Physics 341 Stefan-Boltzmann Law
In the early studies of radiation one of the significant steps forward was the connection between
the radiated energy from a hot body and its temperature. Such a radiating hot body is said to emit
“black body” radiation. Measuring the energy as a function of wavelength and temperature
yielded the “Black body radiation curve”. In 1879, Stefan, using purely thermodynamic
arguments, was able to deduce that the total emitted radiation is proportional to the fourth power
of the temperature of the radiator – the hotter the body, the more it radiates:
I (T )   T 4
The constant in the equation above is known as Stefan’s constant and has a value of
  0.56686 107 Wm 2 K 4
In this part of the experiment, you will measure the radiation from a hot tungsten filament as a
function of its temperature. The radiation will be measured with a radiometer. The radiometer
will measure the radiation from objects at room temperature to the radiation from direct sunlight.
The temperature of the filament will be determined from the resistance of the tungsten filament
wire, which can be described by:
R  R293 1   T  293 
Where room temperature is assumed to be 20oC (293K), α is the temperature coefficient of
resistance which is 0.0045 for tungsten, and T is the temperature in K. The resistance of the
filament is determined by measuring the voltage across the bulb and the current running it.
Experiment
See figure below for experimental setup.
1. Set the source and radiometer facing each other on the lab bench. The source should be
taped to the surface since the leads tend to move the lamp assembly. Measure the distance
from the filament to the middle of the shutter control rod of the radiometer and adjust it
to 10 cm. This distance does not actually matter as long as it remains constant.
Remember that the radiation received by the detector varies as the inverse square of the
distance, so that small changes in position make a large difference in measured radiation.
2. Without plugging the power supply into an outlet, connect the power supply and turn the
output dial to zero. This step is important. If the power supply is turned on with the
output turned up, the lamp could burn out. Also connect the ammeter and voltmeter.
3. Turn on the power supply.
4. Measure the resistance of the filament at a very low current value, between 50 and
70 mA, the voltage will be around 0.03 V. The resistance of the filament measured from
these values will be considered the room temperature resistance.
5. Measure the room temperature.
6. Set the radiometer to the 1 scale and with the shutter closed set the scale to zero.
7. Set the voltage source voltage to about 1.3-1.5 V. The filament will not glow but it will
be radiating. Record the voltage and current readings. Open the radiometer shutter and
record the meter reading.
8. Repeat the previous step, increasing the voltage in small steps, while recording the
voltage, current, and radiation. Do not exceed a current of 1.7 A through the filament.
You might want to set up a spreadsheet in Origin. Check the scale zero often, especially
if the sensitivity range has changed.
Data Analysis
The calculations are simple and repetitive and the best way to carry them out is using Origin.
You can find the temperature of the filament from arranging the equation for resistance:
R  R293
Ti  i
 293 K
 R293
If the room temperature was not 20oC, adjust the values in the equation accordingly. The
temperature coefficient for tungsten is   0.0045 . For a typical experiment the temperature
varies from room temperature to 2,500 K.
In Origin, plot your data as log E versus log T and determine the slope of the line.
The theoretical considerations used by Stefan in reaching his conclusion were based on a special
form of radiation source. It is described as a “black body radiation from a hollow heated
enclosure”. To produce this radiation, visualize a closed furnace with a small hole drilled into the
wall. Radiation inside the furnace is radiated and absorbed and radiated from the walls. The
small hole lets a bit of this radiation out of the furnace, but not enough to upset the equilibrium
inside. This radiation has the properties of an ideal “black body radiator”. Our filament is far
from the theoretical ideal, so if your exponent is a little off it is not surprising.
Leslie’s Cube
From the previous experiment we have seen that the power radiated by an object is given by the
Stefan-Boltzmann law. However, this relation its simplest form describes a theoretical “black
body”, and ideal material which incorporates the assumption that the object has no reflectance. In
reality, objects do not absorb and emit all the energy they encounter, and so the real energy
emitted must be scaled by the emissivity of the object:
I (T )   T 4 ,  - emissivity
Procedure
1. Fill the cube with hot water (approximately 60oC) and measure the temperature with the
thermometer provided.
2. Set the radiometer to the appropriate range. The white and black painted surfaces will
likely require the 10 W/m2 range, while the brass surfaces are more likely to yield data in
the 1 W/m2 range. Zero the radiometer whenever the range is reset. See figure.
3. Place the radiometer about 5 cm away from the cube facing the cube. Only one side of
the cube should bear on the thermal detector for any given trial.
4. Record the radiation detected from the cube and continue until all surfaces have been
observed. Check that your temperature has not changed.
Analysis
Given that the cube is heated uniformly by the water, we can assume that the ideal blackbody
radiation expected from each side would be the same. This means that the variations in emitted
radiation are due to differing values of ε, the emissivity. There is a clear relationship between the
surface material and this value, as observed in the broad changes in the data.
To better quantify the relationship between the measured emissivities, assume that the black
painted side approximates a blackbody emitter and find the relative emissivities of the other
sides.
Research the emissivities of the materials used in the Leslie Cube, and make reasonable
estimates for any values which you can’t find listed explicitly. How well do they agree? Does
scaling the measured data by a more accurate value of the emissivity of the black paint help?
How might you explain any errors in the data?