W9_parallel_resonance.eps

172 Week 5: Resistance

q

no−field trajectory

trajectory with field

E field

Figure 53: The Drude model: With no field, the chargesqbounce very rapidly between atomic
“bumpers” that maintain a roughly thermal distribution of charge speeds. On average, these colli-
sions form a “random walk” with no direction and zero average displacement. With a field, in the
very short timeτ between collisions these random free trajectories arevery slightly curved in the
direction of the fieldand the random walk is nowbiased, with a net displacement that slowly accrues
in the direction of the field.

the atoms in an unbiased random walk, leading tozero average displacement. The charges
themselves are (recall) strongly repulsive and there is an average of roughly one charge per atom.

The application of a voltage across the conductor is equivalent totipping the furiously vibrating
pinball tableup through a small angle relative to gravity. In a pinball table, this would cause the
balls to gradually drift lower in the direction of the net gravitational field. In a conductor, there is
a small net acceleration in the direction determined by the electrostatic force on the charges during
theshorttime between bounces. The charges all “bounce”’ tens to thousands of times between the
atoms in the time that it would take a charge tofreely fallfrom one atom to the next due only to
the applied field, but this small, asymmetric force is enough tobias the random motion of the
charges so that they slowly drift in the direction of net force.

Thisis whyvdis called the “drift velocity” – it is a velocity that is quite distinct from theactual
speedof the particles as they bounce around vigorously between the bumpers! To make a proper
conduction model out of this, we also need to constantly take pinballs off of the lower end of the
table and elevate them back up to the top to start over so that charge doesn’t accumulate at the
bottom and generate a field that stops the process. This constant lifting is an excellent model of a
battery!

With this insight, let’s generate the algebraic description of the Drude model. We start by
estimating the mean speed of the charges from thermodynamics. If the lattice is at temperature
T, and the free charges are in thermal equilibrium with the lattice (a reasonable assumption), then
from the equipartition theorem we expect the average kinetic energy of the free charges in a three
dimensional space when the “table” is not tipped by an applied electricfield to be:

K=

3

2

kbT=

1

2

m(〈vx〉^2 +〈vy〉^2 +〈vz〉^2 ) =

1

2

m〈v〉^2 (304)

Then we expect〈v〉to be:

〈v〉=

√

3 kbT

m

(305)

This speed is easy enough to estimate at (say) 300 degrees kelvin and for electrons fromkb =

1. 38 × 10 −^23 J/K andme= 9. 1 × 10 −^31 as〈v〉≈ 105 m/sec.

Note that this isvery, very fast, and at that isnot fast enoughbecause the electrons arenot
a classical gas of non-interacting particles, they are quantum mechanical fermions whose effective