We know that when we apply an electric field to a charge that
there is a force exerted on it, and that if the charge is ableto move it will do so. The motion of charge gives rise to an
electric current , which we call
.
The current is a measure of how much charge is passing a givenpoint per unit time (
).
It will be helpful if we have some kind of model of how
electricity flows in a conductor. There are several approacheswhich one can take, some more intuitive than others. The one we
will look at, while not correct in the strictest sense, stillgives a very good picture of how electrical conduction works,
and is perfectly fine to use in a variety of situations. In the
Drude theory of conduction, the initial hypothesis
consists of a solid, which contains mobile charges which arefree to move about under the influence of an applied electric
field. There are also fixed charges of polarity opposite thatof the mobile charges, so that everywhere within the solid, the
net charge density is zero. (This hypothesis is based on themodel of the atom, with a positively charged nucleus and
negatively charged electrons surrounding it. In a solid, theatoms are fixed in position in the lattice, but it is assumed
that some of the electrons can break free of their "host" atomand move about to other places within the solid.) In our model,
let us choose the polarity of the mobile charges to be positive;this is not usually the case, but we can avoid a lot of
"minus ones" this way, and have a better chance of ending up
with the right answer in the end.
Model of a conductor. As shown in
, the model of the
conductor consists of a number of mobile positive charges(represented by the balls with the "+" sign in them) and an
equal number of fixed negative charges (represented by the bare"-" sign). In subsequent figures, we will leave out the fixed
charge, since it can not contribute in any way to the conductionprocess, but keep in mind that it is there, and that the total
net charge is zero within the material. Each of the mobilecharge carriers has a mass,
, and
an amount of charge,
.
Applying a potential to a conductor In order to have some conduction, we have to apply a potential
or voltage across the sample (
). We do this with a battery, which
creates a potential difference,
,
between one end of the sample and the other. We will make thesimplest assumption that we can, and say that the voltage,
, gives rise to a uniform electric
field within the sample. The magnitude of the electric field isgiven simply by
where
is the length of the
sample, and
is the voltage which
is placed across it. (In truth, we should be showing
as well as subsequent forces
etc. as vectors in our equations, but since their direction willbe obvious, and unambiguous, let's keep things simple, and just
write them as scalers.)
Electric potential , or
voltage, is just a measure of the change in potential energy perunit charge going from one place to another. Since energy, or
work is simply force times distance, if we divide the energy perunit charge by the distance over which that potential exists, we
will end up with force per unit charge, or electric field,
. If you are not sure about what
you just read, write it out as equations, and see that it is so.
The electric field will exert a force on the movable charges
(And the fixed ones too for that matter, but since they can notgo anywhere, nothing happens to them). The force is given
simply as the product of the electric field strength times thecharge
The force acts on the charges and causes them to accelerate
according to Newton's equations of motion
or
Thus, the velocity of a particle with no initial velocity will
increase linearly with time as:
The rate of acceleration is proportional to the strength of
the electric field, and inversely proportional to the mass ofthe particle. The particle can not continue to accelerate forever
however. Since it is located within a solid, sooner or later itwill collide with either another carrier, or perhaps one of the
fixed atoms within the solid. We will assume that the collisionis completely inelastic, and that after a collision, the
particle comes to a stop, only to be accelerated again by theelectric field. If we were to make a plot of the particles
velocity as a function of time, it might look something like
.
Velocity as a function of time for charge carrier Although the particle achieves various velocities, depending
upon how much time there is between collisions, there will besome average velocity,
, which will depend upon the details of the collision
process. Let us define a scattering time
which will give us that average
velocity when we multiply it times the acceleration of theparticle. That is:
or
Now let's take a look at just a small section of the conductor (
).
It will have the cross section of the sample,
, but will only be
long, where
is just some arbitrary time interval.
Section of the conductor After a time
has passed, all of the charges within the box will
have left it, as they are all moving with the same averagevelocity,
. If the density of charge carriers in the conductoris
per unit volume, then the
number of carriers
within our
little box is just
times the
volume of the box
Thus the total charge,
, which
leaves the box in time
is just
. The current flow,
,
is just the amount of charge which flows out of the box per unittime
We now have two choices, we can look at our result from a field
quantity point of view, in which case we will be interested inthe
current density ,
, which is
just the current,
, divided by the
cross-sectional area
where
is called the
conductivity of the material. If we look at the
conductor from a macroscopic point of view, then we areinterested in the relationship between the voltage and the
current. The voltage is just the electric field times thelength of the sample, and the current is just the current
density times is cross sectional area. Thus we have
or
where
is the resistance of the
sample. We have discovered
Ohm's law !
Note that
tells us that the resistance
of the sample is proportional to its length (the longer thesample, the higher the resistance) and inversely proportional to
its cross sectional area (the fatter the sample, the lower theresistance). The sample resistance is also inversely
proportional to the conductivity
of the sample. Sometimes, instead of conductivity,
the
resistivity ,
, is specified for a resistive material. The
resistivity is simply the inverse of the conductivity
and thus:
And, in an effort towards completeness, there is one other
quantity which you might run into, and that is the carrier
mobility ,
. The
mobility is just the proportionality factor between the averagevelocity of the particle and the electric field. That is:
You should check that the following two relationships are
correct:
If we take an ordinary conductor (and we will have to define
later what we mean by that) and heat it up, the atoms within thematerial start to vibrate faster due to the elevated
temperature, and the carriers suffer significantly morecollisions. The mean collision time
decreases, and hence the conductivity goes down, and the
resistance of the sample goes up.
