LEP3.5.01

-01Stefan-Boltzmanns law of radiation

PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D-37070 Gttingen 23501-01 1

Related topicsBlack body radiation, thermoelectric e.m.f., temperature depen-dence of resistances.

PrincipleAccording of Stefan-Boltzmanns law, the energy emitted by ablack body per unit area and unit time is proportional to thepower four of the absolute temperature of the body. Stefan-Boltzmanns law is also valid for a so-called grey body whosesurface shows a wavelength-independent absorption-coeffi-cient of less than one. In the experiment, the grey body isrepresented by the filament of an incandescent lamp whoseenergy emission is investigated as a function of the tempera-ture.

EquipmentThermopile, molltype 08479.00 1Shielding tube, for 08479.00 08479.01 1Universal measuring amplifier 13626.93 1Power supply var.15 VAC/12 VDC/5 A 13530.93 1Lamp holder E 14, on stem 06175.00 1Filament lamp 6V/5A, E14 06158.00 3Connection box 06030.23 1Resistor in plug-in box 100 06057.10 1Optical profile bench l = 60 cm 08283.00 1Base f. opt. profile-bench, adjust. 08284.00 2Slide mount f. opt. pr.-bench, h = 30 mm 08286.01 2Digital multimeter 07134.00 3Connecting cord, l = 500 mm, blue 07361.04 4Connecting cord, l = 500 mm, red 07361.01 4

Tasks1. To measure the resistance of the filament of the incandes-

cent lamp at room temperature and to ascertain thefilaments resistance R0 at zero degrees centrigrade.

2. To measure the energy flux density of the lamp at differentheating voltages. The corresponding heating currents readoff for each heating voltage and the corresponding filamentresistance calculated. Anticipating a temperature-depen-dency of the second order of the filament-resistance, thetemperature can be calculated from the measured resis-tances.

Set-up and procedureThe experiment is started by setting up the circuit of Fig. 2 tomeasure the filaments resistance at room temperature. Aresistor of 100 is connected in series with the lamp to allowa fine adjustment of the current. For 100 mADC and200 mADC the voltage drops across the filament are read andthe resistance at room temperature is calculated. The currentintensities are sufficiently small to neglect heating effects.

The experiment set-up of Fig. 1 is then built up. The 100 resistor is no longer part of the circuit. The filament is nowsupplied by a variable AC-voltage source via an ammeterallowing measurement of alternating currents of up to 6amperes. The voltmeter is branched across the filament andthe alternating voltage is increased in steps of 1 volt up to amaximum of 8 V AC.

Fig. 1: Set-up for experimental verification of Stefan-Boltzmanns law of radiation.

LEP3.5.01

-01Stefan-Boltzmanns law of radiation

23501-01 PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D-37070 Gttingen2

Fig. 2: Circuit to measure the resistance of the filament atroom temperature.

Remark: the supply voltage of the incandescent lamp is 6 VAC. A voltage of up to 8 V AC can be applied if theperiod of supply is limited to a few minutes.

Initially, a voltage of 1 V AC is applied to the lamp and theMoll-thermopile, which is at a distance of 30 cm from the fila-ment, is turned (slide-mount fixed) to the right and to the leftuntil the thermoelectric e.m. f. shows a maximum. The axis ofthe cylindrical filament should be perpendicular to the opticalbench axis. Since the thermoelectric e.m. f. is in the order ofmagnitude of a few millivolts, an amplifier has to be used foraccurate readings. The factor of amplification will be 102 or103 when using the voltmeter connected to the amplifier in the10 V range. Before a reading of the thermoelectric e.m. f. istaken, a proper zero-adjustment has to be assured. This isdone by taking the lamp together with its slide-mount awayfrom the bench for a few minutes. The amplifier is used in theLOW DRIFT-mode (104 ) with a time constant of 1 s.

After the lamp has been put back onto the bench, the readingcan be taken if the Moll-thermopile has reached its equilibri-um. This takes about one minute. Care must be taken that nobackground radiation disturbs the measurement.

Theory and evalutionIf the energy flux density L of a black body, e.g. energy emittedper unit area and unit time at temperature T and wavelength Mwithin the interval dM, is designated by dL(T, M)/dM,

Plancks formula states:

= (1)

with: c = velocity of light(3.00 108 [m/s])

h = Plancks constant(6.62 1034 [J s])

k = Boltzmanns constant(1.381 1023 [J K1])

Integration of equation (1) over the total wavelength-rangefrom M = 0 to M = gives the flux density L(T) (Stefan-Boltzmanns law).

L(T) = (2)

respectively L(T) = T T4

with T = 5.67 108 [W m2 K4]

The proportionality L T4 is also valid for a so-called greybody whose surface shows a wavelength-independentabsorption-coefficient of less than one.

To prove the validity of Stefan-Boltzmanns law, we measurethe radiation emitted by the filament of an incandescent lampwhich represents a grey body fairly well. For a fixed distancebetween filament and thermopile, the energy flux G which hitsthe thermopile is proportional to L(T).

G L(T)

Because of the proportionality between G and the thermo-electric e.m. f., Utherm of the thermopile, we can also write:

Utherm T4

if the thermopile is at a temperature of zero degrees Kelvin.Since the thermopile is at room temperature TR it also radiatesdue to the T4 law so that we have to write:

Utherm (T4 T4R)

Under the present circumstances, we can neglect T4R against T4

so that we should get a straight line with slope 4 when repre-senting the function Utherm = f(T) double logarithmically.

lg Utherm = 4 lg T + const. (3)

The absolute temperature T = t + 273 of the filament is calcu-lated from the measured resistances R(t) of the tungsten fila-ment (t = temperature in centigrade). For the tungsten filamentresistance, we have the following temperature dependence:

R(t) = R0 (1 + Bt + Ct2) (4)

with R0 = resistance at 0C

B = 4.82 103 K1

C = 6.76 107 K2

The resistance R0 at 0C can be found by using the relation:

R0 = (5)

Solving R(t) with respect to t and using the relation T = t + 273gives:

T = 273 + (6)

R(tR) and R(t) are found by applying Ohms law, e.g. by volt-age and current measurements across the filament.

12b

c Ba2 4b a R1t 2R0 1 b a dR(tR)

1 a tR b t2R

2p 5

15

k 4

c2 h3 T 4

2c2 hl 5

e hclkt1

dL(l, T)

dl

Stefan-Boltzmanns law of radiationLEP

3.5.01-01

PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D-37070 Gttingen 23501-01 3

1. Using the DC voltage output of the power supply unit, adirect current of 100 mA, respectively 200 mA, was suppliedto the filament via an 100 resistor. The corresponding volt-age drops were found to be 16.5 mV and 33.0 mV. Doublingthe current doubles the voltage drop. This shows that the tem-perature influence on the resistance is still negligably small forthe DC values chosen. We find in this case

R(tR) = 0.165 [] (7)

and hence:

R0 = 0.15 [] (8)

Small variatons in R0 only influence the slope S, which is to befound, in a negligable way.

2. Increasing the AC heating voltage in steps of 1 V AC from 0to 8 volts gave the following results:

U[V] I [A] Utherm[mV] T [K]

1 2.20 0.15 6722 2.80 0.62 9833 3.45 1.30 11604 4.00 2.20 13005 4.45 3.20 14306 4.90 4.45 15407 5.30 5.90 16308 5.70 7.50 1720

The double logarithmic, graphical representation of the ener-gy flux versus absolute temperature is shown in Fig. 3. Theslope S of the straight line is calculated, by regression, to be:

S = 4.19 0.265 (9)

The true value of S, which is 4, is found to be within the limitsor error.

Fig. 3: Thermoelectric e.m. f. of thermopile as a function of the filaments absolute temperature.

LEP3.5.01

-01Stefan-Boltzmanns law of radiation

23501-01 PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D-37070 Gttingen4

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