The Stefan-Boltzmann Law Department of Physics Ryerson University 1 Introduction Thermal radiation is typically considered the starting point in many texts for discussions of old quantum theory and the progression towards modern quantum mechanics. It is also something that many of us have had clear encounters with, and know exactly what it means: if something is glowing red, it s hot! This is an emission of electromagnetic radiation in the visible spectrum which allows us to register that it is indeed quite hot. This does not just apply to the glowing coals of a fire or the white-hot filament of a lamp, but to anything at a temperature above absolute zero. Interestingly enough, the radiation emitted is composed of waves with a distribution of frequencies that span the complete electromagnetic spectrum. The peak and width of this distribution change with the temperature of the emitting body. The character of the actual material does not change this property; rather, as it happens, it is the character of the surface which makes a big difference in how good of a blackbody a material is. The initial understanding of this phenomenon came from the absorption and reflection of materials. If a material is absorbing heat from electromagnetic radiation, yet is in thermal equilibrium with its surroundings, it must also be radiating heat to maintain this equilibrium, i.e. to keep the temperature from changing. J. Stefan found an empirical relation relating the temperature of a body T in Kelvin to the radiancy, or total emissive power (the energy emitted per second per unit area), of a body R T (in units of W/m 2 )of a body, shown in Eq. 1, in 1879. R T = σt 4 (1) Where Stefan s constant σ = 5.67 10 8 W/m 2 K 4. This was confirmed in 1884 using thermodynamical arguments by L. Boltzmann, and is now known as the Stefan-Boltzmann Law [2]. Classical thermodynamics and electricity and magnetism were unable to derive this law and thus explain why this is observed. Rayleigh and Jeans formulated a theory, interpreting the atoms of the blackbody as little electric dipole oscillators which gave the spectral distribution of the radiation. This treatment was successful at long wavelengths (and low temperatures), but the emissive power diverged at small wavelengths and high 1
temperatures. No classical solution existed. In 1900 Planck proposed a new theory based on quantization: the wavelengths which could be occupied were not continuous, but rather, discrete. The oscillators could only operate at particular frequencies. The Planck spectral distribution law did not diverge at wavelengths λ 0, but instead was able to reproduce the empirical result at all temperatures. This is one of the experiments that got the conversation started in modern physics. Classical electromagnetic theory was insufficient to entirely describe this seemingly simple physical phenomena. The concept of quantization was quite revolutionary, and quickly found extension to other problems in physics. 2
2 Experiment The Stefan-Boltzmann law will be verified with two different measurements in two temperature regimes. The objective of this experiment is to determine the relation of the power emitted by a blackbody to its temperature in order to verify if the radiance is indeed proportional to T 4. This will be done using relative measurements of the emissive power over a broad temperature range ( 10 2 K, and 10 3 K). The radiation sensor being used is sensitive to the total emissive power between the wavelengths of 0.6 to 30 µm. The sensor outputs a voltage linearly proportional to the total emissive power incident on the sensor, which is read with a multimeter. This method uses a tungsten bulb filament as the source, allowing temperatures up to 3000 K. The filament is being treated as a point source with which we can vary the current through the filament (and by extension the filament temperature). The temperature T fil as a function of resistance for the filament is a known quantity, but relies heavily on an accurately determined room temperature resistance R 300K. With a perfect sensor at zero temperature, this would be R = σtfil 4. The concern at room temperature is that the radiation sensor itself is at a temperature not insignificant with respect to the temperature of the blackbody, as is made quite apparent by seeing the temperature in units Kelvin. The detector is in fact radiating with the same T 4 law as the ideal blackbody. Thus, the actual emissive power will be instead Eq. 2 and will need to be accounted for in the calculations. In the high temperature regime, the blackbody radiation of the sensor is insignificant compared to the filament temperature, and so can be neglected. However, for the purposes of this experiment, we will account for the room temperature such that our low temperature results will also reconcile with the T 4 law. 2.1 Equipment Equipment required: PASCO Stefan-Boltzmann Bulb PASCO Thermal Radiation Sensor PASCO 3630 Power Supply (for bulb) Dual multimeter box Multimeter Thermometer Aluminized foamcore boards R = R fil R sensor = σ(t 4 fil T 4 sensor) (2) 3
Figure 1: The experimental setup for the high temperature verification of the Stefan- Boltzmann Law, from Ref. [3]. 2.2 Precautions The bulb used in these experiments will cause burns if you touch it. Ideally, don t move the bulb while on, but if necessary, move it by the base only. Place the bulb on good footing; it is light and could tip over easily. 2.3 Procedure 1. Measure the resistance of your lamp filament, R 300K using a four-wire measurement. There is one ohmmeter for the class to share. Be sure to attach voltage leads right at the lamp contacts, while the current leads can go in the banana sockets. It is essential to do this before beginning the experiment. Measure and record the ambient temperature as well. 2. Place the radiation sensor facing the bulb as shown in Fig. 1, leaving 6 cm between the sensor surface and filament. The sensor window should be lined up carefully with the filament of the bulb. Connect one of the paired multimeters to the radiation sensor. 3. Connect the second multimeter of the pair to measure the voltage drop across the lamp, as in Fig. 1. Connect the lone multimeter as the ammeter, measuring current through the lamp. The power supply for your lamp will be the PASCO 3630 24 V DC supply. Before connecting the lamp, ensure the supply is turned to zero. Read all values from the multimeters, not from the supply display. Ohm s law will be used to 4
calculate the resistance of the filament, which will yield its temperature. As with the four-wire measurement, it is best to attach the voltage leads at the lamp connections while the current leads are connected to the banana sockets. 4. Take readings around every 0.5 V of voltage drop across the filament. Record current, voltage drop, and radiation sensor output R, along with your uncertainty in each measurement. Between readings, shield the sensor with the aluminized foamcore board, aluminized side facing the bulb. Be brisk when recording your datapoints; the blackbody radiation of the lamp will begin to heat the sensor, contributing error. Do not exceed voltages of 12 V or currents of 3 A on the bulb filament! When you reach this limit, you can gradually turn down the voltage to 0 V, and turn off the supply. 5
3 Report 3.1 Analysis As mentioned in the Procedure, use Ohm s law to determine R fil from your voltage and current measurements. The temperature of the tungsten filament T, in Kelvin, as a function for resistance R rel in Ohms, is tabulated as part of the PASCO Manual for the Thermal Radiation System [3]. Here, we have an expression which gives the temperature as a function of relative resistance, R rel = R fil /R 300K. Using relative resistance eliminates having to know anything about the geometry of the filament. T = 1.6519R 2 rel + 203.55R rel + 126.18 (3) Use this expression to determine the filament temperature in Kelvin. Show plots of both total emissive power R versus T and total emissive power R versus T 4 T 4 rm, fitting each plot to the appropriate relation. Be sure to include sample calculations of your thorough analysis of the uncertainty in your measurement. 3.2 Discussion Discuss the linearity in T 4 over the entire temperature range studied. Determine if your result is within your uncertainty in your measurement. Comment on the shortcomings and sources of error of the method. At which temperature would it be appropriate to begin to neglect the temperature of the sensor? How would you improve this set-up? An accurate determination of Stefan s constant can be used to calculate Planck s constant. How would you accurately determine the Stefan s constant, were you required to do so? You can consider not just the equipment used here, but any other lab equipment you think might be useful. 6
References [1] Robert Eisberg and Robert Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, Wiley, 2nd Edition, 1985. [2] B. H. Bransden and C. J. Joachain, Quantum Mechanics, Pearson, 2nd Edition, 2000. [3] PASCO Thermal Radiation System Manual, PASCO Scientific, 1991. 7