Chapter 28 – Current and Conductivity
This chapter deals with the way a
current is established in a conducting wire.
A wire with a current running through it will usually get warm or even
hot and always produce a magnetic field that will deflect a compass
needle. These are “tests” that could be
done to check to see whether or not there is a current in a wire.
Scientists were doing experiments
using crude batteries long before any atomic model existed for wires. Those experiments led to the observation that
the current in a wire was proportional to the electric field in the wire,
summarized by the equation J = σ E where J is the
current density, σ (sigma) is the conductivity, and E is the
electric field in the wire. This is in
fact a variant of Ohm’s Law, V = IR. (Note
that we are no longer dealing with electrostatics since charges, electrons are
moving. Consequently electric fields can
exist inside of conductors when charges are flowing.) In later chapters, we will be dealing the
current I, the amount of charge passing through a section of wire per second as
opposed to the current density, J, the current per unit area. J = I/area.
Also we will be using the reciprocal of σ, rho
= 1/σ, where rho is called the resistivity.
Much of this chapter deals with
developing an atomic view of the conductivity or resistivity. That is, what factors make it easier or
harder for a wire to carry a current. From the equation above for J, it is
clear that given the same size electric field, a material with a larger σ
produces a larger J. A higher
conductivity means more current for a given electric field.
The atomic view we use in this
chapter is that a conductor consists of a sea of mobile electrons that can move
through a background of stationary positive ions. Typically each metal atom contributes about one
electron to the sea leaving each atom ionized with a charge of +e. The electron sea acts like an incompressible
liquid (water) in the sense that it is very difficult to push the electrons
closer together because they repel one another.
On the other hand, it is also hard to pull them further apart because
the positive ions in the metal tug them together again. Hence the analogy between a current of
electrons through a wire and water flowing in a pipe is surprisingly useful.
The free electrons are zipping around
with very large velocities, on the order of 105 to 106 m/s,
but on average, since the velocity can be in any direction, there is no net
movement of electrons in the wire. When
an electric field is produced in the wire, the electrons now feel a net force
along the wire, this produces a small drift velocity, vD,
on the order of 10-4 m/s, a billion or more times smaller than the
average speed of the electron! This
velocity remains small because the electrons are continually bumping into the
positive metal ions. The typical time
between collisions is very small, about 10-14 seconds. So the picture we
have is that electrons are continually colliding with the ions but between
collisions, the electric field is accelerating the electrons causing them to
slowly drift through the wire. The
drifting sea of electrons produces a current.
1. The size
of current produced depends on the number of free electrons in the sea. Since each atom of the metal contributes one
electron, it is relatively easy to use the density of the metal, its atomic
weight, and Avogadro’s number to calculate the density of electrons, n = number
of electrons/m3. A typical
metal contributes something like 1029 electrons/m3. The larger this number, the more current a
wire made of that material will be able to carry.
2. The
electrons are accelerated by the electric field as the bounce from collision to
collision in the metal. For each metal,
there is some characteristic time, τ (tau),
which represents the average time between collisions. If the electrons can zip around for a longer
time between collisions, then the electric field has more time to increase the
drift velocity of the electrons in that material. Therefore we expect that the amount of
current in a wire will increase as τ increases.
3. To be a
little more quantitative, imagine a wire with a cross-sectional area of A, n
electrons/m3, and a drift velocity of vD.
Then in a small time Δt,
the electron sea drifts a distance vD Δt and since the wire has a cross-section of A, the
volume of electron sea that moves through any cross-section of the wire is just
A vD Δt. The number of electrons in that little volume
is just n A vD Δt. The amount
of charge that passes through that cross-section in time Δt is just e, the charge on each electron, times
the number of electrons, enA vD
Δt = Δq, but I = Δq /Δt or,
I = enA vD,
and J = I/A = e n vD
4. The drift
velocity is caused by the electric field which produces an acceleration of the
electrons equal to e E/m, where m is the mass of an electron. Using that force and the average time between
collisions, τ, it is possible to show that vD = eE
τ/m. Putting that into the equation
for J, we get
J
= (e2 nτ/m) E,
where the
stuff in the parentheses is just the conductivity, σ = e2 nτ/m.
5. This last
equation allows us to calculate the average time between collisions for various
metals by measuring the amount of current produced by a given electric field
since e, m, and n are known numbers.
6. Because
the electron sea acts like an incompressible fluid, the amount of current the
flows into a junction has to equal the amount of current that leaves the
junction, Σ Iin = Σ Iout.
This is one of Kirchoff’s laws for analyzing
circuits. More about Kirchoff in a later chapter.
7. We will be using “batteries” to produce currents. Since the electrons in the sea are continually colliding with ions, there needs to be a constant source of input energy to keep the electrons moving since the internal friction in the wire would otherwise quickly bring the electrons to a halt. The battery is the source of that energy. Chemical reactions inside the battery act as a “pump” that lifts the electrons which then can move through the wire, losing energy along the wire, and causing the wire to get warm. In general, we are not interested in the details inside of batteries. Those details are left to chemists.