Lab 2
Black Body Spectrum
Bill Chun Wai Hung
Binh An
Nimish Kumar
Friday Section
Black Body Spectrum
Goal:
To find the temperature dependence of the energy radiated by a perfect (black
body) radiator by using the Wien Displacement Law connecting lambda max and
temperature T.
Equipment List:
- Light Bulb Cavity.
- Diffraction Grating
- Photometer
- Slits and Lenses
- Calibrated Platform
- Rotary Motion
- Signal Generator
- Voltmeter
- Computer
- Leslie's Cube
- Thermal Sensor
Background:
The radiation from a solid body at a temperature T due to thermal collisions is
called blackbody radiation. The equation describing the amount of radiation is
Q= . This is called the Stefan Boltman Law with _
the Stefan Boltzman constant = 5.670 X 10 power 8 Wm K. is the emissivity
of the surface (0<e<1) where e=1 means perfect absorption and emissivity
of the surface. One of the most impressive accomplishments of quantum theory is
the formula describing how much intensity is emitted at a particular
wavelength, the integral of which is proportional to T power 4.
Experiment #1:
1) Using the thermal sensor provided, measure the total energy radiated from
Leslie's cube and compare to T power 4 using resistance of the bulb supplying
the energy after the system reaches equilibrium.
2) Determine e for each side of the cube at the same T.
Experiment #2:
1) Determine the conversion factor of the angle of the rotary motion sensor to
the angle of the rotating table.
2) Arrange the zero angle of measurement by taking data of intensity (output of
photometer) versus angle.
3) Set up calculator to calculate wavelength from the angle of measurement.
This depends on whether you have a prism or a diffraction grating as your
wavelength analyzer. Set the calculator to calculate the expected wavelength of
maximum intensity from the current supplied to the bulb.
4) Set the temperature by adjusting the voltage to the bulb to 4V. Graph the
intensity versus the wavelength of the light. This will not look like the
distribution in the book due to the uneven frequency response of the
photometer.
5) Use the cursor to find the wavelength corresponding to the maximum
intensity.
6) Integrate the intensity distribution over all the available data.
7) Write down the current supplied to the bulb supplying the blackbody
radiation.
8) Increase the temperature by increasing the voltage to 5 and 6 V.
9) Take the ratio of the values of lambda max for 5 and 6V relative to 4 V and
raise them to the 4th power. Compare that ratio with the ratio of the areas
under the curve.
10) Calculate the resistance R= V/I. Calculate the temperature from the R
versus T curve supplied by
11) Make a second analysis of the integral of the blackbody spectrum energy
versus T power 4 from the resistance measurement calculating the ratio of the 5
and 6V measurement with the first (4V) measurement.
12) Change the photometer to one sensitive to infrared and place an infrared
filter in front that the max Intensity lies about 1000 nm. Compare the integral
of the intensity at this voltage with that at 1250 nm. Discuss the qualitative
difference between the infrared spectrum and the optical spectrum.
Procedure II
|
Voltage (V) |
Intensity (W/M^2) |
Maxmum Wave Length (m) |
Current IA) |
Ratio of Wave Length Relative to
4V |
Ratio^4 |
|
4 |
2.774 |
0.758 |
0.431 |
1 |
1 |
|
5 |
4.573 |
0.808 |
0.493 |
1.065963061 |
1.291126 |
|
6 |
6.469 |
0.811 |
0.541 |
1.069920844 |
1.310408 |
|
Area (% max microns) |
Ratio of Area Relative to 4V |
Resistance ( |
Temperature, T(K) |
Temperature Relative to 4V |
T^4 |
|
1.507 |
1 |
9.280742 |
2533.00065 |
1 |
1 |
|
2.396 |
1.589914 |
10.14199 |
2760.84334 |
1.08995 |
1.411321 |
|
3.078 |
2.042468 |
11.09057 |
3011.7918 |
1.189021 |
1.99875 |
Calculation.
Sample Calculation of Temperature
T = 
where
= 0.84
= 4.5 x 10^-3
K^-1
= 300K
T = 
= 2533K
Conclusion by Bill Chun Wai Hung:
1. Principles and Results
For Procedure I, under constant temperature, the radiations of 4 different surfaces of a cube (white, black, mirror, and gray) are measured by the Thermal Sensor. The results are as the following
|
Surface |
Amount of Radiation, Q (J/s) |
Ratio Relative to Black, |
|
Black |
5.15 |
1 |
|
Gray |
1.89 |
0.367 |
|
White |
4.80 |
0.932 |
|
Mirror |
0.945 |
0.183 |
The temperature is assumed to be constant at 105 oC.
When the amount
of radiation (Q) from different surface is measured, the amount of radiation of
the black surface is assumed to be 1. Using the value of the black surface to
divide the other 3 surfaces (gray, white, and mirror), the ratio of the
surfaces relative to the black surface is obtained. This ratio is emissivity (
) of the surface.
This is because
in the equation Q=
, 
, A, and T are constant. The only two variables that changes
are Q and, and they are proportional to each other. Getting the ratio
of Q is the same of getting the ratio of the emissivity.
This experiment shows that different surface has different emissivity, that is different ability to release energy.
In Procedure II, voltage, current, intensity of radiation, and maximum wavelength are measured. This experiment uses three different ways to measure the ratio of radation.
By using the value of maximum wavelength of 4V to divide the value of maximum wavelength of 5V and 6V, ratios between them can be obtained. Multiplying the ratio to the 4th power, then the multiplied ratio can be compared with the 4th power of the ratio of the temperature as well as the ratio of intensity.
The value of temperature can be obtained by the formula
T =
where = 0.84, = 4.5 x 10^-3
K^-1, = 300K
Similarly, by using the value of temperature of 4V to divide the value of temperature of 5V and 6V, ratios between them can be obtained. Multiplying the ratio to the 4th power, then the multiplied ratio can be compared with the 4th power of the ratio of the maximum wavelength as well as the ratio of intensity.
Also, the ratio of intensity is obtained by dividing the value of intensity of 5V and 6V by the value of intensity of 4 V.
The result of the ratios are as the following. The ratio of area under the curve is the ratio of intensity.
|
Voltage (V) |
Ratio of Area Relative to 4V |
(Ratio of Maximum Wavelength)^4 |
(Ratio of Temperature)^4 |
|
4 |
1 |
1 |
1 |
|
5 |
1.589914 |
1.291126 |
1.411321 |
|
6 |
2.042468 |
1.310408 |
1.99875 |
The table shows that as the voltage of
the power supply increases, the intensity, the maximum wavelength, and the
temperature increase. The ideal results of the ratios should be very close to
each other. However, the experimental results above shows that the ratios
obtained by the maximum wavelength are generally more off than the ratios by
measuring the intensity and the temperature of the light source. These differences
of results are caused by various errors, and the errors will be discussed in
the ¡°Error Analysis¡± section below.
2. Error Analysis
For Procedure I, the temperature of the apparatus was assumed the be constant, but the temperate was not constant due the heat transferred by the light source, and the fluctuation of surrounding temperature. Or more so, the temperature of the people performing the experiment. One way to improve this error is to put the apparatus in a vacuum environment.
Another possible cause of the error is that the surfaces are not perfect surfaces. That is the surfaces may be dirty, or not a perfect plane.
The measurement error is that the amount of radiation is read off from the device, and the value is constantly changing. The values used in the experiment is a best estimation of the true value.
For Procedure II, the ratio is distorted by multiplying the ratio to the 4th power. In other words, even the error of the ratio was very small originally, and the error of the ratio will become much larger when the ratio is multiplied to the 4th power.
Besides, the integration under the curve in the data sheet generated by the computer is plotted will a limited number of points. That is the limited number of points shaped the area under the curve, and the area is used in the comparisons. This contains inherited error. Furthermore, the a certain amount of the left and the right side of the plotting has to be subtracted from the area because those are not the measurement of the light source that the experiment is testing on, but the background light¡¯s intensity. By subtracting the area of the left and right portion, the amount subtracted is based on estimation, so the subtraction contributes to additional amount of error.
Besides, there is a incomplete integration due to the limited range of the photometer. In other words, the light emitted is not totally measured by the photometer due to the distance between the source and the photometer. Thus, certain amount of the measuring object is not measured.