ (sigma),

J = σE (8)

3 Wert and Thomson, p. 238, listed in the References at the end of this chapter.