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1 Laboratory#2 – Phys4480/5480 (Optics) Dr. Cristian Bahrim Blackbody radiation Blackbody radiation Blackbody radiation Blackbody radiation In this experiment we will investigate the radiation emitted by a glowing object. The experiment uses the phenomenon of dispersion of light through a transparent prism (which is a dielectric material transparent to visible light). The blackbody object is simulated by the filament of a light bulb enclosed in a black cavity. The goal of this experiment is to determine accurate experimental values of the Wien’s constant (of theoretical value K m n × 6 10 898 . 2 ) and of the Stefan-Boltzmann constant (of theoretical value ( ) 4 2 8 10 67 . 5 K m W × - ). The setup is able to provide accurate values of the surface temperature and of the flux density of glowing objects using the wavelength at maximum radiancy. This technique enlarges the domain of applicability of a light sensor. Any object having a body temperature emits radiation. When the radiation emitted by objects is analyzed, the reflected light by their surface should be eliminated. In order to study the radiation which originates from the inside of an object, a theoretical model called “blackbody( or black cavity) at thermal equilibrium can be used. Typically, a black cavity is considered as being a hole in the walls of an empty metal box. We note that the blackbody is the hole itself and not the box! Any radiation entering through the hole in the cavity will have a negligible chance to exit. The blackbody radiation that one observes is formed inside the black cavity and is due solely to the temperature of the object. Figure 1 shows the radiancy of a blackbody object as a function of wavelength at four different temperatures. The radiancy of an object can be defined as the energy emitted per unit area, per wavelength, and per second by a glowing object. Planck’s theory of radiation based on the photon concept successfully reproduces these curves. The Planck’s formula for radiancy is () - = 1 1 8 4 4 kT hc e hc c R λ λ λ π λ (1) where c is the speed of light in free space, λ is the wavelength, h is Planck’s constant, k is the Boltzmann constant, and T is absolute temperature [in Kelvin]. Figure 1 shows that the peak of the radiancy shifts toward smaller wavelengths as the temperature increases. Wien generated an experimental relationship between the wavelength at the maximum radiancy and the temperature of the glowing object know as the Wien’s displacement law K nm T . 10 898 . 2 6 max × = λ . (2)

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Laboratory#2 – Phys4480/5480 (Optics) Dr. Cristian Bahrim

Blackbody radiationBlackbody radiationBlackbody radiationBlackbody radiation

In this experiment we will investigate the radiation emitted by a glowing object. The experiment uses the phenomenon of dispersion of light through a transparent prism (which is a dielectric material transparent to visible light). The blackbody object is simulated by the filament of a light bulb enclosed in a black cavity. The goal of this experiment is to determine accurate experimental values of the Wien’s constant (of theoretical value

Kmn ⋅× 610898.2 ) and of the Stefan-Boltzmann constant (of theoretical

value ( )4281067.5 KmW ⋅× − ). The setup is able to provide accurate values

of the surface temperature and of the flux density of glowing objects using the wavelength at maximum radiancy. This technique enlarges the domain of applicability of a light sensor.

Any object having a body temperature emits radiation. When the radiation emitted by objects is analyzed, the reflected light by their surface should be eliminated. In order to study the radiation which originates from the inside of an object, a theoretical model called “blackbody” ( or “black cavity” ) at thermal equilibrium can be used. Typically, a black cavity is considered as being a hole in the walls of an empty metal box. We note that the blackbody is the hole itself and not the box! Any radiation entering through the hole in the cavity will have a negligible chance to exit. The blackbody radiation that one observes is formed inside the black cavity and is due solely to the temperature of the object.

Figure 1 shows the radiancy of a blackbody object as a function of wavelength at four different temperatures. The radiancy of an object can be defined as the energy emitted per unit area, per wavelength, and per second by a glowing object. Planck’s theory of radiation based on the photon concept successfully reproduces these curves. The Planck’s formula for radiancy is

( )

=

1

18

4 4 kThce

hccR

λλλ

πλ (1)

where c is the speed of light in free space, λ is the wavelength, h is Planck’s constant, k is the Boltzmann constant, and T is absolute temperature [in Kelvin]. Figure 1 shows that the peak of the radiancy shifts toward smaller wavelengths as the temperature increases. Wien generated an experimental relationship between the wavelength at the maximum radiancy and the temperature of the glowing object know as the Wien’s displacement law

KnmT .10898.2 6max ×=λ . (2)

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This law can be found by taking the derivative of the Planck’s formula (1) with respect to the wavelength.

In figure 1, the area below the curve of radiancy represents the flux density of the light emitted by a glowing object at certain temperature, and is related to the temperature by a formula known as the Stefan-Boltzmann law.

4

0

)( TdRF σλλ == ∫∞

(3)

where σ is the Stefan-Boltzmann constant ( 428 /106705.5 KmW−× ). The flux

density is an average measurement of the total amount of energy emitted from each square meter of a light source per second. If the flux density is multiplied by the area of the light source then we have the luminosity of a glowing object, which is a useful quantity for astronomers, and also, it provides one of the seven fundamental units of the international system of units, called candela.

Theorectical Blackbody Curves

0

0.5

1

1.5

2

2.5

0 500 1000 1500 2000 2500 3000 3500 4000 4500

Wavelength (nm)

Radia

ncy(rela

tive u

nits)

2800 K

2600 K

2300 K

1900 K

Experimental setup

The setup is presented in figure 2. An infrared (or broad spectrum) light sensor attached to the light sensor arm is used to analyze the light emitted from a blackbody-like light source. The light first passes through a set of collimating slits and a lens before it reaches the prism. The light is then dispersed by the prism,

Figure 1 The theoretical radiancy emitted by a blackbody versus the wavelength at several temperatures.

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focused by a lens and passes through an aperture slit before it reaches the light sensor. A prism is mounted in the center of a degree plate (rotary table). A light sensor arm is attached to the plate and it can be rotated so that the sensor collects the entire spectrum of radiation dispersed by the prism. The angular position of the arm is recorded by a rotary motion sensor which has a small pinion in contact with the rotary table. The rotary motion sensor and the light sensor transfer the data to a laptop computer via a PASCO Science Workshop Interface (Model 750). Finally, the data is analyzed by the DataStudio™ software (IM-BB 1999) to create accurate plots of the radiancy.

Details about the theoretical model

You will use a DataStudio file already configured for measurements, and also, detectors calibrated for proper analysis. You will need to plot the radiancy versus wavelength (as in figure 1) for several temperatures (T)

and to find the wavelength maxλ at which the radiancy reaches a maximum

value. This wavelength is a function of temperature as predicted by the Wien’s displacement law (2). The temperature of the blackbody is varied by adjusting the voltage supplied across it. A voltmeter and an ammeter are used to find the temperature as a function of the supplied voltage and the current going through the blackbody, as described below.

The resistance of the filament depends linearly on temperature as

( )[ ]00 1 TTRR −+= α (4)

Figure 2 The PASCO setup for the study of the radiation emitted by a blackbody. The path of the light is shown by two thick arrows. In inset we show the position of the prism with respect to the incoming and outgoing light.

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where R is the resistance at temperature T and α is the temperature coefficient of

resistivity. Our light bulb, which is used as a blackbody, has 13105.4 −−= Kxα .

The subscript “0” in (4) labels the parameters at room temperature (300 K). The

resistance at room temperature is Ω= 84.00R . Solving equation (4) for T leads to

0

00

1

α

+=R

R

TT (5)

The resistance R can be found using Ohm’s law ( RIV = ):

I

VR = (6)

Both the voltage and current are measured using meters. Finally, the expression for temperature is:

3105.4

184.0

300−×

+=I

V

KT (7)

During this experiment, the temperature given by equation (7) is compared with the value predicted by Wien’s law. The agreement is typically within 15%. The blackbody radiation is incident on a 60 degree triangular prism (see figure 3).

By using Snell’s law at each face of the prism, one can show that the index of refraction is given by the expression

4

3

2

1sin

3

22

+

+= θn (8)

The Cauchy equation gives a relationship between the index of refraction and λ:

Figure 3 The path of the light through the prism.

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( ) BA

n +=2λ

λ (9)

The coefficients A and B are specific to the dispersive medium. The coefficients A and B will be derived in a later lab. For our acrylic prism these coefficients are A=13,900 and B=1.689.

Solving equation (9) for wavelength gives

Bn

A

−=2λ . (10)

Substituting the index of refraction n from equation (8) into equation (10) leads to the following equation which is necessary for getting the wavelength from the angular position

689.14

3

2

1sin

3

2

13900

2

−+

+

=

θ

λ (11)

Remark: The entire domain of λ , from 1000 nm to1500 nm, of interest in our

study (see figure 4.a) corresponds to a change in θ (which is the table-angle) of only 0.02 radians or 1.146 degrees (!!!) (see figures 4 below). Therefore, our measurement demands a very precise calibration and alignment of the setup.

Figure 4.a Experimental values of

wavelength versus the angle θ. The region between the dashed lines indicates the range of interest for the present study: (1000-1500) nm.

Figure 4.b An enlarged view of the range for the wavelengths used in this experiment.

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Details about the experimental procedure

Some special considerations need to be made related to formula (11). The

deviation angle θ from figure 3 is not the same with the angle measured by the rotary motion sensor. There are two measurements which need to be done:

(1) The angular ratio which will be simply called “Ratio” represents the ratio between the angle of rotation of a small pinion attached to the rotary motion sensor and the angle of the degree plate:

Ratio = angleplate

anglepin (12)

(2) Figure 3 shows that the angle θ needed in equation (11) is measured from the optical axis or the normal to the back side of the prism (where the light exits). However, the angular position measured by the rotary motion sensor

(which is rotated clockwise during measurements) is the angle αααα as shown in figure 5. We name INIT (see figure 5), the angle between the Stop (or the initial position of the rotary arm) and the normal to the back face of the

prism. From figure 5 we see that θ = (INIT – α ). In figure 5 all the angles are

Figure 5 Diagram of the angles used during the experiment. Note that all angles are measured with respect to the stop.

αααα

θ

INIT

Light

Sensor

Stop

Prism

Incident

Light

Initial

position

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measured with respect to the degree plate. However, the Data Studio software measures the angles using the pinion of the rotary motion sensor. So, the actual angle displayed by Data Studio is the pin-angle. Therefore, all angles should be divided by Ratio (which is given in equation (12)):

Ratio

INIT αθ

−= (13)

The value of INIT is critical for this experiment. Our study proved that the success of the entire experiment depends on the accuracy of the INIT angle. In order to have a good measurement, we need an angular precision that should reach a hundredth of a radian. This difficulty will be overcome by using a calibration technique for a particular voltage (7 volts) where the setup is the most sensitive.

Experimental Steps

using DataStudio (DS) and an Excel spreadsheet.

Phase 1. Measuring the RATIO.

1. Turn on the PASCO interface, the Power Amplifier and the two meters. The

Power amplifier delivers dc currents. Set the voltmeter on scale DC V 20 and the

ammeter on DC A 20A.

2. Open the “Blackbody_Data_Studio.ds” file located on the Desktop.

3. Open the “Blackbody_spreadsheet.xls” file located on the Desktop.

4. Set the light sensor at 60 degrees away from the 0-180 degrees axis, placing it

opposite to your position with respect to the central axis (as indicated in figure 2).

5. Press Start in DS and look to the window “Angular position”. This window

displays the pin-angle in radians.

6. Record the pin-angle on the first Excel spreadsheet called “Ratio”. Go in steps of

5 degrees on the table-angle as indicated in the first column of the spreadsheet.

7. After reading all the angles from 0 down to 60 degrees, and calculating the pin-

angle in degrees open from “Tools”, the Data Analysis and Regression and

perform a linear regression of the Pin Angle (y) vs. Table Angle (x) and set the

intercept to zero (for that click “Set Constant is zero”).

Record the Ratio with uncertainty: ________±_________ degrees

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Phase 2. Measuring the INIT angle.

8. Rotate the arm of the light sensor counterclockwise until it stops. A STOP is

fixed under the rotary arm. Delete all the data runs in the Data Studio windows.

9. Access the window called ”Radiancy vs. Alpha” in DS. Enlarge the window.

For this step you will need to rotate the light sensor arm in the clockwise

direction from STOP until you align it with the prism and the light source.

How to do the measurement?

a. Turn ON the blackbody source: click ON in the “Signal Generator” window of

DS which is initially set up at 7 volts.

b. Check if you see a rainbow on the aperture bracket in front of the light sensor.

c. Press the START button on DS and before you start rotating the rotary arm,

TARE the light sensor. This will zero the plot for “Radiancy vs. Alphas”. The

Tare button is on the light sensor. This action eliminates the background

radiation from your measurements.

d. Rotate the light-sensor arm in the clockwise direction SMOOTHLY and

STEADLY until you get two peaks on the ”Radiancy vs. Alpha” window at

about 14 and 72 degrees.

e. After recording all the data press STOP and turn off the “Signal Generator”.

10. Using a cross-hair read the angles αmax = _______________ degrees

and INIT = _______________ degrees

Remark: The peak at smaller angle represents αmax (see figure 5 and equation

(13) ) and the peak at larger angles represents INIT (equation (13)). This value of

INIT is just an approximated value. You will find a more precise value for INIT

later on.

11. Go on EXCEL and open the second spreadsheet called “Initial” (2nd

page of the

Excel file). Introduce the voltage V and the current I measured by the two meters:

V = ___________ volts

I = ___________ amps

On the line called “run#1” you have to evaluate the temperature according to

equation (7) and calculate λλλλpredicted according to the Wien’s law (2):

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Report: λpredicted = _______________ degrees

12. On the “Initial” spreadsheet, type in the value for Ratio (from item 7) and αmax

(from item 10) in the first two columns. Next, in the 3rd column introduce a range

of angles with a step of 0.1 degrees near the estimated value for INIT from item

10. The first value should be the closest lower integer to the estimated INIT

(example: type in 73 for INIT = 73.6). Next, calculate λ in column 4 (according to

formula (11)). Adjust the value of the INIT angle until λλλλ associated matches

the value of λλλλpredicted (from item 11) within a precision of 0.1 nm. Typically, the

angle should have three decimals for the best agreement with λpredicted. This is the

most precise value of INIT that you can measure. This measurement for INIT

takes care of the imprecision in the alignment of the prism.

Record the INIT : ____________ degrees

13. Go on DS look under DATA (upper left window) for the formula “theta=…”.

Open a “Calculator” window (clicking on the formula for “theta =…”) and insert

the Variables INIT and RATIO. Next, press “Accept” on the “Calculator”. Look

to the changes in the plot “Radiancy vs. wavelength” (on DS) when these

variables are introduced. Close the “Calculator”.

14. In the window “Radiancy vs. wavelength” check with a cross-hair if the value of

λmax for the maximum radiancy at 7 volts agrees with the value of λ after the best

adjustment of INIT using the spreadsheet (item 12) within 1 %. If not, you have

first to check if the two meters still display the same numbers as when you have

recorded them (there is a chance that the readings has changed because the circuit

needed some time to reach thermal stability), and next, please check the formulas

for “theta” (given in equation (13)) and “wavelength” (given in equation (11))

which are typed in “Calculator”. If the problem still persists, you will need to re-

start collecting new data again.

Now you are ready to start collecting the data for measuring the Wien’s constant and the Stephan-Boltzmann’s constant.

Phase 3. Measuring the Wien’s constant.

15. In the window called “Radiancy vs. wavelength” you have the experimental plot

for R=f(λ). Read λmax @ 7 volts measured with the cross-hairs.

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16. Open the “Data-Wien constant” spreadsheet and type in “run 1” the voltage and

the current indicated by meters, for 7 volts delivered by the “Signal Generator”,

and type in λmax @ 7 volts measured by DS.

17. Now you will start setting up the voltage on the “Signal Generator” at 4, 6, 8, and 10 volts, taking one voltage at the time and collecting the data. For that

please start with the lowest voltage (4 volts). The rotation of the arm should be

done smoothly and steadily in a clockwise direction. After each run you have to

rotate back (counter-clockwise) the rotary arm until it reaches the STOP position.

BEFORE you start rotating the arm of the light sensor you need to TARE the light

sensor!!! For each voltage you need to collect the current and the voltage

read by meters and to measure λλλλmax at maximum radiancy using the cross-hairs tool on “Radiancy vs. wavelength”. Rotate the light sensor until the DS

plot reaches 4,000 nm (on the horizontal axis), and so it completely includes

the peak of the radiancy.

You collect four runs in addition to the measurement used for calibration

(run#1 @ 7 volts). After you set up a new voltage wait about 1 minute before

you will re-start collecting new data. The voltage and current should be

stable when you do the readings.

18. On the “Data-Wien constant” spreadsheet for each run you will evaluate the

temperature measured with meters and indicated as T=f(V,i). Applying the Wien’s

law (2) you will calculate maxλ from the measured temperature (in the column “λ

- from T”). This represents the expected value for λmax at each T.

19. Using the data from column F called “λ - from DS” you need to calculate “T =

f(λ - DS)” based on the Wien’s law (2). The precision of this measurement for

getting λ is compared with “λ - from T”. You will see that the error is the largest

at 4 volts (and is expected to be about 20%). For the rest of the voltages, the

precision for λ should be within 10%.

20. Plot λλλλ = f( 1/T ) using the DS values for λ and the 1/T calculated based on the

measurements using meters ( T = f(V,i) ). You should get ideally a straight line.

21. Run a “Linear regression” (using “Tools” - “Data analysis” in Excel) of λλλλ from DS vs. 1/T from f(V,i). The slope represents the Wien’s constant, which

should be about 5% of the theoretical value (which is 2,898,000 nm.K).

Report the Wien’s constant: (_______±_______) x 106 [nm.K]

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22. Estimate the percentage error between the theoretical and the experimental values

for the Wien’s constant. For that, copy your value of the Wien’s constant in the

cell next to “Report”.

Estimate the precision in measuring the Wien’s constant = ____%.

Phase 4. Measuring the Stephan-Boltzmann’s constant.

23. Go on “Planck Radiancy vs. Wavelength” (enlarge the window) and take each of

the five runs at the time. The window is set up so that you can run an integral of

the radiancy over λ. This represents “the flux density” defined in (3), and is

indicated graphically as the shaded area under the curve of radiancy. For getting

this area, you have to activate “ΣΣΣΣ - Show selected statistics” – “Area”.

24. Go on the Excel spreadsheet called “Flux density” and write the temperature measured with meters ( T = f(V,i) ) and next, calculate T

4. The temperatures T

= f(V,i) should be copied from “Data – Wien constant” spreadsheet (Attention:

when you “Paste”, you have to select “Paste Special” and click “Values”).

25. For each run you need to update the value of the temperature (T) in the formula

for “planck radiancy=…” using the “Calculator”, and next, you need to calculate

the area below the curve of radiancy on “Planck Radiancy vs. Wavelength”.

26. Enter in the “Flux density” spreadsheet the area calculated with DS for each

run, and next, convert this area in units that will be cancelled by dλλλλ [in nm] from equation (3). This corresponds to column F (labeled “area*10^-9”).

27. The error bar can now be estimated by comparing the two areas. It should be

within a few percents.

28. Plot “Flux density” (given by DS) versus T4 (given by meters). For this plot

you need to have the data in order (i.e. the increasing order of temperatures: move

run#1 between runs #3 and #4).

29. Run a linear regression of the “Flux density” (column F) given by DS with the T

4 (column C). The agreement between theory and experiment should be

within 5%.

30. Estimate the deviation error (in percentage) between the experimental and the

theoretical Stephan-Boltzmann’s constant (which is ( ( )4281067.5 KmW ⋅× − ).

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Report the Stephan-Boltzmann’s constant: (_____±_____) x 10-8

[W/m2.K

4]

Estimate the precision in measuring the Stefan-Boltzman constant = _____ %.

You need to print out and show the four Excel spreadsheets

with the results and the plots you have done.

SETUP – BLACKBODY RADIATION

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