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PHYS235 Thermodynamics Experiment

STEFAN-BOLTZMANN LAW

References

1. Tipler & Llewellyn, Modern Physics, 4th ed, pp 132-1412. Blatt, Modern Physics, pp 65-723. Young & Freedman, University Physics, 11th ed, p 6694. Kittel & Kroemer, Thermal Physics, 2nd ed, pp 90-98

AimTo verify the Stefan-Boltzmann Law for black body radiationTo determine the value of the Stefan-Boltzmann constantTo verify inverse square law

Apparatus

Demonstrating StandOptical benchElectric oven on standBlack body accessory (2 pieces)Iris diaphragm in frameMoll's thermopileMillivoltmeter Keithley Model 1774 Riders for optical benchDigital Thermometer -10 to 750 C

Principles

The Planck black body distribution is given by

Fd=2hc

2

5

d

ehc/ kT

1

where F = power radiated per unit area per unit wavelength interval at ;

T = temperature, in Kelvin;h = Planck's constant;c = velocity of light;k = Boltzmann's constant.

If follows, by integration, that the power per unit area emitted by the blackbody is,

0

Fd= T4

where =2k

4

5

15h3c2

.

This is the Stefan-Boltzmann Law and is the Stefan-Boltzmann constant, which has the current

(2008) CODATA value of

= (5.670400 0.00004) x 10-8 W m-2 K-4

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It also follows that, as Tincreases, the wavelength at which maximum energy is radiated decreases.

Tmax = 0.28978 cm K = 2898 m K.

This is the Wein displacement law. Due to asymmetry of the curve, the mid-energy point lies atslightly longer wavelengths.

T mid-energy 0.42 cm K.

For all uncollimated radiation, the flux (energy per cm2) decreases as 1/d2 where d is the distancefrom the source. This is a simple consequence of the rectilinear propogation of light and the

conservation of energy - the radiation is spread over an area which is increasing as d2.

Description of Apparatus

The ceramic electric oven (shown in fig 1.) has a cylindrical heating chamber of approx. 37 mm insidediameter and is 100 mm in length. When connected to the mains voltage marked on its case, theelectric oven reaches, after about one hour, its maximum permissible working temperature of approx.600 C; power input approx. 200 W.

It is connected to the mains via an autotransformer to regulate the temperature. 150 V gives atemperature of approx. 330 C. Higher voltages may be used to accelerate heating to this temperature.

Fig.1. Oven and stand

The Black body accessory (shown in fig 2) is intended for use with the electric oven. It consists of abrass tube, blackened on the inside, which can be inserted into this oven and heated, and a water-cooled stop with funnel-shaped front surface and an opening of 20 mm diameter. The cooled stopensures that only radiation emitted by the opening, and not by the side walls,emerges.The brass tube is

approx. 10 cm long, and has a diameter of 36 mm. One end is provided with an opening for inserting athermometer.

The stop, dia. approx. 120 mm, is provided with two hose nipples for the circulation of cooling water.Water should flow in at the bottom and out at the top.

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Fig.2. Black body accessory

Moll's thermopile (shown in fig. 3) is a sensitive instrument for measuring radiant energy. Over anarea of 10 mm diameter, the thermopile contains 16 linearly arranged constantan-manganinthermocouples, silver-soldered and blackened on the front; internal resistance approx. 10 , responsetime 2 to 3 seconds; wavelength range from 150 nm to 15 m.

The thermopile is enclosed in a massive metal case, 8 cm long, dia. approx. 34 mm, with polishedfunnel, aperture angle: 22 . The front opening of the funnel is covered with a glass window whichshould be taken out when measuring thermal radiation.

Fig. 3. Moll's thermopile

The voltage generated by the thermopile is read by a Keithley Millivoltmeter Model 177 (Use D.C.

20 mV range).

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Method

1. The oven, stop, iris, and thermopile should be already mounted on the optical bench.

2. Note:a) The position of the end of the brass tube (edge of the oven) with respect to the optical bench.

b) The size of the hole in the stop.c) The aperture of the thermopile lies behind the protective window. Its diameter is 25 mm and it

is located about 25 mm from the edge of the rider on the optical bench. Note the position ofthis aperture; DO NOT stick anything into the polished funnel of the thermopile!

3. With components close together, adjust them to same height. The thermopile must have a clearview of the stop through the iris and only the hole in the brass tube must show through thestop.

4. Set thermopile about 40 cm away from the stop with the iris about one third of the way fromthe thermopile1. Note the exact distance from the aperture of the thermopile to the aperture ofthe brass tube.

5. Connect thermopile to Keithley Millivoltmeter Model 177 and use D.C. 20 mV range.

6. Start water flow.

7. Remove protective window from the thermopile.

8. Note the room temperature (T0) and the voltage reading before the oven is switched on. This isdue to the temperature difference between the thermopile reference junctions and the ambientradiation. (If the thermopile is equilibrated at room temperature, this should be zero.)

9. Turn oven voltage with autotransformer to 240 V.

10. As Tincreases, note it and the thermopile voltage every ten degrees.

11. When Treaches about 330 C, turn the voltage on the oven back to 150 V. Note there is someovershoot. Take further measurements until the temperature stabilises.

12 Vary the distance of the thermopile from the stop. Keep the iris approximately half waybetween. Note the voltage at distances from about 30 to 50 cm in 2 cm intervals. (Work as

quickly as possible to avoid the effect of unsteady temperature. Note the temperature at thebeginning and end. If it changed more than 5 C discard and repeat your measurements whenthe temperatue is steady). Repeat the measurements going in opposite direction and average.What error does a 5 C change in temperature make?

13. When finished: turn voltage to 0 and switch off autotransformer, turn Keithley Millivoltmeteroff, replace window in thermopile, turn water off.

1The iris is to block out ambient light, especially that coming from you as you move around. You may be able to adjust its

position to do this best.

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Analysis of results

1) Plot Thermopile output voltage (minus background reading) vs (T4 T04) (Kelvin).

Comment on the shape of the curve.

Note the specification of the thermopile - its sensitivity is between 150 nm and 15 m - willthis have an effect on your results? At the lowest temperature for which you got a signal, what doesthe Wein displacement law tell you about the wavelength of the radiation?

Calibration of thermopile: You should have verified that the energy received by the

thermopile is proportional to (T4 - T04). The full relationship between the flux observed (Fobs) at the

thermopile at a distance dfrom any source of flux (Fs) is given by

Fobs = Fs (Projected area of Source)/(d2)The energy received by the thermopile is this value times the collecting area of thethermopile. Because the source is a black body, and the ambient temperature is T0, its flux is

Fs =(T4 T04)

Thus

Pobs = (T4 T04) (Projected area of Source) (Collecting area of thermopile)/( d2)

Given that the rated sensitivity of the thermopile is (0.16 0.02) mV/mW use the slope of your

Vvs (T4 T04) graph to find a value for the Stefan-Boltzmann constant, . Be sure to include the

uncertainty in your result. How well does it agree with the accepted value?

2) Plot Thermopile output voltage vs 1/d2; d = distance from the aperture of the thermopile to theaperture of the brass tube.

Suggest reason(s) for any non-linearity in yourVvs 1/d2 relation.

Questions

1. What is a black body, and how is a black body approximated in the apparatus used?

2. What basic assumption did Planck make in order to derive the wavelength distribution of theradiation? Describe how this assumption leads to avoidance of the ultraviolet catastropheShow that Planck's frequency distribution reduces to the Rayleigh-Jeans law in the lowfrequency limit.

3. Why is the stop cooled?

Modified April 2008