W9_parallel_resonance.eps

Week 5: Resistance 173

“speed” is determined by something called the fermi energy that is only weakly dependent on the
termperature and leads to average speeds roughly an order of magnitude larger:

〈v〉∼ 105 m/sec (306)

Note that this is an appreciable fraction of the speed of light and twoorders of magnitude larger
than escape velocity from the surface of the earth!

Next, we define themean free pathdas the average distance a free charge travels in some
unbiased random direction between atomic “bumpers” at this average speed. We’ll assume (for
our purpose of estimation) thatd=〈v〉τtherm≈ 10 −^10 meters or one angstrom, a typical distance
between atoms in a metal. Then we can estimate the average time between “bumper events” when
the charges interact violently with the atoms in the lattice as:

τtherm=

d

〈v〉

∼ 10 −^16 seconds (307)

In passing, let’s compare this to the timeτEit might take for a charge to start at rest and to
move a distanceddue to the electric field only in the passive pinball model above. Since we want a
quantitative estimate, let’s assume a (fairly strong) field, one we might expect to find in the filament
of an incandescent light bulb with a 100 volt potential difference across a 1 cm filament:

E=∆V

ℓ

= 100 volts

0 .01 meters

≈ 104 volts/meter. (308)

Thislargenumber is fairly representative of the field strength in significant resistive loads, and
is orders of magnitude larger than the field strength one would expect in a halfway decent conductor
such as household copper wiring. FromEwe can easily estimate the expected accelerationa:

a=qE/m= 1. 6 × 10 −^19 ∗ 104 / 9. 1 × 10 −^31 = 1. 76 × 1015 m/sec^2 (309)

To estimateτE, we recall that (neglecting factors of order unity):

τE=

√

2 md

qE

=

√

2 d

a

∼ 10 −^13 seconds (310)

which is orders of magnitude greater thanτthermas expected even for very strong fields.

During the short timeτthermwe expect the force exerted by any reasonable electric field in-
side the material to be “small” compared to the force exerted by the thermally vibrating atomic
“bumpers” so thatbiasedaccumulation of momentum in the direction of the field during the time
τis approximately differential. Also, the strong interaction between lattice and charge maintains
near thermal equilibrium with the much more massive lattice; any momentum gained duringτtherm
is (on average)lost again(transferred to the lattice) in each lattice collision so that any givencharge
doesn’t systematically accumulate kinetic energy as it moves in the direction of the field but rather
heats into the entire material while remaining in thermal equilibrium withit.

We are finally ready to build the model. We expect the equation of motion of our “thermalized
pinball”between collisionsto be something like:

∆〈~p〉=m∆v=F~τtherm=qE~τtherm (311)

so that the average acceleration of a particle between collisions will be:

〈~ac〉=

∆〈~p〉

mτtherm

=

qE~

m

(312)

Since this acceleration only applies for the average timeτthermbefore the charge is redirected in a
random direction with the unchanged thermal distribution of speeds, the average velocity during
this time is:

〈~v〉=

1

2

〈~ac〉τtherm=

1

2

qE~

m

τtherm=~vd (313)