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PHYSICS EXPERIMENTS 133 7-1

Experiment 7

Temperature Dependence of Electrical Resistance

OBJECTIVE. To determine the temperature-

dependence of the resistance of a metallic conductorand of a semiconductor.

APPARATUS. Ring stand with insulatedsupport, water bath with heating element,

thermometer, stopper, metallic conductor assembly

(wire wrapped on a cylinder), semiconductorassembly (a disk held by stiff wires), ammeter, and

voltmeter.

THEORY. Electrical resistivity for a material

may be defined as = E/J, where E is the electricfield 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 ofcharge q) per unit volume and with average (drift)

velocity v. Thus the resistivity = E/nqv. In

general, as E is increased the velocity v increases

because the field accelerates the charges to a highervelocity 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 vreached 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 islarge and constant. Metals obey Ohms 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 ratioof 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 theprobability that an electron will be slowed down byinteractions with the thermal vibrations of the atoms

of the metal increases with temperature, becoming

proportional to the absolute temperature at highertemperatures. For this reason (and since for a wire

of a given cross-section and length the resistance R

is proportional to the resistivity ) the metal shows

an approximately linear relationship between

resistance and temperature which may be written as

(1) Rt = R20 [1 + 20 (t - 20C)]

where Rt and R20 are the resistance values a

temperatures tC and 20C, respectively, and isthe temperature coefficient of resistance. Eq. 1 can

be rewritten to yield:

(2) 20 = (Rt - R20)/[R20 (t - 20C)]

=1

R20

!R

!t

for a reference temperature of 20C.

Experimentally, a series of readings for t and the

corresponding values of Rt are measured. When

these values are plotted the resulting curve will benearly straight. The slope of the line divided by R20

is the coefficient of resistivity, .

b) SEMICONDUCTORS are materials such as

the carbon in a carbon incandescent lamp filamentor 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 differentdependence 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 asinsulators. This suggests that almost all of the

electrons are bound to individual atoms or atomic

bonds and are not free to conduct a current untilthey have been given an initial energy by heating orother 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 isBoltzmanns constant (see your textbook) and T is

the absolute temperature in Kelvin given by:

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7-2 PHYSICS EXPERIMENTS 133

T(K) = t(C) + 273(C).

The value of n in a semiconductor may change sorapidly 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 becomeconduction electrons by receiving an amount of

energy U:

(3) n = noe -U/kBT

where no is the maximum number of electrons

which could take part in this process at very hightemperatures 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 basicequations of physics, such as the law for the

decrease of the earths atmospheric pressure with

altitude, the Planck formula for the energy

distribution in heat radiation, the Maxwell-Boltzmann 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 givenapproximately by an expression of the form

(4) R = R0e

U

kBT-

U

k BT0

!"##

$%&&,

where T0 is a reference temperature (say, 20C =

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 anatomic 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 warningcircuits, 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 orextruded 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 ofequation (4) we obtain:

(5)ln

R

R0

!

"##

$

%&& =

U

kB

!

"##

$

%&&1

T-

U

kBT0

.

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 forcomputed values of ln(R/R0) and 1/T.

PROCEDURE. In order to measure the temper-

ature dependence of the resistance of a sample we

need to reliably measure its resistance. It is commonin physics, engineering, and materials science to

characterize the electrical properties of samples

using an "I-V tester" arranged as in Fig. 1. The "I-Vtester" 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 measuredby ammeter A while voltmeter V measures the

potential difference (V) across the sample.

A

V

I-V tester sample

Figure 1.

The experiment is shown in Figure 2 below.

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PHYSICS EXPERIMENTS 133 7-3

water

sample

bindingpost

stirringrod

thermometer

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 ofthe 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 intemperature) read and record the following data:

a) the temperature t to the nearest 0.1C,

b) the current (I) through the sample of metallic

conductor,

c) the potential difference (V) across the sample.

d) Calculate the resistance (R = V/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 ofRt. 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 maximumtemperature 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.

REPORT. a) Plot the resistance (vertical) vs

temperature t (horizontal) for the metallic conductor(copper in this experiment) choosing a suitable

scale including 20C. Use the computer or a full

sheet of graph paper with a horizontal scale from

0C to 80C.From the best straight line that you can draw

through the data points read off the value of R20(the resistance at t = 20C) and calculate the slope

Use the line you draw (not your data points) to

determine the slope. Next calculate the temperaturecoefficient 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 writeupComment 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 )

!(1/T)=

U

kB

Calculate this slope (do not forget units) and thencalculate the value of U in electron volts which are

common for materials science.

c) Along with your conclusion include in yourreport answers to the following questions:

What would be the ideal internal resistance of an

ammeter? Why?

What would be the ideal internal resistance of avoltmeter? Why?

Why is it OK if the current is not the same for

every temperature?

Does the data plotted for the thermistor supportthe theory as represented by equation (4)? Explain.

rev. 8/05

Possibly useful information:

Cu, 20 =3.9x10-3oC-1

kB = 1.38x10-23 J/K

1 eV = 1.6x10-19 J

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7-4 PHYSICS EXPERIMENTS 133