Download Temperature Dependence of Electrical Resistance

PHYSICS EXPERIMENTS — 133
7-1
Experiment 7
Temperature Dependence of Electrical Resistance
OBJECTIVE. To determine the temperaturedependence of the resistance of a metallic conductor
and of a semiconductor.
an approximately linear relationship between
resistance and temperature which may be written as
(1)
APPARATUS. Ring stand with insulated
support, water bath with heating element,
thermometer, stopper, metallic conductor assembly
(wire wrapped on a cylinder), semiconductor
assembly (a disk held by stiff wires), ammeter, and
voltmeter.
THEORY. Electrical resistivity for a material
may be defined as ρ = E/J, where E is the electric
field in the material and J is the current density
which E produces. J itself is given by J = nqv, here
n is the number of conduction charges (each of
charge q) per unit volume and with average (“drift”)
velocity v. Thus the resistivity ρ = E/nqv. In
general, as E is increased the velocity v increases
because the field accelerates the charges to a higher
velocity before they collide with the atoms of the
conductor. High resistivity of a given material
results from a small value of n or a large likelihood
of atomic collisions which reduces the velocity v
reached for a given electric field E.
a) METALS have many electrons (often one per
atom) that are able to move freely as conduction
charges at all temperatures; in other words, n is
large and constant. Metals obey Ohm’s Law; this is
equivalent to saying that (at a fixed temperature) as
E increases the velocity v increases proportionately.
However, variations in temperature change the ratio
of E/v and thus the resistivity changes with
temperature, assuming the value of n is unchanged.
The change in v for a given E occurs because the
probability that an electron will be slowed down by
interactions with the thermal vibrations of the atoms
of the metal increases with temperature, becoming
proportional to the absolute temperature at higher
temperatures. For this reason (and since for a wire
of a given cross-section and length the resistance R
is proportional to the resistivity ρ) the metal shows
Rt = R20 [1 + α20 (t - 20°C)]
where Rt and R20 are the resistance values at
temperatures t°C and 20°C, respectively, and α is
the temperature coefficient of resistance. Eq. 1 can
be rewritten to yield:
(2)
α20 = (Rt - R20)/[R20 (t - 20°C)]
= 1 !R
R2 0 ! t
for a reference temperature of 20°C.
Experimentally, a series of readings for t and the
corresponding values of Rt are measured. When
these values are plotted the resulting curve will be
nearly straight. The slope of the line divided by R20
is the coefficient of resistivity, α.
b) SEMICONDUCTORS are materials such as
the carbon in a carbon incandescent lamp filament,
or germanium and silicon used in making
transistors, or the “thermistor” to be used in this
experiment. These materials have much higher
resistivities than metals; they also have a different
dependence of resistance on temperature and this
reveals their fundamentally different nature. The
resistance of these materials may become so high at
very low temperatures that they can be used as
insulators. This suggests that almost all of the
electrons are bound to individual atoms or atomic
bonds and are not free to conduct a current until
they have been given an initial energy by heating or
other means. Thus n may change rapidly with
temperature if this initial energy is of the same
order of magnitude as the average thermal energy
kBT/2 per degree of freedom. Here kB is
Boltzmann’s constant (see your textbook) and T is
the absolute temperature in Kelvin given by:
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PHYSICS EXPERIMENTS — 133
T(K) = t(°C) + 273(°C).
The value of n in a semiconductor may change so
rapidly with temperature that in comparison the
change in the ratio E/v is quite small and
unimportant. The Boltzmann equation gives the
number n of electrons which will become
conduction electrons by receiving an amount of
energy U:
(3)
n = noe -U/kBT
(5)
where no is the maximum number of electrons
which could take part in this process at very high
temperatures and e is the base of natural logarithms.
U is known as the band-gap energy of the
semiconductor. This same exponential function of
energy divided by kBT appears in many basic
equations of physics, such as the law for the
decrease of the earth’s atmospheric pressure with
altitude, the Planck formula for the energy
distribution in heat radiation, the MaxwellBoltzmann law for the distribution of the velocities
of the molecules of a gas, the formula for the
specific heat of a solid, and the equilibrium number
of excited electrons in the energy levels of a laser.
Since the resistance is inversely proportional to n
we can expect the resistance of a semiconductor
over a suitable temperature interval to be given
approximately by an expression of the form
(4)
R = R 0e
! U
U $&
#
#k T - k T &
B 0%
" B
of manganese, nickel and cobalt mixed in the
desired proportions with a binder and pressed or
extruded into shape. They are sintered under
carefully controlled atmospheric and temperature
conditions to produce a hard ceramic-like material.
If natural logarithms are taken of both sides of
equation (4) we obtain:
,
where T0 is a reference temperature (say, 20°C =
293 K) at which the resistance is R0. The
“thermistor” or thermally sensitive resistor used in
this experiment is made of material which requires
an energy of roughly ten times the value of kBT at
room temperature to remove an electron from an
atomic bond and free it to conduct a current. As a
result, e+U/kBT and thus the resistance will change
rapidly with ordinary temperature changes;
consequently the thermistor is very useful for such
applications as temperature measurement and
control, voltage regulation, safety and warning
circuits, time-delay switches, flow metering and
sequence switching. Thermistors are made of oxides
! R$
! U$1
U
ln ## && = ## &&
.
k BT0
" R0 %
" kB % T
y= m x + b
A plot of ln(R/R0) as a function of 1/T should give a
straight line. Add columns to your data table for
computed values of ln(R/R0) and 1/T.
PROCEDURE. In order to measure the temperature dependence of the resistance of a sample we
need to reliably measure its resistance. It is common
in physics, engineering, and materials science to
characterize the electrical properties of samples
using an "I-V tester" arranged as in Fig. 1. The "I-V
tester" can be considered everything to the left of
the dashed line in the diagram below. A variable
DC power supply is used to drive a current I
through both a limiting resistor and the sample (to
the right of the dashed line). Current I is measured
by ammeter A while voltmeter V measures the
potential difference (ΔV) across the sample.
A
V
I-V tester
sample
Figure 1.
The experiment is shown in Figure 2 below.
PHYSICS EXPERIMENTS — 133
thermometer
stirring
rod
binding
post
water
sample
Figure 2.
1. Fill the vessel with water to within 2 to 3 cm
of the top and support it in the fiber ring on the
tripod. Insert the thermometer in the 1-hole stopper
and place it in the black bakelite cover support of
the metallic-conductor unit. Insert the unit in the
vessel, clamping it in place.
Stir the water; when water and apparatus have
come to thermal equilibrium (no change in
temperature) read and record the following data:
a) the temperature t to the nearest 0.1°C,
b) the current (I) through the sample of metallic
conductor,
c) the potential difference (ΔV) across the sample.
d) Calculate the resistance (R = ΔV/I) of the
sample.
Use ice to start near 10 oC.
Next, plug in the 115 V AC cord and heat the
water so that its temperature is increased about 5 or
so degrees. In each instance be sure that the
temperature is constant during the measurement of
Rt. To do this it will be necessary to turn off the
heater one or two degrees before the desired
temperature is reached, and then stir until maximum
temperature is obtained. Record t and Rt as before.
Go from 10 oC to about 60 oC.
2. Repeat PROCEDURE 1 for the semiconductor
(thermistor unit) starting again with tap water as
before.
7-3
REPORT. a) Plot the resistance (vertical) vs.
temperature t (horizontal) for the metallic conductor
(copper in this experiment) choosing a suitable
scale including 20°C. Use the computer or a full
sheet of graph paper with a horizontal scale from
0°C to 80°C.
From the best straight line that you can draw
through the data points read off the value of R20
(the resistance at t = 20°C) and calculate the slope.
Use the line you draw (not your data points) to
determine the slope. Next calculate the temperature
coefficient of resistance using Eq. 2, showing your
calculation with the slope and R20 from your graph.
Calculate the percent difference between your value
and that given for copper at the end of this writeup.
Comment on possible reasons for any differences.
b) Plot the data for the thermistor as ln(R/R0)
(vertical) vs. 1/T (horizontal) on a full sheet of
graph paper. The slope of this graph is
! ln(R / R0 ) U
=
!(1/ T )
kB
Calculate this slope (do not forget units) and then
calculate the value of U in electron volts which are
common for materials science.
c) Along with your conclusion include in your
report answers to the following questions:
What would be the ideal internal resistance of an
ammeter? Why?
What would be the ideal internal resistance of a
voltmeter? Why?
Why is it OK if the current is not the same for
every temperature?
Does the data plotted for the thermistor support
the theory as represented by equation (4)? Explain.
rev. 8/05
Possibly useful information:
Cu, α20 =3.9x10-3 oC-1
kB = 1.38x10-23 J/K
1 eV = 1.6x10-19 J
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PHYSICS EXPERIMENTS — 133