Page 134 - Engineering Electromagnetics, 8th Edition

116 ENGINEERING ELECTROMAGNETICS

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where σ is measured is siemens per meter (S/m). One siemens (1 S) is the basic
unit of conductance in the SI system and is defined as one ampere per volt. Formerly,
the unit of conductance was called the mho and was symbolized by an inverted .
Just as the siemens honors the Siemens brothers, the reciprocal unit of resistance that
we call the ohm (1 is one volt per ampere) honors Georg Simon Ohm, a German
physicist who first described the current-voltage relationship implied by Eq. (8). We
call this equation the point form of Ohm’s law; we will look at the more common
form of Ohm’s law shortly.
First, however, it is informative to note the conductivity of several metallic con-
7
ductors; typical values (in siemens per meter) are 3.82×10 for aluminum, 5.80×10 7
7
for copper, and 6.17 × 10 for silver. Data for other conductors may be found in
Appendix C. On seeing data such as these, it is only natural to assume that we are be-
ing presented with constant values; this is essentially true. Metallic conductors obey
Ohm’s law quite faithfully, and it is a linear relationship; the conductivity is constant
over wide ranges of current density and electric field intensity. Ohm’s law and the
metallic conductors are also described as isotropic, or having the same properties in
every direction. A material which is not isotropic is called anisotropic, and we shall
mention such a material in Chapter 6.
The conductivity is a function of temperature, however. The resistivity, which
is the reciprocal of the conductivity, varies almost linearly with temperature in the
region of room temperature, and for aluminum, copper, and silver it increases about
5
0.4 percent for a 1-K rise in temperature. Forseveral metals the resistivity drops
abruptly to zero at a temperature of a few kelvin; this property is termed super-
conductivity. Copper and silver are not superconductors, although aluminum is (for
temperatures below 1.14 K).
If we now combine Equations (7) and (8), conductivity may be expressed in terms
of the charge density and the electron mobility,

σ =−ρ e µ e (9)
Fromthedefinitionofmobility(6),itisnowsatisfyingtonotethatahighertemperature
infersagreatercrystallinelatticevibration,moreimpededelectronprogressforagiven
electric field strength, lower drift velocity, lower mobility, lower conductivity from
Eq. (9), and higher resistivity as stated.
The application of Ohm’s law in point form to a macroscopic (visible to the naked
eye) region leads to a more familiar form. Initially, assume that J and E are uniform,
as they are in the cylindrical region shown in Figure 5.3. Because they are uniform,

I = J · dS = JS (10)
S

4 This is the family name of two German-born brothers, Karl Wilhelm and Werner von Siemens, who
were famous engineer-inventors in the nineteenth century. Karl became a British subject and was
knighted, becoming Sir William Siemens.
5 Copious temperature data for conducting materials are available in the Standard Handbook for
Electrical Engineers, listed among the References at the end of this chapter.