Resistivity and Temperature Dependence of Resistivity
Introduction
The resistance of a resistor depends on the geometry of the resistor and the resistivity of the
material. For a resistor whose cross sectional area does not vary significantly, its resistance,
R, can be calculated as follows:
[1]
R = ρL/A
L = length
A = cross sectional area = πd2/4 for a circle
ρ = resistivity
Resistivity is primarily a function of the material. Materials vary in their order at the atomic
level and conduction-electron density, among other things.
The resistivity of a material varies a bit with temperature. This variation can be simplified to
a linear function for a reasonable temperature range as follows:
[2]
ρ = ρo[1 + α(T – To)]
ρo = resistivity at temperature of To
ρ = resistivity at temperature of T
α = temperature coefficient of resistivity
The temperature coefficient of resistivity is positive for most substances, so resistivity
increases with temperature. Higher temperature corresponds to higher translational kinetic
energy per particle in a substance. Higher translational kinetic energy per particle
corresponds to higher root mean squared speed. Higher root mean squared speed increases
the likelihood of a collision between a free electron and a particle.
Another effect of increasing temperature is that the conduction-electron density can
increase, thus decreasing the resistivity in substances where this effect dominates.
Semiconductors exhibit this behavior, though we will not investigate this in lab.
Using the above two equations, the temperature dependence of resistance can be derived as
follows:
[3]
[4]
[5]
[6]
R0 = ρ0L/A (resistance at temperature T0)
R = ρL/A (resistance at temperature T)
R = ρ0[1 + α(T – To)]L/A (equation [2] in [4])
R = R0[1 + α(T – To)] (equation [3] in [5])
Substance Resistivity (ρ) at 20°C (Ωm) Temperature Coefficient of Resistivity (°C-1)
Copper
1.72E-8
0.0039
Nichrome 1.10E-6
0.0004
Copper has a low resistivity due to its high conduction electron density and ordered
structure. Nichrome has a high resistivity for a metal due to the high resistivity of nickel and
chromium plus the disorder created by the mixing of the two materials.
Wire Gauge Diameter (mm)
28
0.321 ± 0.001
30
0.255 ± 0.001
32
0.202 ± 0.001
34
0.160 ± 0.001
36
0.127 ± 0.001
38
0.101 ± 0.001
Experimental Procedure
Resistivity at Room Temperature
1)
2)
3)
4)
5)
6)
7)
8)
Measure the resistance of the ohmmeter and probes by connecting the two probes.
Measure or use a given value for the length of a piece of wire.
Measure or use the above table for the diameter of the wire.
Measure the resistance of the wire with the digital multi-meter configured as an
ohmmeter. Make good contact with the wire by pressing firmly to either the ends of
the wire for the nichrome or the bolts in the PVC for the copper.
Subtract the value obtained in step 1 from the value obtained in step 4.
Solve equation [1] for resistivity and calculate your experimental value.
Scientifically compare your experimental value for resistivity to the theoretical value
in the first table.
Repeat steps 2 through 7 with 7 other wires. Use a variety of lengths, wire gauges,
and materials.
The temperature dependence of resistance
1) Set aside all of your data except for the resistance of the ohmmeter and probes.
Don’t even be tempted to look at it or use it in this part of the experiment.
2) Place a beaker of water on the hot plate.
3) Plug in the hot plate and turn it to “HI”.
4) Fill a second beaker with ice water.
5) Place your highest resistance copper wire in the ice water. Leave the ends of the wire
sticking out of the ice water.
6) Measure the resistance of the wire every minute or so (make firm contact with the
probes) until you obtain a fairly constant value. Subtract the resistance of the
ohmmeter and probes from your last measurement. Record this as R0.
7) Use your knowledge of physics to infer the temperature of the ice water and hence
the temperature of the wire, T0.
8) Place the wire in the hot water. Leave the ends of the wire sticking out of the hot
water.
9) Wait for the water to come to a boil, and then turn down the hot plate to simmer.
10) Measure the resistance of the wire every minute or so (make firm contact with the
probes) until you obtain a fairly constant value. Subtract the resistance of the
ohmmeter and probes from your last measurement. Record this as R.
11) Use your knowledge of physics to infer the temperature of the boiling water and
hence the temperature of the wire, T.
12) Solve equation [6] for the temperature coefficient of resistivity (α) and calculate your
experimental value based on To in ice water and T in boiling water. Don’t forget to
subtract the resistance of the ohmmeter and probes from measured values for
When calculating alpha, do not use any
data obtained in the room temperature
experiment except for the resistance of the
ohmmeter and probes.
resistance.
13) Scientifically compare your single experimental value for temperature coefficient of
resistivity to the theoretical value.
14) Repeat steps 5 through 13 for your highest resistance nichrome wire.
When
calculating alpha, do not use any data
obtained in the room temperature experiment
except for the resistance of the ohmmeter and
probes.
15) Re-read and obey the above sentence in gigantic bold font. If you disobey those
instructions, you risk a zero on your report.