Chapt. 27
Currents and Resistance
209
3. The probability of electron collision (with an ion, impurity, or defect) in a metal happening in
the time span from last collision to time
t
is
P
(
t
) = 1
−
e
−
t/τ
where
τ
is the
e
folding time for collisions.
a) Take the derivative of
P
(
t
) with respect to time.
b) The derivative from part (a) is a probability density
ρ
for collisions:
i.e.,
ρdt
is the
probability for a collision happening in the differential time interval
dt
. The time interval
over which the density applies is from time 0 (immediately after the last collision) to time
infinity.
If you have
N
electron flights between collisions,
Nρdt
is the mean number of
them that end in
dt
. Say you have a quantity
Q
associated with the electron flights: in
general
Q
depends on
t
. The mean total value of
Q
for flights that end in
dt
is
QNρ
(
t
)
dt
.
The mean total value for all
N
flights is
N
integraldisplay
∞
0
Qρdt.
The average value of
Q
per flight (i.e.,
(
Q
)
) given by
(
Q
)
=
integraldisplay
∞
0
Qρdt.
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 Fall '06
 Buchler
 Physics, Electron, Current, Resistance, Electric charge, Fundamental physics concepts, extra velocity, mean total value

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