BLACKBODY RADIATION
- Any Modern Physics text.
- CRC manual, Experimental Physics book, or internet for properties of tungsten.
- To investigate the temperature dependence of the total power radiated from a heated filament.
- To investigate the extent to which an incandescent light bulb filament emulates a blackbody when heated.
Plank's radiation law describes the intensity per unit wavelength emitted by a blackbody f(l) as,
,
where h is Plank's constant, c is the speed of light, and k is Boltzmann's constant.
Exercise 1: A regular 100 Watt tungsten filament lightbulb is heated to less than 3000 K, while a 100 W halogen bulb burns at around 5000 K. Why are halogen bulbs cheaper to power, or why are they brighter, than the tungsten bulbs? To help answer this, graph f(l) throughout the spectrum over which is emits appreciably for both bulbs (on the same graph).
Exercise 2: From Plank's radiation law, determine a general relationship for the wavelength of the peak of the curves in Exercise 1.
What is important about blackbody radiation? Why was the correct theoretical description so important? Discuss this in terms of Plank's law and the Rayleigh-Jeans law. Are the specific theories of blackbody radiation relevant to current scientific research?
We want to look at blackbody radiation as it pertains to a lightbulb. The electrical energy delivered to the bulb is predominately radiated as electromagnetic energy by the filament posing as a blackbody. Knowing the power delivered to the bulb from Ohm’s law and knowing the temperature of the filament (based on its resistive properties), we would like to determine the power law relationship between the two.
The main measurements in this lab is to very carefully determine the Voltage vs. Current characteristics of an incandescent lightbulb throughout the entire range of possible voltages. An incandescent light bulb is wired in series with a resistor R (does not need to be exactly 99.3 ohms) as shown below. This assumes a 12 V bulb, but that, too can be different. Just be sure to go up to a voltage just above the rating of the bulb.
Exercise 3: A slightly simpler circuit would be to use the multimeter as a ammeter (current meter) in series with the bulb and eliminate the 99.3 ohm resistor. Why is the above setup more accurate and/or precise?
The tungsten filament of the lamp is to be assumed to be a blackbody that radiates all of the electrical power delivered to it (IV) as electromagnetic radiation. The resistivity of tungsten as a function of temperature is well known, and can be found in a number of sources (see referemces). The temperature of the filament can be determined by measuring the resistance of the lamp, converting it to resistivity, and from resistivity to temperature using the known data. A log-log plot of power verses temperature can confirm or deny the fourth order power relationship predicted by Stefan’s Law. Measurements of the length and radius of the filament can also allow Stefan's constant s to be calculated.
Exercise 4: Using Stefan's Law, determine what the y-intercept and slope should be of a log-log plot of Power vs. Temperature. What does A from this law represent for the lightbulb?
Recall that the relationship between resistance and resistivity is R = rL/Ac, where this area is the cross sectional area of the object through which current is moving (not the same area in Stefan's Law).
- Determine V vs I as above: The data collection will consist of recording the voltages across the resistor R and the corresponding voltages across the light bulb while the voltage source is incremented from zero to about just above the voltage rating of the bulb. It will be desirable for the incremental changes across the filament to be small (about 1 millivolt) for values less than 0.1 volts, but larger incremental changes (about 0.1 volt) for the larger values. Graph V vs. I to see how the resistance changes as the bulb gets hotter.
- Calculate Power (=IV) for each I,V pair, which is eventually to be plotted against temperature (the latter will be determined from the resistances, steps 4-6 below.)
- Calculate Resistance (=V/I, not slope) for each I,V pair.
- Determine L/Ac from a determination of R at small voltages: at the lowest voltages (very small currents), the filament’s temperature can be approximated to be room temperature (our only known temperature), consequently allowing L/Ac to be determined, since it equals R/r, where you can determine R at the lowest V/I values, and you have looked up the value for resistivity of tungsten at room temperature.
- Now all the resistances of the lightbulb should be multiplied by Ac/L to determine the corresponding resistivities.
- Finally, the resistivities can be converted into temperatures by indexing their value with a resistivity-temperature table for tungsten (look it up). This can be done by plotting the known resistivity verses temperature and taking a best-line-fit analysis in order to find a function which describes the resistivity as a function of temperature. This function can then be used in the spreadsheet/table to create a column of temperature values which correspond to your measured values of resistance.
- With power vs. temperature now determined, Stefan’s power law can be tested by plotting power vs. temperature on a log-log plot. The low voltage data may not be like the data at higher voltages. Can you guess why not? Test the power law for the higher voltage data, does it give the expected slope?
What about the y-intercept? Using the result of exercise 4, and assuming e = 1, if we can determine A, we can determine a value for Stefan's contant:
- Break open the bulb (when it is cool) and carefully measure the length of the filament.
- Knowing L/Ac (step 4) and the length of the filament (step 8), you can then find the surface area A of the filament.
- Your y-intercept will then yield a value for s which can be compared to the standard value.
The End
