174 Week 5: Resistance
where the term “drift velocity” is now formally justified as it is the differentialbiasof a much more
rapid and violent random process of the charged particles bouncingaround between the atoms.
You will note that this isexactly the same as the expression we obtained in the pinballmodelex-
cept that instead of trying to computeτusinga=qE/mstartingfrom rest, we use the average time
τbetween collisions driven by the strong interaction of the charged particles with the surrounding
material. This seems like a small change, but it is a very important one!
In this “active pinball” Drude model, then, we expect that:J~=nq~vd=nq(^2) τtherm
m
E~=σE~ (314)
which scaleslinearly with the electric field strength.
If we used timeτEfrom the naivepassivepinball model in exactly the same argument, we obtain
an average speed of〈v〉=^12 ~aτE=^12 q
E~
mτE=√
dqE
2 m=vdorJ=nq〈v〉=nqA√
dqE
2 m(in the direction
of the applied field). The current density would then scale with thesquare rootof the field, which
does not empirically agree with Ohm’s Law! The passive pinball model fails to empirically scale
correctly in several ways.
If we compare this to the equilibrium conditionqE=bvdwe see that the “linear drag coefficient”
is given by:
b=
m
τtherm(315)
or:
F~d= m~vd
τtherm=∆~p
∆t (316)This is conceptually perhaps the easiest way to see what’s going on. The “drag force” equals the
average momentum change per unit timeof the free charges as they move through the lattice
of atoms. We’ve simply done a very simple quantitative estimate of that momentum change in
terms of our active pinball model. Sadly, this also reveals that we haven’t actually made much
useful progress as we’ve exchangedoneunknown, dimensioned number characteristic of the material
(b) for another (τthermand/orm).
Ohm’s Law
At last we are set to establish the connection between the (empirical!) Ohm’s Law and the Drude
conduction model. We just showed above that the current densityJ~is proportional to the applied
electric fieldE. In so doing, we wrapped up all of the complexity – all the unknown stuff about a
conductor, including,b,n,q,m,τtherm– into a single parameter called theconductivity:
σ=nq^2 τtherm
m =nq^2
b =qρfree
b =1
ρ (317)where we have also defined its reciprocalρ, called theresistivity^52
The resistivity and/or conductivity^53 is a characteristic of the material of the conductor in
question, and as the Drude model suggests, depends on many things. Its most important dependence
(^52) Unfortunately the common symbol for resistivity isρ, which you can easily confuse with the charge density which
alsois given the symbolρin most physics books. I’ve tried pretty hard to label the charge densityρe, read “the
density of ELECTRIC charge” orρfree=nq, read “the density of the FREE electric charge”, and will continue to do
so whereever there is any chance of confusing the two. Sadly,things are no better if we use the conductivityσ, easily
confused with the surface charge densityσe. As you gain experience, the meaning of any symbol intended/needed
will generally be clear from context without the need for hints such as this.
(^53) Wikipedia: http://www.wikipedia.org/wiki/Electrical resistivity and conductivity. As usual, follow this
wikipedia link to learn more about resistivity and conductivity than this short treatment allows, as well as to access
tables of resistivities and temperature coefficients of resistivity.