Object

Measure how the current through an electric light bulb varies as the applied voltage is changed.

An Excel spreadsheet has been prepared to help you complete the following data analysis:

1. Calculate the electrical resistance and electrical power for each current/voltage measurement.
2. Plot a graph of electrical power (y-axis) versus electrical resistance (x-axis) and estimate the heat loss due to conduction (see graph 1).
3. Determine the radiated power and plot a graph of log_e (radiated power) against log_e (resistance) to verify Stefan's Law (see graph 2).

Introduction

When an electric current flows through the filament in a light bulb the filament heats up. The filament loses heat in two ways: electromagnetic radiation (mainly visible light and invisible heat radiation) and conduction (through the base of the bulb). The heat conducted away from the filament increases linearly with filament temperature. The air in the bulb is pumped out during manufacture so no heat is lost by convection.

It is difficult to measure the temperature of the filament directly, however the filament resistance is approximately proportional to its temperature. Therefore the filament resistance is used as a measure of filament temperature. The graph of electrical power against resistance should be like graph 1. At low temperatures (AB) most of the heat is lost by conduction and little of the heat is lost due to radiation. Therefore this part of the graph is extrapolated to estimate the conducted power at higher temperatures. Since this is an extrapolation (dotted line on graph 1) the data in region AB of the graph must be accurately measured. The radiated power is simply the electrical power minus the conducted power (PQ on the graph).

Stefan's Law states that the radiated power is proportional to the absolute temperature T raised to the fourth power so the equation is:
Equation 1 radiated power = sT⁴ (s is a constant).

Taking natural logs we obtain:
Equation 2 log_e (radiated power) = 4 log_e (T) + log_e (s)

The temperature (T) is proportional to the filament resistance (R) so:
Equation 3 T = kR (k is a constant)

Replacing T in equation 2 with kR (from equation 3) we obtain:
Equation 4 log_e (radiated power) = 4 log_e (R) + log_e (s) + 4 log_e (k)

Therefore a graph of log_e (radiated power) against log_e (resistance) with a gradient of 4 (within the error in the gradient) confirms Stefan's Law (Graph 2). The last two terms in equation 4 are constants.

Apparatus

Two digital multimeters, variable D.C. power supply, light bulb, various leads.

Method

1. Connect the circuit as shown in figure 1. The power supply, ammeter and bulb must be in series. The voltmeter must be across (in parallel with) the light bulb.
2. Measure the current and voltage for voltages between 0 and 12 V.
3. Make sure you have at least six measurements for electrical powers of less than one watt and filament resistance of less than one ohm (measured voltage less than about one volt). This corresponds to the region AB on graph 1 where most of the heat is lost by conduction.
4. Make sure you have at least ten measurements for electrical powers greater than one watt (measured voltages between about one and twelve volts). This corresponds to the region BP on graph 1 where most of the heat is lost by radiation. The radiated power is equal to PQ.

Analysis

1. Open the Excel spreadsheet called "stefan.xls" (Download "stefan.xls", 54.5kB) as a read only file (you may save a copy using another name).
2. Type in your voltage and current measurements into columns B and C respectively (label the columns with the appropriate units).
3. Calculate and type into cell E10 the filament resistance for your first measurement. The spreadsheet will calculate the filament resistances for the rest of your measurements if you enter the correct resistance value (label the column with the appropriate units).
4. Calculate and type into cell G10 the electrical power for your first measurement. The spreadsheet will calculate the filament resistances for the rest of your measurements if you enter the correct electrical power value (label the column with the appropriate units).
5. The chart sheets marked "Low Power" and "Full Power" will now show graphs of Electrical Power (y-axis) versus Electrical Resistance (x-axis) calculated from your data.
6. At low power, most of the heat is lost due to conduction. Adjust the resistance and power values in cells H10:H11 and I10:I11 respectively so that the conducted power line corresponds to your data.
7. The "Full Power" graph will now show the estimate for the conducted power extrapolated from your low power regime data.
8. Print out copies of your "Low Power" and "Full Power" graphs (at this point please ask a demonstrator to check your data and graphs).
9. Type into cell M10 your estimate for the conducted power for the largest electrical power you have measured. The spreadsheet will calculate the conducted power for the rest of your measurements if you enter the correct conducted power value.
10. Type into cell M16 your estimate for the radiated power for the largest electrical power you have measured. The spreadsheet will calculate the radiated power for the rest of your measurements if you enter the correct radiated power value.
11. Type into cell S10 the natural logarithm of the filament resistance for the maximum radiated power. The spreadsheet will calculate the natural logarithm of the filament resistance for the rest of your measurements if you enter the correct value.
12. Type into cell S16 the natural logarithm of the radiated power for the maximum radiated power. The spreadsheet will calculate the natural logarithm of the radiated power for the rest of your measurements if you enter the correct value.
13. The chart sheet marked "nat. log graph" will now show a graph of log_e (Radiated Power) on the y-axis versus log_e (resistance) on the x-axis calculated from your measurements and using your estimate for the conducted power.
14. Print out copies of your "nat. log graph" graph (at this point please ask a demonstrator to check your data and graph).
15. The graph of log_e (radiated power) against log_e (resistance) should be a straight line, measure the gradient and error in the gradient of the line. Does this agree with Stefan's Law?

©  Mark Davison, 1997,  give feedback or ask questions   about this experiment.