166 Week 5: Resistance
proportional to that force, where the forces balance.
A Simple Linear Conduction Model
We now use this model for “linear drag” to build a working description of voltage changes, electric
fields, and electrical currents and current densities inside conductors that arenotin electrostatic
equilibrium. They are not really static because charge is being pushedby electric fields and is moving,
but they are still in a kind ofdynamicequilibrium where forces on the charges balance. This model
will also work for“slowly varying” currents– currents we can treat as beingapproximatelyconstant
on small time intervals ∆t– but ultimately it willfailwhen we take into account the possibility of the
conductorradiating energy and momentuminto space for rapidly varying currents (a consequence
of the Maxwell Equations we haven’t learned yet inelectrodynamics). It is thus a “quasi-static”
theory and should not be taken too seriously or considered to be completely general or correct.
vdvdvdvdvdvdvd∆tq A
qqF = b F = qEqFigure 50: The simple linear model for conduction in a resistive lattice.In figure (50) we see a model for a conducting wire. This wire has a cross-sectional area ofA
and contains an “inexhaustible” supply of free charged particles (recall, order of one free charge per
atom) each with chargeq. An electric field is created within the wire by abattery(not shown) that
exerts a force on any given charge carrier to the right ofF=qE. The wire resists the flow of that
charge carrier with a “drag force”bvto the left, wherebis a phenomenological “drag coefficient”
characteristic of the imperfect conductor. Microscopically, we can initially mentally picture this
drag force as being the result of an ongoing average loss of momentum as each free charged particle
speeds up in the direction of the electric field for a time but then is suddenly slowed down enough
to “start again” as it collides with the atoms or molecules of the material (incidentally heating the
material).
In “dynamic equilibrium” (steady, or nearly steady currents) we require these two forces to
balance:
vd=qE
b(287)
where we introduce thedrift velocityvd, defined to be the average “terminal velocity” of charges in
the conductor^50. It is important to keep in mind that in a typical normal metal our charge carriers
are negatively chargedelectrons(recall “Franklin’s mistake”) and all of the vectors are reversed for
a current and field that still go from left to right, but this makes no difference in anything we care
about (yet!); the argument given below works for either sign of thecharge carrier.
(^50) We will give a particular, simple, classical model called theDrude modelfor the drift velocity that will give us
an actual functional form forbin a more advanced section below that can safely be omitted bystudents uninterested
in majoring in physics or more advanced studies in e.g. engineering (although it is not terribly difficult and is a
worthwhile exercise in mechanics).