Week 5: Resistance 171
was effectively impossible to drop a ball in such a way that the final outcome could be predicted or
controlled.
ball bearings
trajectory
gravity
pins/bumpers
Figure 52: An early pinball machine. Balls (typically small ball bearings)dropped in at the top fall
into an array of “pins” that function as bumpers but are “stopped” by the pins after falling a short
timeτso they only build up a finite downward average speed.
Note well that the pinballs cannot escape through the sides, and toavoid complications such
as a ball striking a side and fallingstraight down to the bottomalong a side, we will assume that
the sides are perfectly elastic bumpers that effectlyreflecta ball back into the lattice of pins in the
horizontal direction without affecting its vertical motion.
Physicists and mathematicians got involved in the game at a very earlypoint – for example the
Wikipedia: http://www.wikipedia.org/wiki/Bean Machine was built specifically to demonstrate the
central limit theorem, an important result in the theory of probability and statistics. This sort of
machine is equally useful in the context of understanding classical resistance. Let us build a very
simple “pinball” model for conduction where the electric field that pushes charge through a lattice
of atoms is replaced by gravity pulling down ball bearings and where the atoms in a lattice are
replaced by the pins. One can still sometimes find simple pinball machines of this sort (sometimes
called Pachinko machines) sold as toys.
Let’s use this pinball model to make a simple conduction model, replacing the balls with free
charges and the pins with the lattice of atoms through which the charges move. There is just one
catch – in the passive pinball model above, the balls fall between pinsonly due to the force of gravity,
and the pins themselves effectively stop their downward motion so they have to build up speed again
after each collision. In a lattice of atoms, the atoms are at afinite temperatureand the electrons are
inthermal equilibrium(more or less) with the lattice. This means that the average thermal kinetic
energy of the electrons is much, much larger than the energy theymight gain from the field between
collisions!
To put it another way, if it takes an electron a timeτEto “fall” (say) some average distance
between atoms/collisions and a timeτthermfor the atom to travel that same distance due to their
average speed due to their temperature:
τE≫τtherm (303)During the shorter timeτtherm, the component of the velocity of an electron in the direction of the
force due to the field isslightlyincreased so that – on average – the electron “drifts” in the direction
of this force as it otherwise bounces around randomly and rapidly in all directions.
The resulting “pinball model”, pictured in figure 53, is called theDrude model. With no
voltage/field, this is basically a horizontal “active” pinball table filled with pinballs (charges) with
bumpers (atoms) that are firing/vibrating at a rapid rate so that charges constantly bounce between