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Chapter 27
Current and Resistance (Cont.)
Dr. Jie Zou
PHY 1361
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Outline
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A model for electrical conduction (27.3)
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Derivation of the drift velocity vd
Conductivity and resistivity in terms of
microscopic quantities
Resistance and temperature (27.4)
Electrical power (27.6)
Dr. Jie Zou
PHY 1361
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A model for electrical
conduction
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(a)
A classical model of electrical
conduction in metals: Drude model in
1900
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(b)
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Dr. Jie Zou
In the absence of an electric field, the
conduction electrons move in random
directions through the conductor with
average speeds v ~ 106 m/s. The drift
velocity of the free electrons is zero.
There is no current in the conductor since
there is no net flow of charge.
When an electric field is applied, in
addition to the random motion, the free
electrons drift slowly (vd ~ 10-4 m/s) in a
direction opposite that of the electric
field.
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Derivation of the drift velocity,
vd, using Drude model
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Electric force on an electron: F = qE, where q = -e.
Acceleration of the electron: a = F/me = qE/me.
Define the following:
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t = 0: the instant just after one collision has occurred;
t : the instant just before the next collision occurs;
vi: velocity of the electron at t = 0; vf: velocity of the
electron at time t.
Apply Newton’s 2nd law: vf = vi + at = vi + (qE/me)t.
Average vf over all possible values of vi and collision
time t: v  v  qE t  qE  ; so, v  qE 
f
i
me
me
d
me
: average time interval between successive collisions =
mean free time = relaxation
time.
Dr. Jie Zou
PHY 1361
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Conductivity and resistivity in
terms of microscopic quantities
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According to Drude model:
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Conductivity  = (nq2)/me.
Resistivity:  = 1/ = me/(nq2).
n: charge carrier density = the number of charge carriers
per unit volume.
q: the charge on each carrier. For electrons, q=-e.
me: electronic mass.
: mean free time or relaxation time
According to Drude model, conductivity and resistivity
do not depend on the strength of the electric field, a
feature characteristic of a conductor obeying Ohm’s
law.
Mean free path = average distance between
collisions: l  v 
Dr. Jie Zou
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Example 27.5 Electron
collisions in a wire
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(A) Using the data and results from Example 27.1
and the classical model of electron conduction,
estimate the average time interval between collisions
for electrons in household copper wiring. (For copper,
resistivity  = 1.7 x 10-8 m -see Table 27.1; Charge
carrier density n = 8.49 x 1028 electrons/m3).
(B) Assuming that the average speed for free
electrons in copper is 1.6 x 106 m/s and using the
result from part (A), calculate the mean free path for
electrons in copper.
Dr. Jie Zou
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Resistance and temperature
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For a metal, such as copper
Over a limited temperature range, the
resistivity of a conductor varies
approximately linearly with temperature:
 = 0[1 + (T – T0)].
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For a pure semiconductor,
such as silicon
Dr. Jie Zou
: temperature coefficient of resistivity (see Table
27.1)
Variation of resistance with temperature:
R = R0[1 + (T – T0)].
Example 27.6 A Platinum resistance
thermometer: A resistance thermometer, which
measures temperature by measuring the change
in resistance of a conductor, is made from
platinum and has a resistance of 50.0  at
20.0°C. When immersed in a vessel containing
melting indium, its resistance increases to 76.8
. Calculate the melting point of the indium.
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Electrical power
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Image following a positive charge Q
moving clockwise around the circuit
from point a through the battery and
resistor back to point a.
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In typical electric circuits,
energy is transferred from a
source such as a battery, to
some device, such as a light
bulb.
Dr. Jie Zou
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From a to b through the battery: electric
potential energy increases while the
chemical potential energy in the battery
decreases by the same amount.
From c to d through the resistor: electric
potential energy is transformed to the
internal energy of the resistor.
When the charge returns to point a, the
net result is that some of the chemical
energy in the battery has been delivered
to the resistor and resides in the resistor
as internal energy associated with
molecular vibration.
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Power: Rate of energy transfer
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Rate at which the system loses electric
potential energy as the charge Q passes
through the resistor:
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Power P, the rate at which energy is delivered
to the resistor: P = I V
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Dr. Jie Zou
dU/dt = d(QV)=(dQ/dt) V = I V
The above equation can be used to calculate the
power delivered by a voltage source to any device
carrying a current I and having a potential difference
V between its terminals.
For a resistor: P = I V = I2 R = (V)2/R.
Quick Quiz: For the two light bulbs shown,
rank the current values carried by each light
bulb and their resistance.
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