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 /m) E,

 

where the stuff in the parentheses is just the conductivity, σ = e2 /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.