## Temperature Coefficient of Resistance

#### PURPOSE:

To investigate the change in the resistance of a coil of wire as the temperature of the coil is varied.

#### WHAT'S THE POINT?

To see that resistance really does change with temperature, and gain some understanding for the size of that change for a typical conductor.

Cutnell and Johnson 20.3. Note Equation 20.5.

#### EXPERIMENT CHECKLIST

1. Set up the equipment (beaker, stirrer, power supply)
2. Prepare the Logger Pro software to collect data
3. Make measurements of resistance and temperature difference as water warms up from room temperature to near boiling
4. Save data from this run
5. Make measurements of resistance and temperature difference as water cools down afterwards
6. Save data from this run
7. Make graphs showing resistance versus temperature difference for each run; fit a straight line to the appropriate data on each graph
8. Use the parameters of the linear fit to calculate the temperature coefficient of copper

#### THEORY:

When a potential difference, V, exists across a length of metallic conductor it is observed that negative charges flow in the conductor, from low to high electric potential, due to the electric field associated with the potential difference. A charge flow, called a current, I, is taken to be positive in the direction positive charges move in response to an electric field. In real life, many electrical circuits employ currents caused by the motions of electrons, which have negative charge; therefore, the actual direction of motion of the charged particles runs opposite to the direction of the current. A linear relationship is observed between the current through and the potential difference across a conductor. This relationship, called Ohm's Law, is often stated as

```             V = I * R
```
but may also be written as
```                  V
R = ---                                                 (1)
I
```
where R is the resistance in ohms:
```                  1 volt
1 ohm = ---------
1 ampere
```
For a conductor of uniform cross-section (such as a wire or rod), one can calculate the resistance from its material properties:
```                   rho * L
R = ---------                                          (2)
area
```
where
```              rho  =  resistivity of material (Ohm-meters)
L    =  length of conductor (meters)
area =  cross-section area (meters*meters)

```
For most materials, the resistivity changes with temperature. If the temperature range is not too large, the resistivity is a linear function of the temperature, T, and can be expressed as
```            rho(T) =  rho(T0) * [ 1 + a(T - T0) ]                    (3)
```
where
```           T0      =  reference temperature (deg Celsius)
T       =  temperature of interest (deg Celsius)
rho(T0) =  resistivity at reference temperature (ohm-meters)
rho(T)  =  resistivity at temperature of interest (ohm-meters)
a       =  temperature coefficient of resistivity (1/deg Celsius)
```
When a tabulated for several different materials, as it is in your textbook, the reference temperature is usually 20 °C. We will use the same value in this experiment.

As the temperature of a conductor is varied, its resistance changes due to

• the temperature dependence of the resistivity
• the thermal expansion of the conductor

For copper, which you will be using in this experiment, the thermal expansion effects are more than 200 times smaller than the effect of the resistivity change. We may ignore the thermal expansion effects completely in this experiment, considering the precision of our measuring devices. Since only the change in resistivity of the material is important, the overall resistance of the conductor has a similar dependence on temperature:

```            R(T)   =  R(T0) * [ 1 + a(T - T0) ]

=  R(T0) + R(T0)*a*(T - T0)                     (4)
```
where
```           T0      =  reference temperature (deg Celsius)
T       =  temperature of interest (deg Celsius)
R(T0)   =  resistance at reference temperature (ohm)
R(T)    =  resistance at temperature of interest (ohm)
a       =  temperature coefficient of resistivity (1/deg Celsius)
```
The temperature coefficient a is the same as that given above for the resistivity.

If the resistance of a conductor is measured at several different temperatures, a graph of the data will be linear with a slope of a * R(T0) and a vertical intercept at T0 of R(T0).

#### PROCEDURE:

Safety Precautions: Don't touch portions of the apparatus which are hot!

In this experiment, you will determine the resistance of a coil of copper wire while its temperature is varied from room temperature to near boiling temperature. The resistance will be calculated from a measurement of the potential difference across the coil and a measurement of the current passing through the coil. The potential difference and the temperature will be measured automatically by the computer through sensors connected to the interface. The current will be supplied by a constant-current source (a DC power supply) and is measured using a digital multimeter.

1. Place approximately 200 ml of tap water in the beaker and place it on the heating plate. Replace the cap holding the resistor and the temperature probe making sure that the resistor and temperature sensor (located at the end of the probe) are near each other and sufficiently submersed in the water.

2. The temperature probe measures the temperature of the water. The temperature of the coil will be the same as the temperature of the water assuming that it remains in thermal equilibrium with the water. Turn on and adjust the magnetic stirrer to a speed that will properly mix the water while it warms, without producing turbulence.

3. Turn on the power supply. Verify that the digital multimeter is set to read DC current.
The stockroom staff should have set up the power supply for you, so you should not have to make any adjustments. However, if it has been mis-adjusted, you may need to adjust the controls as follows:
• turn all knobs fully counterclockwise (to zero)
• turn the upper voltage knob about 1/2 of a turn clockwise (so the little white line on the knob moves to about 12 o'clock)
• slowly turn the "fine adjust" current knob (far left) clockwise, while you watch the multimeter's display
• stop when the multimeter display reads about 80. The display's units are milliamps at this setting, so the current will be about 80 milliamps.
The constant-current indicator (red light) should be lit up on the power supply's front panel. If the green light is lit, your circuit is receiving constant voltage, not a constant current; ask your lab instructor to fix the situation.

4. Look on the desktop for a folder labelled Experiments; inside, you should find an item called Resistance vs. Temperature. Use it to launch the application which reads the sensors.

The sensors should have been set up by the stockroom staff. Just in case they need to be adjusted:

1. From the menu, choose Setup, then Sensors.
2. Click on Ch1 and set sensor to "Raw Voltage", in the range 0-5 Volts.
3. Click on Ch2 and choose "Stainless Steel Temperature." from the sensor menu.
You may see an error message which says "Sensor Conflict". Correct the Ch1 sensor by using the pull down menu to the right to choose "Raw Voltage, 0-5V", then click "OK." The error message should disappear.

You must calculate values of

• (T - T0), where T0 is the reference temperature (usually room temperature, around 20 - 25 degrees Celsius)
• the resistance R(T) as a function of temperature

The Logger Pro program should have been set up for this experiment when you start. You will need only to make two small adjustments to its default values.

1. The sensors will measure temperature, T, but you need to use temperature difference, T - T0, in your analysis. So, create a new data column for "Temperature Difference":
• From the menu, choose Data, New Column, then Formula.
• Fill in the Options (Long Name, Short Name, and Units) for temperature difference.
• Choose Definition to build an equation to calculate (T - T0). Variable names from previous columns in the spreadsheet are listed under Variables. Notice that the defining equation does not use an equal sign and variable names are in quotes. You should build an entry like this: "Temperature"-20.0

2. Second, you may need to adjust slightly the formula used to calculate resistance.
• Check your multimeter's reading of the current through your circuit. It will show the current in milliamps, which should be around 80 mA.
• Look for the data column called Resistance. Choose Modify Column.
• Choose Definition: it should include an equation to calculate R(T). Your entry should look something like this: "Potential"/0.080 The denominator is supposed to be the current running through your circuit, measured in Amps (not milliAmps). If your current isn't exactly equal to the value in the denominator, modify the equation so that it contains your actual current, in Amps.

At this point, you should see a Red Collect button at the top of the screen, alongside the menu controls File,Edit, etc. If you do not see a Collect button, then Save the current session into a file on disk, then quit and re-start Logger Pro. This is a known bug :-(

#### Data Collection Run #1 (Resistor warms up)

1. Turn on the heater to setting 5 to begin warming the water. Wait for approximately one minute so that the temperature change of the water is well under way. Begin taking data by clicking on the Collect button, or choosing (from the menu) Experiment, then Data Collection.
2. After 10 to 12 minutes, the temperature of the water should reach roughly 35 degrees C. That's a good point to stop. The Stop button will end data acquisition before the specified time, if necessary. Do not allow the water to heat up past 50 degrees C!

#### Data Collection Run #2 (Resistor cools down)

1. Don't touch hot items!
2. Turn off the heat.
3. Wait 3 minutes for the water to start to cool down.
4. Since the cooling-off process is slower than the heating process, decrease your sampling rate to about a fourth of the previous value.
5. Take data over a period of approximately 20 minutes.
6. If the apparatus can be adjusted so that there is a gap between the beaker and the lid (while keeping the resistor and temperature sensor submerged), the temperature will drop more rapidly.
7. Obtain another data set while the water (and resistor) cool.

#### GRAPHING

You should make graphs of the resistance as a function of temperature. Create separate graphs for the warming-up and cooling-down processes. You may use computer-generated graphs in your report for this lab, or you may plot them by hand.

If you wish to make plots on the computer, you may do so within the Logger Pro program. Keep in mind:

• Do not allow the computer to "connect the dots" (an unfortunate default setting). Plot your data using "point protectors."
• Clicking on the axis label allows a different variable to be plotted.
• Axes can be scaled automatically or manually; double-click on a graph's axis to change the scale or range. Choose limits which are appropriate for your data.
• Curve fitting is available.
• Graphs may be printed on the printers in the lab.

You may also export your dataset to another program and make plots with it, if you wish. You should consider saving each "run" to a floppy disk. Do not save your file on the computer's hard drive. Save As produces a file which can later be opened by Logger Pro

Data may be exported as a text file to be used in other applications. The best way to do this is to cut-and-paste: click on any entry in the Data Table area in Logger Pro, then ¨Select All" and "Copy" from the Edit menu. Open another program (such as Excel or Notepad) and "Paste" the columns of data into it.

Logger Pro's command Export should produce a nice simple ASCII text file with all the data ... but it appears to ignore some of the data columns!

#### ANALYSIS:

Make a plot of resistance vs. temperature on linear graph paper. Fit a line to the data on this graph, preferably with some computer program that makes a least-squares fit using many data points. From the slope of this line and its vertical intercept at T0, determine R(T0) and the temperature coefficient of resistivity a. In your report you should compare your value for a with that found in your textbook. There should be close agreement between these values, but slight differences can occur. To improve the mechanical strength of wire, copper is usually alloyed with small amounts of other metals. The temperature coefficients of resistivity for these other materials are similar to that for copper.

Calculate the temperature at which the resistance is zero, assuming that the linear relationship you have found remains valid for extended temperature ranges. Comment on this value -- how does it compare to the value of "Absolute Zero" quoted in your textbook?