Thermal Radiation of Blackbodies Lab Partner 1 & Lab Partner 2 12 May 2011 We report on experiments investigating the thermal radiation from a blackbody. By finding the electromagnetic spectra emitted from a tungsten bulb, an approximate blackbody, at 1600 and 2600 K, we were able to confirm the accuracy of Planck s Law, describing the shape of the spectra, and the Stefan- Boltzmann Law, describing the total power emitted from the source. We found the peak wavelengths transmitted were 1.2 ±.17 µm and 1.4 ±.17 µm for the 2600 and 1600 K source, respectively, which are in close agreement with Wien s Law after the consideration of error in corrections taken for imperfections in our setup. Introduction It has long been observed that when matter is very hot, it can begin to emit light, just as coals in a fire glow. It was eventually observed that this phenomenon was not limited to visible light light of all wavelengths was emitted from all bodies of matter. The spectrum of wavelengths emitted from a body was very well described experimentally, but no physicist was able to explain it theoretically. The major unsuccessful theory to describe this, called blackbody radiation, was introduced by Lord Rayleigh and Sir James Jeans in 1900, derived from classical wave theory, which consisted of intensity dependent on the inverse fourth power of wavelength. Although accurate at long wavelengths, the Rayleigh-Jeans equation predicted an intensity approaching infinity at small wavelengths, called the ultraviolet catastrophe, which was clearly not observed in experiment. The first accurate description also came in 1900, developed by Max Planck. Planck s proposal fit the experimental results exactly, and would come to imply the quantization of energy. Planck s development would later prove to be one of the most important experiments in the development of quantum mechanics and modern physics In this study, we wish to find the radiation spectrum of a blackbody, in our case a tungsten light bulb, at various temperatures to assess the validity of Planck s Law and its consequences, as well as prove the invalidity of the Rayleigh- Jeans equation. Theory The first few equations regarding blackbody radiation that physicists discovered in fact described various aspects of the spectrum of wavelengths, rather than describe the spectrum itself. This first description came in 1879 from Jozef Stefan and Ludwig Boltzmann, who found that the radiated power density of a blackbody, R, was governed by:, (1) where T is the temperature of the blackbody in Kelvin, and σ is the Stefan-Boltzmann constant, equal to. In a plot of energy density per wavelength, like figure 2.a., this value of power is equal to the area under the curve (energy density), multiplied by c/4, where c is the speed of light (see Tipler 122). By looking at the energy density vs. wavelength plot for several temperatures, we should see this effect: as temperature increases, the integral of the curve increases by a factor of T 4. The next important description of blackbody radiation came in 1893 at the hand of Wilhelm Carl Werner Otto Fritz Franz Wien, who noticed that these experimental plots of energy density vs. wavelength each had a characteristic peak (maximum intensity) wavelength. Wien found
Page 2 that this maximum wavelength was related to the temperature by: (2) In order to describe the electromagnetic spectrum emitted from a blackbody, one must first define what a blackbody is. A blackbody is an idealized physical body whose main characteristic is that it absorbs all incident electromagnetic radiation. This makes the body ideal for emitting thermal radiation of a continuous spectrum. Often a blackbody is described by a cavity in a material with a small hole open to the outside environment all incident radiation is transmitted through the hole, and is absorbed before it can be reflected back out of the hole. This heats the body, and the body will emit electromagnetic waves corresponding to the standing waves available within the cavity. In 1900 Lord Rayleigh and James Jean first made an attempt to describe the electromagnetic spectrum emitted from a blackbody using this generalization. They theorized that each standing wave present within the cavity carried energy at equilibrium equal to, where k B is Boltzmann s constant, equal to. Using this theory, they were able to calculate the energy corresponding to each wavelength released from the blackbody, which gave them a relationship of:, (3) where u is the radiated energy density per wavelength. This relationship fit the experimental data well for large wavelengths, but at small wavelengths, the Rayleigh-Jeans equation predicted energy density to approach infinite, which is not represented in the data gathered by experiment. This rapid increase at low wavelength is due to the increasing density of normal modes of the standing waves as wavelength decreases. The Rayleigh-Jeans hypothesis of equally distributed energy among the modes dictated that a great deal more energy is emitted at small wavelengths, something that is not indeed witnessed. Very soon after Rayleigh and Jeans proposed their equation, Max Planck found the equation which would fit the experimental plots. Planck found that: (4) where h is a constant which made the equation fit the experimental data. h would become known as Planck s constant, and is equal to. This equation is based off of the same approximation made of a blackbody above, but includes a correction for the decreasing probability of higher modes of waves being filled. This avoids the ultraviolet catastrophe seen with the Rayleigh-Jeans equation. It is worth noting that this derivation was done simply by searching for the equation that would fit the experimental data. The consequences, that of quantized energy and the birth of quantum mechanics, would only be realized later. It is also worth noting however that Planck s Law (eq. 4), will give the relationships already derived from experiment: the Stefan-Boltzmann Law (eq. 1) and Wien s Law (eq. 2). Experiment To find the temperature dependent electromagnetic spectrum emitted from a blackbody, we designed an experiment to find the spectrum of a tungsten bulb, which can be approximated as a black body. Tungsten bulbs shine white light, which consists of all wavelengths. To measure the spectrum, we must measure the intensity of each wavelength being emitted. We can do this by splitting up the tungsten white light into its component
Page 3 Figure 1: Setup used in our experiment to measure the thermal radiation spectrum from a tungsten bulb. Light is emitted from a tungsten source in the lower left, passed through a chopper (a fan), reflected off mirror 1, passed through a collimator, reflected off the grating to break the light into different wavelengths, reflected off mirror 2, and finally absorbed by the photocell. wavelengths and shining each wavelength onto a photocell. To split the white light into its component wavelengths, we reflected the light off of a diffraction grating of regular separation of 16700 Å. Different wavelengths reflect off of the grating at different angles. To measure this spectrum, we used a setup similar to that seen in fig 1. Light is shone from the tungsten bulb, controlled by a power supply, in the lower left. The light is then reflected off of a slightly concave mirror, with the tungsten source at its focal length, to collimate the light. In order to ensure we are only measuring one wavelength at a time, all of the light must be striking the grating parallel, which can be accomplished using this convex mirror. The light passes through a width collimator to ensure that the same total amount of light is reaching the grating at all times. The light reflects off of the diffraction grating, located on a rotatable platform, to another mirror, and is reflected back to the photo detector. The second mirror is also convex, to direct all the incident light back to a point source like the tungsten source. The photo cell is also located at the focal length of the second mirror. In order to decrease the effects of noise from the environment, a chopper, here a fan seen in fig 1, is used to chop up the light. The photocell is designed to only detect periodic signals, to eliminate constant environmental noise; this chopper ensures the signal we wish to inspect is being observed. To find the electromagnetic spectrum, we rotate the grating through a range of angles to cover the wavelengths we wish to investigate. The wavelength corresponding to the angle of rotation is given by:, (5) where n is the reflection order, d is the separation of the grating, θ 0 is half the angle between the light rays incident and reflected from the grating in the zeroth order position (where the zeroth order reflection is on the detector), and is the angle the grating is rotated from the zeroth order position. We found θ 0 to be 13.5 in our setup. At each wavelength, we read the intensity from the photocell on an oscilloscope through an amplifier. These values of intensity are then corrected for imperfections in the components: differences in emissivity of the bulb, reflectivity of the mirrors and grating, response through various filters and response by the photocell. Filters are used to prevent orders higher than one from striking the photocell, and therefore falsely increasing the intensity. Once we read these
Page 4 Figure 2. a. (left): Experimentally calculated electromagnetic spectrum emitted from our tungsten light source, at 2600 and 1600 K. Intensity here is the corrected signal read from an oscilloscope. b. (right): Energy density vs. wavelength for blackbodies of 2600 and 1600 K, calculated using Planck s Law. We can clearly see similarities in the experimentally and theoretically calculated spectra. intensities from the oscilloscope, we corrected these values for these impurities with correction factors unique to the devices used. Finally, upon getting the spectrum, we calculated the temperature of the bulb using the voltage and current through the power supply to calculate the resistivity of the bulb. We have data for resistivity of tungsten versus temperature, and used this to find the temperature of the bulb at a given voltage and current. We then found the electromagnetic spectrum at another temperature, to compare the temperature dependent effects of Planck s Law. Results We found the electromagnetic spectrum at two temperatures, 2600 ± 50 K and 1600 ± 30 K. These spectra appear in fig 2.a., while the theoretical spectra, calculated from Planck s Law, appear in fig 2.b. Clearly, the plots are very similar, and we can already see that Planck s Law fits the experiment well. From our spectra, we can find the maximum wavelength to assess Wien s Law. For the 2600 K spectrum, we found the peak wavelength was 1.2 ±.17 µm, while the spectrum at 1600 K gave a wavelength of 1.4 ±.17 µm. For the theoretical plots, the peak wavelengths are 1.11 µm and 1.8 µm for the 2600 and 1600 K spectra, respectively. By matching each theory curve with its experimental curve, and scaling it appropriately, as seen in fig 3, we can see these differences. Analysis Using fig 2, we can see qualitatively see the validity of the Stefan-Boltzmann Law. The cooler temperature spectrum, at 1600 K, has significantly less area beneath the curve, in agreement with the Stefan-Boltzmann Law. We can also see this is in close agreement with the relationship between the two spectra in theory, by comparing fig 2.a. to fig 2.b. We next compared our experimental data to that predicted by Wien s Law. As written, we found max wavelengths of 1.2 ±.17 µm and 1.4 ±.17 µm for 2600 and 1600 K respectively, where theory predicts 1.11 µm and 1.8 µm. The theoretical peak wavelength at 2600 K falls within our error, but the second does not. By taking a closer look at fig 3.b., we see that the
Page 5 Figure 3. a. (left): Experimental and theoretical spectra for a 2600 K blackbody, in arbitrary units. We can clearly see the experimental plot does closely follow theory. b. (right): Experimental and theoretical spectra for a 1600 K blackbody, in arbitrary units. Again, the experimental curve is close to theory, but it drops off significantly more at higher wavelengths. experimental curve, although having the correct general shape to it, does not fit the theory all that well. In fact, the experimental spectrum falls off much more quickly at higher wavelengths than theory. Our first explanation for this was absorption due to the atmosphere: observing a plot of solar emission reaching the Earth, we see absorption lines at 1.1-1.15 µm, 1.35-1.5 µm and 1.8-1.95 µm. These absorption lines line up very well with places of decreased intensity on both spectra in fact not only could it explain the poor fit of the 1600 K spectrum, but it could explain the large dip at about 1 µm on the 2600 K spectrum. Upon further consideration of this hypothesis, we found that this did not in fact explain our observations. The absorption of electromagnetic waves in a medium is given by:, (6) where I is the transmitted intensity, I 0 is the intensity incident on the medium, α is the absorption or extinction coefficient unique to the medium, and x is the depth of the medium. The value e -αx will give us the fraction of light transmitted through a medium. The coefficient α for air as a medium varies with wavelength, and is shown in fig 4 for relevant wavelengths. The absorption peaks at about 1.4 and 1.8 µm both Figure 4: Extinction Coefficient vs. wavelength for air as a medium. The absorption peaks relevant to our measurements occur at about 1.1, 1.4 and 1.9 µm. reach values of about 10 km -1. The light in our setup travelled about one meter from source to detector, so using these and the absorption equation, we find that 99% of the light from the source reaches the detector due to atmospheric absorption. This effect is too small to be noticed on our data, so our differences in our spectra must come from another source. The most likely source of error in our measurement, now that atmospheric absorption is ruled out, is the correction factors we used. Small errors in the correction factors could easily change the spectra noticeably. By looking at the correction data supplied, we noticed that at small wavelengths, approaching 1 µm, the correction factor for the photocell response begins having two values, one a theoretical
Page 6 Figure 5. a. (left): Experimentally found electromagnetic spectrum emitted from a 2600 K blackbody with theoretical curves from Planck s Law and the Rayleigh-Jeans Equation. It can be clearly seen that both theories fit well at large wavelengths, but at small wavelengths, the Rayleigh-Jeans approaches infinity while Planck s Law and the experiment do not. b. (right): Experimentally found electromagnetic spectrum emitted from a 1600 K blackbody with theoretical curves from Planck s Law and the Rayleigh-Jeans Equation. It can be clearly seen that both theories fit well at large wavelengths, but at small wavelengths, the Rayleigh-Jeans approaches infinity while Planck s Law and the experiment do not. response, and another larger response calculated from previous experiment. We used the experimentally calculated value but clearly there is a good deal of uncertainty in the correction at those wavelengths. This deviation could very easily explain the small dip right at the theoretical maximum wavelength in the 2600 K spectrum. At larger wavelengths, approaching 2 µm, the emissivity of the tungsten bulb, another correction factor, drops dramatically while the error increases to about 5% the value. This uncertainty could also account for the large difference in theoretical and experimental curves at high wavelengths. If these corrections were corrected, the values of the 1600 K spectra around the peak wavelength could very well be larger, fitting Planck s and Wien s Law more accurately. Finally, we plotted the Rayleigh-Jeans equation for 2600 and 1600 K and compared them to both experiment and Planck s Law. This appears in fig 5. We see what is expected: similar behavior at high wavelengths, but with Rayleigh-Jeans increasing to infinity at low wavelengths, diverging from Planck s Law and our experiment. Conclusions The description of the thermal radiation from a blackbody will forever stand in the history of physics as one of the key experiments in making the transition from classical theory to quantum mechanics and modern physics. Classical theory could not describe it; only with the idea of quantized energy, the cornerstone of quantum mechanics, could this phenomenon be explained. In this study, we were able to confirm what these experiments found at the beginning of the 20 th Century that only a quantum mechanically derived equation can describe blackbody radiation. We also confirmed Wien s Law and the Stefan-Boltzmann Law, both consequences of Planck s Law. The understanding of blackbody radiation is not only important for historical reasons, but it can be used in modern day astronomy research. Using a method similar to what we used, astronomers can determine the temperature of various stars, planets and other objects in space. Researchers were even able to
Page 7 find the temperature of the universe (about 2 K), which is in agreement with theoretical cooling due to expansion after the Big Bang. Now we fully understand the effect and consequences of blackbody radiation, and perhaps we will one day use it in our research. References Taylor, John. An Introduction to Error Analysis. Second Edition. University Science Books. Sausalito CA, 1997. Tipler, Paul A & Llewellyn, Ralph A. Modern Physics. Fifth Edition. WH Freeman and Company. New York NY, 2008. Thermal Radiation. Phys 0560 Manual, Brown University Department of Physics. 2011. https://wiki.brown.edu/confluence/display/p hysicslabs/phys+0560+experiments+in+ Modern+Physics. (accessed 25 April 2011).