Lab 2

Black Body Spectrum

 

 

 

 

 

 

Bill Chun Wai Hung

Binh An

Nimish Kumar

 

 

Friday Section

 

23 April 2004


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 Pasco.
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.