Chapter XXVII
Electric Current Properties
At this point
it will be helpful to review the status of the various electrical quantities
from the standpoint of their relationship to the basic entities, space
and time. Electrical quantity, Q, has been identified as space, s,
and current, I, has similarly been identified as velocity, s/t.
Resistance, R, is defined in such a manner as to make it equal
to mass per unit time, t³/s³ × 1/t = t²/s³.
Electrical energy is interchangeable with energy in other forms and like
energy in general is t/s, the reciprocal of velocity.
Energy per unit
time is power, hence power is t/s × 1/t = 1/s.
Power divided by current is electromotive force, which makes this quantity
equal to 1/s × t/s = t/s² . This is the
general expression for force, and the electromotive force, F, IR,
or emf, therefore has the same basic characteristics as other forces,
gravitational, mechanical, etc. In many respects it is analogous
to gas pressure, which is also a force phenomenon; that is, force per
unit area. The magnitude of the emf, or potential, at any point may be
increased in the same manner that gas pressure is increased, either by
the introduction of more electrons of the same average velocity or by
imparting a greater velocity to the electrons already present. If this
location is connected by means of a conductor with a region which does
not participate in the increase in potential, the force difference which
is created will cause a flow of current from the high potential region
to the region of lower potential. This flow will persist until the potentials
are equalized. Ordinarily we deal with currents which are produced by
some agency that creates a continuing potential difference, and the current
flows in a circuit starting and terminating at the generating agency.
It is not essential, however, that such a circuit exist; a current will
flow between any two points of different potential if the necessary conductor
is available.
When the potential
difference which caused the flow of current is eliminated the directional
movement of the electrons which constitutes the current ceases. The electrons
remaining in any particular volume, however, continue to react with the
moving atoms of matter and since the electrons as well as the atoms are
free to move the eventual result is an equilibrium wherein the thermal
motion is divided between motion of mass through space and motion of space
(electrons)through matter. The nature of the equilibrium; that is, the
division of motion between the electrons and the mass, is determined by
the average resistance in all directions. At a given temperature the atoms
of a low resistance conductor such as copper impart less velocity to the
electrons than the atoms of a conductor such as iron which has a greater
resistance. Under the same conditions the various conductors therefore
have different electron velocities and where the density of electrons
is the same the electron pressure or potential depends upon the characteristics
of the conductor. It should be noted, however, that the resistance of
a conductor to the thermal motion of the electrons is not necessarily
identical with the resistance which this conductor offers to the flow
of electric current, inasmuch as there is no requirement that the directions
of these motions be coincident. The possibility of a difference in flow
direction is obvious in the case of anisotropic substances but even where
the conductor is isotropic the directions of movement relative to the
line of action of the interatomic forces may not coincide.
If we place two
conductors with different electron potentials, copper and zinc for example,
in contact the higher potential of the electrons in the zinc will cause
a flow from the zinc to the copper until the density of electrons in the
copper becomes great enough to equalize the potential. We then have an
equilibrium of potential between a smaller number of high velocity electrons
in the zinc and a greater number of low velocity electrons in the copper.
This diff erence in potential which becomes apparent when two dissimilar
conductors are placed in contact is known as a contact potential.
The effect of
temperature on any aggregate of electrons is more complex than the corresponding
effect on an aggregate composed of material atoms. The latter exists in
free space and consequently the reaction to addition or removal of thermal
energy is determined by the properties of the material aggregate itself
without any modification by the environment. The behavior of the electron
aggregate, on the contrary, is determined not only by its own properties
but also by the properties of the conductor in which it is located. As
a result the electronic effects show a range of variation both in magnitude
and direction which is totally foreign to the analogous phenomena involving
material atoms.
If a conductor
is heated, the primary effect on the electrons within the conductor is
a decrease in potential, the reaction of the electrons (units of space)
being the inverse of the reaction of the material atoms (units of time)
to the same addition of energy. Simultaneously, however, the heating of
the conductor causes an increase in resistance and a corresponding increase
in the potential per electron, as explained in the preceding paragraphs.
The net result depends on the relative magnitude of the two effects. In
a low resistance conductor such as copper or silver, the increase in potential
due to the heating of the conductor is smaller than the direct effect
of the temperature on the electrons and there is a net loss in potential
as the temperature rises. When a conductor of this type is heated at one
end only the cold end acquires a higher (more negative) potential: a phenomenon
known as the positive Thompson effect. In a conductor such as
iron or mercury which has a higher resistance the increase in potential
due to the change in resistance may be greater than the direct effect
on the electrons, in which case the hot end of the conductor acquires
the higher potential: the negative Thomson effect. The effective resistance
for this purpose is, of course, the resistance to thermal motion of the
electrons and it does not necessarily coincide with the resistance to
directional flow, as previously pointed out, but there is a general qualitative
correspondence between the two, as would be inferred from the examples
cited.
Now let us construct
a circuit of two different conductors as in Fig. 41 and cause a current
to flow in this circuit. At junction A where the electrons flow from zinc
to copper they leave the zinc with the relatively high potential which
represents the equilibrillim condition in the zinc conductor. In the copper
conductor the equilibrium potential is lower, and the electrons therefore
reduce their potential in the process of attaining equilibrium. This reduction
in electrical potential corresponds to an increase in thermal energy and
consequently the electrons absorb heat from the surroundings. The flow
of electrons thus results in a cooling effect at junction A. Where the
electrons return to the zinc conductor at junction B the reverse process
takes place and heat is given up to the environment. This phenomenon is
known as the Peltier effect.
The inverse of
the Peltier effect is the Seebeck effect or thermoelectric
effect. Here heat is applied to junction A. This lowers the potential
per electron and since there are more electrons in the copper than in
the zinc the effective potential of the copper drops below that of the
zinc, causing a current flow from zinc to copper. If both junction A and
junction B are at the same temperature the flow is only momentary until
the necessary potential equilibrium is established but if one junction
alone is heated a continuous current is produced and heat is transferred
from the hot junction to the cold junction through the agency of the current.
In view of the
free motion of the electrons in conductors and theestablishment of thermal
equilibrium between the electrons and matter it is obvious that the thermal
energy will similarly tend to equalize in all parts of any system which
is interconnected electrically. It follows that the electron movement
constitutes a means of heat transfer whenever a conductor is available.
This type of heat transmission is called conduction.
It should be
noted particularly that the motion of the electrons through matter is
an integral part of the total thermal motion, not something separate.
A mass m reaches a certain temperature T when the thermal
velocity attains a specific average value v. It is immaterial from
this standpoint whether the velocity consists entirely of motion of the
mass through space or partly of such motion and partly of motion of space
(electrons) through the mass. In either case the total velocity corresponding
to the temperature T remains the same. In previous discussions
of the theory that metallic conduction of heat is due to the movement
of electrons the objection has been raised that there is no indication
of any increase in the specific heat due to the thermal energy of the
electron movement. The answer lies in the foregoing explanation that the
thermal motion of the electrons is not an addition to the thermal motion
of the atoms; it is an integral part of the atomic motion and hence has
no effect on the specific heat.
Since the conduction
of heat is accomplished through the same agency as the conduction of electric
currentthe movement of electronsit follows that the resistance to
the flow of heat is the same as the resistance to the flow of current.
There is, however, a difference in the mechanism of conduction which introduces
an additional factor into the heat flow. The force causing the flow of
current is a directional force imposed from the outside and unaffected
by the conditions within the conductor. The application of heat to one
end of a conductor does not introduce any such directional force; it merely
changes the average velocity of the electrons in the heated zone and the
directional force gradient is a secondary effect. The greater the velocity
gradient in the electron aggregate the more rapidly the velocity will
be transferred, hence the effective force causing the heat flow is proportional
to the temperature. The actual rate of heat transfer is the result of
a combination of these two factors. As the temperature increases the resistance
also increases, reducing the rate of heat flow. The same rise in temperature
increases the rate of flow by reason of the greater thermal gradient.
In the neighborhood of room temperature these two influences are nearly
equal and the thermal conductivity, C, therefore has only a relatively
small temperature variation. The general relation in thistemperature range
is expressed in approximate terms by the wellknown WiedemannFranz
Law:
For greater accuracy
it is necessary to take the initial level into account in both the thermal
and electrical factors. Expressing the resistance in terms of temperature
and introducing the initial level in both cases, we have the revised equation
C = k (TI_{T})/(T  I_{E})
 (121)

At the higher
temperatures where T is much larger than I_{T} or
I_{E}, a variation in T has relatively little effect
on C and equation 121 gives substantially the same results as the
WiedemannFranz Law (equation 120). As the temperature decreases the difference
between I_{T} and I_{E} becomes increasingly
effective and since I_{E} is normally larger than I_{T}
the value of C rises, slowly at first and then more rapidly. When
T  I_{E} approaches zero, however, the resistance
diverges from the linear relation and follows a probability curve, as
we have seen in our examination of the resistance relations. Beyond this
point, therefore, further change in the denominator of the conductivity
equation is relatively slow and the decrease in the numerator becomes
the controlling factor. The thermal conductivity thus passes through a
maximum and then drops gradually to zero at zero temperature. Here there
is no resistance to the electron flow but nevertheless there is no heat
conductivity because the electrons have no thermal motion.
The general shape
of the experimental thermal conductivity curves agrees with the theoretical
curve as described in the foregoing paragraph. In view of the uncertainties
in the measurements at low temperatures, however, it has not appeared
worth while to set up any detailed comparisons for this low temperature
range, and since ample supporting data for the WiedemannFranz relation
at the higher temperatures are available elsewhere, no heat conductivity
comparisons are included in this presentation.
