Recommended problems: FGT 27.18, 27.32, 27.60
Recommended reading: FGT pp747-754
Electric current
Electric current describes the flow of charge. Thus we define
current as charge per unit time, I = dQ/dt. When we write
this formula we consider the net amount of charge that crosses a surface
per unit time. Its units
are C/s, however current is so important it has its own
unit, the amp (A). The direction of positive current is
the direction of flow of positive charge. This is
the natural definition in terms of the electrostatics
definitions that we have used. However it is a little
unnatural in that in many cases it is the electrons which
flow. Electron flow in a specified direction
is negative current flow in that direction.
Current in terms of current density
Current density is the current per unit area. If all of the
current density flows in the same direction, then the current
is the current density times the area, ie. I=jAwhere A is the area. However if the current density has a variety
of directions we have,
(1) |
Current in terms of individual charges
Consider that we have a number density (number per unit volume), nq,
of charged carriers which each have charge q. Suppose that
these charge carriers have net ``drift'' velocity
in a specified
direction. The current density is then given by,
(2) |
Physical origin of drift velocity
The physical origin of the drift velocity is most
easily seen using the so-called free electron
model. In this model, metals are composed
of a sea of free electrons which floats in
between positive nuclei whose positions are
essentially fixed (there are only some small vibrations
about these fixed positions). The electrons move
at high speed and in random directions. They
scatter off each other, off the positive nuclei and
from impurities. The time between scattering events
is
the scattering time of the material. This
varies from metal to metal. Now consider applying
an electric field in a given direction. The electric
field leads to an acceleration ma = - eE of each
electron. Over a time ,
the velocity of the
electron thus increases by
in a direction
opposite to the direction of the applied field. This is
the drift velocity,
(3) |
(4) |
Resistance, conductance
Ohm's, V=IR is famous, and was found experimentally. Using
it we define the resistance R=V/I which has the unit the
Ohm (). The resistor is a second circuit element which
we need to learn how to analyse. Conductance is,
in the simplest case, simply 1/R. We shall study predominantly
linear resistors which obey Ohm's law. However many
circuits use non-linear resistors and other non-linear
circuit elements such as diodes and transistors.
Resistivity, conductivity
Resistivity is defined from .
From this we see that
(5) |
(6) |
Resistivity and conductivity are the ``intrinsic'' properties of a material. They do not depend on the geometry of a sample. In contrast, the resistance and conductance do depend on geometry (i.e. length and cross-sectional area). Material science and physics typically concentrate on designing atomic structure and morphology to control the resistivity and conductivity. Engineers typically engineer the geometry. However these days these area are merging in nano-engineering. Actually integrated circuits already combine engineering of the intrinsic properties (through silicon doping levels) as well as geometry.
Within the free electron model, using
,
and Eq. (4),
we have the elegant formula,
(7) |
Electric power
When a small amount of charge dq moves through a potential difference V, the change
in potential energy is dU = dq V. In the case of resistors,
this potential energy is dissipated as heat. The amount of
power generated as heat is,
(8) |