PHYSICS EXPERIMENTS — 133 7-1
Temperature Dependence of Electrical Resistance
OBJECTIVE. To determine the temperature-
dependence of the resistance of a metallic conductor (1) R = R [1 + ? (t - 20?C)] t2020
and of a semiconductor.
where R and R are the resistance values at t20 APPARATUS. Ring stand with insulated temperatures t?C and 20?C, respectively, and ? is support, water bath with heating element, the temperature coefficient of resistance. Eq. 1 can thermometer, stopper, metallic conductor assembly be rewritten to yield: (wire wrapped on a cylinder), semiconductor assembly (a disk held by stiff wires), ammeter, and (2) ? = (R - R)/[R (t - 20?C)] 20t2020voltmeter. ?R1 = R ?t 20 THEORY. Electrical resistivity for a material
may be defined as ? = E/J, where E is the electric for a reference temperature of 20?C. field in the material and J is the current density Experimentally, a series of readings for t and the which E produces. J itself is given by J = nqv, here corresponding values of R are measured. When tn is the number of conduction charges (each of
these values are plotted the resulting curve will be charge q) per unit volume and with average (“drift”)
nearly straight. The slope of the line divided by R 20velocity v. Thus the resistivity ? = E/nqv. In general,
is the coefficient of resistivity, ?. as E is increased the velocity v increases because
the field accelerates the charges to a higher velocity
b) SEMICONDUCTORS are materials such as before they collide with the atoms of the conductor.
the carbon in a carbon incandescent lamp filament, High resistivity of a given material results from a
or germanium and silicon used in making transistors, small value of n or a large likelihood of atomic
or the “thermistor” to be used in this experiment. collisions which reduces the velocity v reached for a
These materials have much higher resistivities than given electric field E.
metals; they also have a different dependence of a) METALS have many electrons (often one per
resistance on temperature and this reveals their atom) that are able to move freely as conduction
fundamentally different nature. The resistance of charges at all temperatures; in other words, n is
these materials may become so high at very low large and constant. Metals obey Ohm’s Law; this is
temperatures that they can be used as insulators. equivalent to saying that (at a fixed temperature) as
This suggests that almost all of the electrons are E increases the velocity v increases proportionately.
bound to individual atoms or atomic bonds and are However, variations in temperature change the ratio
not free to conduct a current until they have been of E/v and thus the resistivity changes with
given an initial energy by heating or other means. temperature, assuming the value of n is unchanged.
Thus n may change rapidly with temperature if this The change in v for a given E occurs because the
initial energy is of the same order of magnitude as probability that an electron will be slowed down by
the average thermal energy kT/2 per degree of interactions with the thermal vibrations of the atoms B
freedom. Here k is Boltzmann’s constant (see your of the metal increases with temperature, becoming B
textbook) and T is the absolute temperature in proportional to the absolute temperature at higher
Kelvin given by: temperatures. For this reason (and since for a wire
of a given cross-section and length the resistance R
T(K) = t(?C) + 273(?C). is proportional to the resistivity ?) the metal shows
The value of n in a semiconductor may change so an approximately linear relationship between
rapidly with temperature that in comparison the resistance and temperature which may be written as
7-2 PHYSICS EXPERIMENTS — 133
change in the ratio E/v is quite small and carefully controlled atmospheric and temperature unimportant. The Boltzmann equation gives the conditions to produce a hard ceramic-like material. number n of electrons which will become If natural logarithms are taken of both sides of conduction electrons by receiving an amount of equation (4) we obtain: energy U:
???????? U1UR????????ln= - .-U/kT B????????e (3) n = no (5) RkTkT????????0BB0
y?mx?bwhere n is the maximum number of electrons o
which could take part in this process at very high
A plot of ln(R/R) as a function of 1/T should give a 0temperatures and e is the base of natural logarithms.
straight line. Add columns to your data table for U is known as the band-gap energy of the
computed values of ln(R/R) and 1/T. semiconductor. This same exponential function of 0
energy divided by kT appears in many basic B
equations of physics, such as the law for the PROCEDURE. In order to measure the temper-
decrease of the earth’s atmospheric pressure with ature dependence of the resistance of a sample we altitude, the Planck formula for the energy need to reliably measure its resistance. It is common distribution in heat radiation, the Maxwell-in physics, engineering, and materials science to Boltzmann law for the distribution of the velocities characterize the electrical properties of samples of the molecules of a gas, the formula for the using an "I-V tester" arranged as in Fig. 1. The "I-V specific heat of a solid, and the equilibrium number tester" can be considered everything to the left of of excited electrons in the energy levels of a laser. the dashed line in the diagram below. A variable Since the resistance is inversely proportional to n DC power supply is used to drive a current I we can expect the resistance of a semiconductor through both a limiting resistor and the sample (to over a suitable temperature interval to be given the right of the dashed line). Current I is measured approximately by an expression of the form by ammeter A while voltmeter V measures the
potential difference (?V) across the sample. ????UU???? - ????kTkTBB0???? (4) R = Re,0 A
where T is a reference temperature (say, 20?C = 0
293 K) at which the resistance is R. The V0
“thermistor” or thermally sensitive resistor used in
this experiment is made of material which requires I-V tester samplean energy of roughly ten times the value of kT at B room temperature to remove an electron from an Figure 1. atomic bond and free it to conduct a current. As a +U/kTBresult, e and thus the resistance will change The experiment is shown in Figure 2 below. 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
of manganese, nickel and cobalt mixed in the
desired proportions with a binder and pressed or
extruded into shape. They are sintered under
PHYSICS EXPERIMENTS — 133 7-3
including 20?C. Use the computer or a full sheet of
graph paper with a horizontal scale from 0?C to stirring thermometer rod80?C. binding From the best straight line that you can draw post
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 watercalculation with the slope and R from your graph. 20
Calculate the percent difference between your value
and that given for copper at the end of this writeup.
sampleComment on possible reasons for any differences.
Figure 2. b) Plot the data for the thermistor as ln(R/R) 0 (vertical) vs. 1/T (horizontal) on a full sheet of 1. Fill the vessel with water to within 2 to 3 cm graph paper. The slope of this graph is of the top and support it in the fiber ring on the tripod. Insert the thermometer in the 1-hole stopper ?ln(R/R)U0 ?and place it in the black bakelite cover support of ?(1/T)kBthe metallic-conductor unit. Insert the unit in the vessel, clamping it in place. Calculate this slope (do not forget units) and then Stir the water; when water and apparatus have calculate the value of U in electron volts which are come to thermal equilibrium (no change in common for materials science. temperature) read and record the following data: a) the temperature t to the nearest 0.1?C, c) Along with your conclusion include in your b) the current (I) through the sample of metallic report answers to the following questions: conductor, What would be the ideal internal resistance of an c) the potential difference (?V) across the sample. ammeter? Why? d) Calculate the resistance (R = ?V/I) of the sample. What would be the ideal internal resistance of a oUse ice to start near 10 C. voltmeter? Why? Next, plug in the 115 V AC cord and heat the Why is it OK if the current is not the same for water so that its temperature is increased about 5 or every temperature? so degrees. In each instance be sure that the Does the data plotted for the thermistor support temperature is constant during the measurement of the theory as represented by equation (4)? Explain. R. To do this it will be necessary to turn off the t heater one or two degrees before the desired rev. 8/05 temperature is reached, and then stir until maximum temperature is obtained. Record t and R as before. tPossibly useful information: oo-3o-1Go from 10 C to about 60 C. Cu, ? =3.9x10 C 20-23 k = 1.38x10 J/K B-19 2. Repeat PROCEDURE 1 for the semiconductor 1 eV = 1.6x10 J(thermistor unit) starting again with tap water as
REPORT. a) Plot the resistance (vertical) vs.
temperature t (horizontal) for the metallic conductor
(copper in this experiment) choosing a suitable scale
7-4 PHYSICS EXPERIMENTS — 133