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PROFESSOR CIMA: Today we're going to talk about properties of an electron
while a current is flowing. First, we're going to address how many electrons
we're talking about. And we'll use the example of copper.
Copper has an electronic structure shown here, with the completed d
subshell and one electron in the 4s orbital. So the electronic structure of metallic copper
will consist primarily of a 4s band.
The molecular weight of copper is 63.55. And its density is 8.94.
The molar density of copper is just given by 8.94, the mass density,
divided by the molecular weight. And so you end up with 0.141 moles per cc.
If you wanted to know the atomic density, the number of atoms per cc,
you have to multiply this by Avogadro's number. And you get 8.49 times 10 to the 22 atoms
per cc. If you wanted that per cubic meter, you have
to multiply this by 1 million.
And we'll use that number in a moment. Now, each one of these atoms contributes one
electron to the s band. And that being the case, we can calculate
the electronic density in the s band as being exactly the same as this
number, except that instead of atoms per cubic meter, it'd be electrons per
cubic meter. And that's fairly typical for a metal.
Now over here is a piece of copper that we're going to
apply a voltage to. We'll apply 10 volts across the long dimension
of this piece of copper. It's 1 meter long.
And it has cross-sectional area of 1 square millimeter.
That's the same as 1 times 10 to the minus 6 square meters.
Now of course, the current that we generate will be V, the voltage that
we apply, divided by the resistance of this piece of metal.
We know that the resistance is related to the resistivity of the material, in
this case copper, times L over A. Now the resistivity for copper it is 1.68
times 10 to the minus 8 Ohm * meters. This is the resistivity at room temperature.
When you plug this in here and these parameters into the other factors
here, you end up calculating that the current that we'll generate with this
10 volts is around 600 amps. Quite a bit of current.
Of course, that's just what will happen when we
first apply the voltage. You probably know from your experience, when
you pass current through metals, they start to heat up.
One of the things we haven't talked about, but is very true, and we'll
learn why, is that as you heat up a metal, the resistivity increases with
temperature. There are two questions that arise from this.
Why does passing a current cause the metal to heat up?
And why does a metal's resistivity increase with temperature?
We'll address this in an upcoming screencast. But for right now, let's just look at that
first instance, when we apply the voltage and the 600 amps is generated.
We'll ask the question, how fast are the electrons moving?
Get an estimate of that number, we can first calculate two other factors that
are related to this. First is, the number of electrons per time
that are actually coming out the ends of this piece of metal as we pass the
current. Well, that's just the current divided by the
charge on the electron. This is in coulombs per second.
If I divide by the coulombs on a single electron, I'll get the number
per time that are actually coming out the ends.
The other factor that we can calculate is the number of electrons per length.
Well that's just the density of electrons, the number of electrons per
unit volume times the area. Now these two factors can be combined to get
how fast the electrons are moving by just taking the ratio.
To see that, note that the number of electrons per time divided by the
number of electrons per length gives you length per time, or velocity.
We'll call this the drift velocity in the subsequent screencast.
So what is the drift velocity? Well if you take these expressions over here
and actually take the ratio, you'll find that the drift velocity is I over
rho_e * A * e. So that if I plug-in, for this case here,
that's 600 amps, the (8.49 times 10 to the 28) * (1 times 10 to the minus 6)
area, and then of course 1.6 times 10 to the minus 19 for the charge on
the electron. You find that the drift velocity is 0.044
meters per second, or on the order of 0.01 meters per second.
In other words, it's not incredibly fast. It seems pretty slow, in fact.
And we'll answer the question as to why it's so slow in the next
screencast.