Experiment

22

Stefan-Boltzmann Law

1. Introduction

When you turn on an electric heater you may observe that it has changed its color to red or orange. At high temperatures all objects emit visible radiation. But when the heater is set to work at low power, you will not observed any visible radiation, however you may check that it is working by simple holding your hand near the burner. In both cases the energy is emitted by the burner, and it is called radiation.

The Stefan-Boltzmann law relates the power, P, per area, A, radiated by an object to the absolute temperature, T, of the object as shown below

P/A = s T⁴ (1)

where P is in watts, A is in meter², T in degrees Kelvin, and the Stefan-Boltzmann constant s (sigma) = 5.67 10^-8 W/m²K⁴. The law has been found experimentally by J. Stefan and explained theoretically by L. Boltzmann. This law describes only the total amount of energy emitted each second by the object, and does not provide any information about the distribution of energy as a function of frequency.

2. Procedure

In this experiment you will do relative measurements of the power per unit area emitted by a lamp at different temperatures. Since we do not know the surface area of the filament in the light bulb, all measurements will be relative, and we will not be able to determine absolute values of emitted radiation. However, from your data you will be able to test the Stefan-Boltzmann law.

Set up the equipment as shown in Fig. 1. The voltmeter should be connected directly to the binding posts of the lamp. The sensor should be at the same height as the filament of the lamp, with the front face of the sensor approximately 6 cm away from the filament. There should be no objects other than the lamp close to the sensor. In order to find the emitted energy you will use a radiation sensor. It is a miniature thermopile that produces a voltage proportional to the intensity of the radiation. It measures radiation in the infrared and visible region, and the

Fig. 1. Experimental set-up.

voltage produced ranges from 1 microvolt up to about 100 millivolts. The sensor is mounted on a stand and its position should be fixed during the experiment. The sensor is protected by a shutter that can be open by sliding the shutter ring back or forward. It can also be open by momentarily pressing the horizontal arm of the shutter. After you record the data release the shutter. The shutter should be closed when the measurements are not being taken. Please note, that when closing or opening the shutter it is possible to change the sensor position and it may affect your measurements.

The source of radiation in this experiment is a 12 V lamp. By adjusting the power of the lamp, filament temperatures up to 3000⁰C can be obtained. The filament temperatures can be determined by carefully measuring the voltage and current flowing through the lamp. From the Ohm's law you may find the resistance of the filament

R = V/I (2)

For small temperature changes the temperature of the tungsten filament can be calculated from the equation relating the change in the resistance to the increase of the temperature

T = T_room + (R_T - R_room)/aR_room (3)

where T_room is the reference temperature, usually room temperature, R_room is the resistance at the reference temperature, and a is the temperature coefficient of resistivity. The temperature coefficient of resistivity for tungsten is a = 4.5 10^-3 K^-1.

Eq. 3 is true only for small temperature changes, and for large temperature differences, as in this experiment, the temperature of the filament can be estimated using experimental data on resistivity of tungsten listed in Table 1 and depicted in Fig. 2. Use the following procedure.

a) Accurately measure the resistance R_room of the tungsten filament. Accuracy is important. A small error in R_room will result in large errors in your estimation of the filament temperature. It is possible that the lab assistant measured this value for you and recorded on the lamp’s stand. Use that value for R_room.

Table. 1. Resistivity of tungsten as a function of temperature

R/R300K	Temp [K]	Resistivity mW cm	R/R300K	Temp [K]	Resistivity mW cm	R/R300K	Temp [K]	Resistivity mW cm	R/R300K	Temp [K]	Resistivity mW cm
1.0	300	5.65	5.48	1200	30.98	10.63	2100	60.06	16.29	3000	92.04
1.43	400	8.06	6.03	1300	34.08	11.24	2200	63.48	16.95	3100	95.76
1.87	500	10.56	6.58	1400	37.19	11.84	2300	66.91	17.62	3200	99.54
2.34	600	13.23	7.14	1500	40.36	12.46	2400	70.39	18.28	3300	103.3
2.85	700	16.09	7.71	1600	43.55	13.08	2500	73.91	18.97	3400	107.2
3.36	800	19.00	8.28	1700	46.78	13.72	2600	77.49	19.66	3500	111.1
3.88	900	21.94	8.86	1800	50.05	14.34	2700	81.04	26.35	3600	115.0
4.41	1000	24.93	9.44	1900	53.35	14.99	2800	84.70
4.95	1100	27.94	10.03	2000	56.67	15.63	2900	88.33

b) Turn on the power supply. Set the voltage, V, to each of the settings listed in Table 2. When filament is hot, measure the voltage and current in the filament and find the resistance R_T. Record the radiation, RAD, the reading on the millivoltmeter.

Important: The voltage to the lamp should never exceed 13 V. Make each reading of the sensor as quickly as possible so that the temperature of the sensor stays relatively constant.

c) Divide R_T by R_room to obtain the relative resistance.

d) Using the value for the relative resistance of the filament, use Table 1, or associated graph, to determine the temperature of the filament and record the result in Table 2.

3. Report

Calculate T⁴ for each value of T and enter your results in Table 2. On a separate sheet of paper, construct a graph of RAD versus T⁴. Use RAD as your dependent variable (y-axis).

In the introduction discuss the Stefan-Boltzmann law. It is true only for ideal, perfectly black body radiation. A black body is any object that absorbs all the radiation that strikes it. Is the filament of the lamp a true black body? Also, in the introduction list different units of energy commonly used to measure electric power, such as erg, Joule, kilo-watt-hour, electron-volt, calorie, BTU (British thermal unit) and show relationships

In the discussion section answer the following questions.

1. Is your graph a straight line? Is it straight over the entire range of measurements?

2. What sources of thermal radiation, other than the lamp, might have influenced your measurements? What effect would you expect these sources to have on your results?

Fig. 2. Temperature dependence of relative resistivity for tungsten.

Table 2. Experimental data

Potential [V]	Current [A]	Radiation, RAD [mV]	R [ohms]	Temperature [K]	T^4
1.0
2.0
3.
4.0
5.0
6.0
7.0
8.0
9.0
10.0
11.0
12.0