Stefann Boltzman Experimnet Viva

8/10/2019 Stefann Boltzman Experimnet Viva

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Boltzmann, Ludwig (1844-1906), an Austrian physicist, made a major contribution to the foundation and development of statistical

mechanics, the study of temperature, pressure, and related phenomena by a mathematical analysis of movements of molecules that

are assumed to obey the laws of mechanics.

The son of an Austrian taxation official, Ludwig Eduard Boltzmann grew up in Wels and Linz. He earned his doctorate in 1866 from

the University of Vienna. His doctoral thesis, written under the direction of Josef Stefan, was on the kinetic theory of gases. Hebegan his academic career the following year with an appointment at the Physikalisches Institut in Vienna, which was followed by a

series of professorial positions in different related disciplines. He left a professorship in theoretical physics at the University of Graz

(1869-1873), in southeast Austria, for a professorship in mathematics at the University of Vienna (1873-1876), and then returned to

the University of Graz (1876-1879) as a professor of experimental physics. After serving as director of the Physikalisches Institut

(1879-1889), Boltzmann took a position as professor of theoretical physics at the University of Munich. In 1894, he returned to

Vienna as professor of theoretical physics at the university, succeeding his mentor, Josef Stefan. The following year, Ernst Mach

(whose name today is used to measure the speed of sound, as the Mach I) was appointed to the chair of history and philosophy of

science at Vienna. The two men not only were scientific opponents, with Boltzmann championing the point of view of the atomists

and Mach of the energeticists, but also were on bad personal terms. Boltzmann's dislike of working alongside Mach led him to

move to the University of Leipzig in 1900, where he became professor of theoretical physics. Even though the move made him the

colleague of his most vigorous scientific opponent, Wilhelm Ostwald, Boltzmann and Ostwald had a good personal relationship.

Nonetheless, a bout of severe depression led Boltzmann to make a suicide attempt while in Leipzig.

Boltzmann at least partly in jest, attributed his peripatetic nature to the fact that he was born during the dying hours of a Mardi Gras

ball. He recognized early on that he was subject to what today is called bipolar disorder, characterized by periods of mania

alternating with periods of depression, usually separated by lengthy intervals of normal mood. How much of his depression was

attributable to chemical imbalances and how much to the lack of acceptance of his ideas can never be known for sure.

After Mach's retirement from the University of Vienna in 1901, Boltzmann returned there in 1902 to his chair of theoretical physics,

which had remained unfilled in the interim. The university also honored him with a second chair, which Mach had just vacated.

Responsible now for teaching Mach's course in the philosophy of science, Boltzmann became famous as a lecturer in this field. The

audience for his lectures soon grew too large to be accommodated in the largest available lecture hall. Boltzmann's reputation even

reached the imperial court, and Austrian emperor Franz Josef invited him to the royal palace.

Boltzmann is best known for establishing statistical mechanics, the branch of science independently invented by the American

mathematical physicist Josiah Willard Gibbs. Their work connected the properties and behavior of molecules, atoms, or elementary

particles with the large-scale properties of the substances of which they are the building blocks. Boltzmann used probability to

describe how atomic properties determine the properties of matter.

Over the course of his career, Boltzmann made significant contributions to mechanics, dynamics, and electromagnetism, all of which

were being developed during his student years. He was one of the first European scientists to recognize the importance of Scottish

physicist James Clerk Maxwell's research in the field of electromagnetism. Theory of Electricity and Magnetism, a book based on

Boltzmann's lectures on Maxwell's theories, which was published in 1897 with a preface by Boltzmann, helped disseminate

Maxwell's work.

Boltzmann's first published paper (1865) was stimulated by a lecture given by Josef Stefan on electrical theory. Three years later, he

published a paper on thermal equilibrium in gases, which both cited and extended Maxwell's work on this topic. In 1871, he obtained

an equation for the distribution of atoms due to collision. According to this equation, which became known as the Maxwell-Boltzmann distribution, the average amount of energy required for atomic motion in all directions is equal. In 1884, Boltzmann

expanded on the work of his mentor by demonstrating how the empirical law for black-body radiation that Stefan formulated in 1879

could be derived from the principles of thermodynamics. According to the Stefan-Boltzmann law, the total energy radiated from a

black body is proportional to the fourth power of its absolute temperature. The law is illustrated by the fact that a blue star that is the

same size as a red star is very much brighter. In the 1890's, Boltzmann derived the second law of thermodynamics from the

principles of mechanics. He asserted that entropy increases almost always, rather than always.


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electromagnetism, on which he wrote a two-volume treatise. Boltzmann also worked on a derivation for black-body

radiation based on the Stefans law, which was later termed by Hendrik Antoon Lorentz as a true pearl of theoretical

physics. His work in statistical mechanics was vocally criticized by Wilhelm Ostwald and the energeticists who

disregarded atoms and based physical science exclusively on energy conditions. They were unable to understand the

statistical nature of Boltzmanns logic.

His ideas were supported by the later discoveries in atomic physics in the early 1900, for instance Brownian motion,

which can only be explained by statistical mechanics.

Later Life and Death:

Ludwig Boltzmann was greatly demoralized due to the harsh criticism of his work. He committed suicide on

September 5, 1906 at Duino, Italy by hanging himself. He was 62 years old.


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JOSEPH STEFAN

Both Josef Stefan's parents, although living near Klagenfurt in Austria-Hungary (now Austria), were of

Slovenian origin and spoke Slovenian. His father, Ales Stefan (1805-1872), worked as a miller of flour and

as a baker. Josef's mother, Marija Startinik (1815-1863), was employed as a maidservant. They were

both illiterate and were not married. Josef showed his brilliance when at elementary school in

Klagenfurt and he showed himself to have both the desire and ability to do well at the Gymnasium

which was recommended by his teachers. However, as an illegitimate child he would not be allowed to

attend a Gymnasium so, when he was eleven years old, his parents married to give Josef to opportunity

of a good education. Stefan entered the Gymnasium in Klagenfurt in 1846.

On 13 March 1848, eleven days before Stefan's thirteenth birthday, a Revolution began in Austria. It was

prompted by the Paris Revolution in February of the same year. People sought basic freedoms but the

country was divided and revolutionary and counter-revolutionary groups fought for power. Stefan was

at an impressionable age and the Revolution made him much more aware of the various ethnic

groupings and his own Slovenian origins. He reacted by writing Slovenian poetry which he published. His

poetry touched on scientific topics as well as sometime being fiercely patriotic while at other times it

was romantic. In 1853 he completed his studies at the Gymnasium as the top student in his class and,although having a range of interests which he could have chosen to study at university, nevertheless was

quite certain that mathematics and physics were for him. He did consider joining the Benedictine Order

for a while but soon gave up the idea.

Stefan entered the University of Vienna in 1853. He graduated four years later with a degree in

mathematics and physics. He continued to write Slovenian poetry and prose throughout his student

years but after criticisms by the Slovenian literary experts, he gave this up around the time he graduated

from the University of Vienna. For the next year he taught physics for pharmacy students, then accepted

a position with Karl Ludwig at the Physiology Institute of Vienna University. Here he carried out

experimental work on the flow of water through tubes. During this period he was preparing to habilitate

which he did in 1858. Appointed a lecturer in mathematical physics at the University of Vienna in 1858,he was elected to the Austrian Academy of Sciences in 1860, then he became a professor at the

University of Vienna in 1863. In 1866 he became director of the Physical Institute at Vienna. This

Institute had been founded byDopplerin 1850.

His career at the University of Vienna included a spell as dean of the Philosophy Faculty in 1869-70, and

rector in 1876-77. We noted his election to the Austrian Academy of Sciences in 1860. He became a full

member in 1865, was secretary of the Mathematical Sciences Class of the Academy from 1875, and was

vice-president of the Academy from 1885 until his death.

The programme of research that Stefan embarked on was wide ranging across a number of different

areas. He was a great admirer ofMaxwell's contributions and was a major player in making his work

known on the Continent. It was inMaxwell's papers that he came across the following:-

It would be almost impossible to establish the value of the conductivity of a gas by direct experiment, asthe heat radiated from the sides of the vessel would be far greater than the heat conducted through theair, even if current could be entirely prevented.
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He used the tools of advanced mathematics and understood how to present the most difficultdevelopments in the clearest and most lucid form without ever having to resort to mathematical

formalism. ... [He]never tried to flaunt [his]mental superiority. [His]uplifting humour, which turned themost difficult discussion into an entertaining game for the student, made such a deep impression on me.

Boltzmannpainted a picture of a wonderful research environment:-

Nothing diminishes the excellence of [Stefan's]character, the magic [he]worked on the youngacademics. That magic could only be experienced personally. [The experience]stayed with me my wholelife as a symbol of serious, inspired experimental activity.

For most of his life Stefan was unmarried, too dedicated to his profession to have space for wife or

family. However, in 1891 when he was 56 years old, he married Marija Neumann who was a widow. He

lived only a little over a year after his marriage, suffering a stroke. He was buried in the Zentralfriedhof

in Vienna.
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Blackbody Concepts

Incident energy striking an object from the surroundings, can be absorbed by the object,reflected by the object, or transmitted through the object (if it is not opaque) as seen in

Figure 2-1. If the object is at a constant temperature, then the rate at which it emits energymust equal the rate at which it absorbs energy, otherwise the object would cool (emittancegreater than absorption), or warm (emittance less than absorption). Therefore, for bodies atconstant temperature, the emittance (absorption), the reflection and the transmittance of

energy equals unity. Central to radiation thermometry is the concept of the blackbody. In

1860, Kirchhoff defined a blackbody as a surface that neither reflects or transmits, butabsorbs all incident radiation, independent of direction and wavelength. The fraction of

radiation absorbed by a real body is called absorptivity, For an ideal blackbody, the

absorptivity is 1.0 . For non-blackbodies, the absorption is a fraction of the radiation

heat transfer incident on a surface, or Hence, in term of radiation heat transfer, q":

In addition to absorbing all incident radiation, a blackbody is a perfect radiating body. To

describe the emitting capabilities of a surface in comparison to a blackbody, Kirchoff defined

emissivity of a real surface as the ratio of the thermal radiation emitted by a surface ata given temperature to that of a blackbody at the same temperature and for the same

spectral and directional conditions.This value also must be considered by a non-contact temperature sensor when taking a

temperature measurement. The total emissivity for a real surface is the ratio of the total

amount of radiation emitted by a surface in comparison to a blackbody at the same

temperature. The tables beginning on p. 72 give representative emissivity values for a widerange of materials. If precise temperature measurements are required, however, the

surface's actual emittivity value should be obtained. (Although often used interchangeably,the terms emissivity and emittivity have technically different meanings. Emissivity refers to

a property of a material, such as cast iron, whereas emittivity refers to a property of aspecific surface.)

In 1879, Stefan concluded based on experimental results that the radiation emitted from

the surface of an object was proportional to the fourth power of the absolute temperature ofthe surface. The underlying theory was later developed by Boltzmann, who showed that theradiation given off by a blackbody at absolute temperatureTs(K) is equal to:

where ( is the Stefan-Boltzmann constant The heat transfer rate byradiation for a non-blackbody, per unit area is defined as:

where Tsis the surface temperature and Tsuris the temperature of the surroundings.


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Figure2-2:Spectral Distributions

Although some surfaces come close to blackbody performance, all real objects and

surfaces have emissivities less than 1. Non-blackbody objects are either graybodies, whose

emissivity does vary with wavelength, or non-graybodies, whose emissivities vary withwavelength. Most organic objects are graybodies, with an emissivity between 0.90 and 0.95(Figure 2-2). The blackbody concept is important because it shows that radiant powerdepends on temperature. When using non-contact temperature sensors to measure the

energy emitted from an object, depending on the nature of the surface, the emissivity must

be taken into account and corrected. For example, an object with an emissivity of 0.6 isonly radiating 60% of the energy of a blackbody. If it is not corrected for, the temperaturewill be lower than the actual temperature. For objects with an emissivity less than 0.9, the

heat transfer rate of a real surface is identified as:

The closest approximation to a blackbody is a cavity with an interior surface at a uniform

temperature Ts, which communicates with the surroundings by a small hole having a

diameter small in comparison to the dimensions of the cavity (Figure 2-3). Most of theradiation entering the opening is either absorbed or reflected within the cavity (to ultimatelybe absorbed), while negligible radiation exits the aperture. The body approximates a perfect

absorber, independent of the cavity's surface properties.


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Figure 2-3: An Isothermal Blacbody Cavity

The radiation trapped within the interior of the cavity is absorbed and reflected so that the

radiation within the cavity is equally distributed--some radiation is absorbed and some

reflected. The incident radiant energy falling per unit time on any surface per unit area

within the cavity is defined as the irradiance If the total irradiation G (W/m2)represents the rate at which radiation is incident per unit area from all directions and at allwavelengths, it follows that:

If another blackbody is brought into the cavity with an identical temperature as the interior

walls of the cavity, the blackbody will maintain its current temperature. Therefore, the rate

at which the energy absorbed by the inner surface of the cavity will equal the rate at whichit is emitted. In many industrial applications, transmission of radiation, such as through a

layer of water or a glass plate, must be considered. For a spectral component of theirradiation, portions may be reflected, absorbed, and transmitted. It follows that:

In many engineering applications, however, the medium is opaque to the incident radiation.

Therefore, and the remaining absorption and reflection can be treated as surface

phenomenon. In other words, they are controlled by processes occurring within a fraction of

a micrometer from the irradiated surface. It is therefore appropriate to say that theirradiation is absorbed and reflected by the surface, with the relative magnitudes

of depending on the wavelength and the nature of the surface.

Radiation transfer by a non-blackbody encompasses a wide range of wavelengths and

directions. The spectral hemispherical emissive power, is defined as the rate at

which radiation is emitted per unit area at all possible wavelengths and in all possible

directions from a surface, per unit wavelength and per unit surface area.

Although the directional distribution of surface emission varies depends on the surfaceitself, many surfaces approximate diffuse emitters. That is, the intensity of emitted radiationis independent of the direction in which the energy is incident or emitted. In this case, the

total, hemispherical (spectral) emissive power, is defined as:


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or

where Ieis the total intensity of the emitted radiation, or the rate at which radiant energy is

emitted at a specific wavelength, per unit area of the emitting surface normal to the

direction, per unit solid angle about this direction, and per unit wavelength. Notice that is

a flux based on the actual surface area, where is based on the projected area. In

approximating a blackbody, the radiation is almost entirely absorbed by the cavity. Anyradiation that exits the cavity is due to the surface temperature only.

The spectral characteristics of blackbody radiation as a function of temperature and

wavelength were determined by Wilhelm Wien in 1896. Wien derived his law for thedistribution of energy in the emission spectrum as:

where (b for blackbody) represents the intensity of radiation emitted by a blackbody at

temperature T, and wavelength per unit wavelength interval, per unit time, per unit solidangle, per unit area. Also, h = 6.626 x 10-24Js and k = 1.3807 x 10-23JK-1are the

universal Planck and Boltzman constants, respectively; co= 2.9979 x 108m/s is the speed of

light in a vacuum, and T is the absolute temperature of the blackbody in Kelvins (K). Dueto the fact that deviations appeared between experimental results and the equation, Plancksuggested in 1900 a refinement that better fit reality:

It is from this equation that Planck postulated his quantum theory. A more convenientexpression for this equation, referred to as the Planck distribution law (Figure 2-4), is:

where the first and second radiation constants are

and Planck's distribution shows that as wavelength varies,emitted radiation varies continuously. As temperature increases, the total amount of energyemitted increases and the peak of the curve shifts to the left, or toward the shorter

wavelengths. In considering the electromagnetic spectrum, it is apparent that bodies withvery high temperatures emit energy in the visible spectrum as wavelength decreases.

Figure 2-4 also shows that there is more energy difference per degree at shorterwavelengths.


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Figure 2-4: Planck Prediction of Blackbody Emissive Power

From Figure 2-4, the blackbody spectral distribution has a maximum wavelength value,

lmax, which depends on the temperature. By differentiating equation 2.12 with respect to

and setting the result equal to zero:

where the third radiation constant, C3 = 2897.7 K.This is known as Wien'sdisplacement law. The dashed line in Figure 2-4 defines this equation and locates the

maximum radiation values for each temperature, at a specific wavelength. Notice thatmaximum radiance is associated with higher temperatures and lower wavelengths.


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THE CONCEPTS MAP OF BLACK BODY


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Abstract

There are several mechanisms whereby thermal energyheat- can be transferredbetween

two bodies of different temperature. The flow of heat is of great interest and use in

engineering applications and thermodynamical processes. This is the mechanism

responsible for the absorption of thermal energy from the sun. One of thesemechanisms is

Thermal Radiation. This was the most complex to understand of the three. In light ofmodern

Physics and electromagnetic radiation theory, this flow of thermal energy is nowunderstood

to be electromagnetic radiation from the infra-red and visible light sections of thespectrum.

The Stefan-Boltzmann law of radiation was the first successful model of theexperimental

data obtained by heating a radiating blackbody and analysing the colour and thermal

energy

flux at higher temperatures. This law describes a dependency upon energy per unitarea of

output radiation on the 4th power of the objects temperature. Qualitatively warmer

objects

emit more radiation with smaller wavelengths. This empirical relationship of P~T4was

investigated in this experiment by using an incandescent Tungsten filament lightbulband

heating it by applying a higher current to it in successive steps, whilst analysing thevoltage


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(and hence Temperature of) a thermocouple stack referred to as a thermopile placedlevel to

the radiating bulb. It was found that the colour of the filament became whiter as its

temperature increased, and that in fact the output energy per unit area is related to the

filaments temperature to the power 4 by means of a plot of thermopile potential ( ameasure

of its temperature) versus filament temperature converted via its resistance. Indeed

the

Stefan Boltzmann Law of radiation is a good description of the radiative flow of heatfrom a

hot object to its surroundings.

PHY 263 Practical Experiment :

Stefan-Boltzmann Law of Radiation

ALISSA KRIEL

11123002

Practical performed on 18 October 2013

(Group 2 , Friday session)

Immediate Laboratory partner : S. Congolo

Other group members : M.L. Jacobs, K. Van Der Laarse

The analysis of the power per unit area emitted by a

Tungsten filament as a function of filament temperature

to verify the 4th power relation of the Stefan-Boltzmann

Law of radiation

I ntroduction


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The movement and flow of heat is a cardinally important thermodynamical process

that is at

the same time a life-source for Earth and a property that can be used in many modern

applications of technology and industry. The heat from the sun provides the surface ofthe

earth with the necessary light and heat for nature to thrive. The important question ishow

this heat travels through space to our atmosphereor generally how heat istransferred

between two bodies of differing temperature. ([4], p100)

There are three mechanisms whereby thermal energy is transferred between objects :

Conduction (by direct contact) , Convection (by means of a fluid medium around theobjects)

and Thermal Radiation, the mechanism of interest in this practical investigation. ([1],p I-4)

The radiation of thermal energy has been a field of great interest and a source of great

confusion for many scientists over the years. It was found that an object that absorbed

thermal energy also emitted thermal energy. An ideal radiator of heat (and hence atotal

absorber of heat) was labelled aBlackbody Radiator. Experimentally acquired data of

intensity of radiation from a heated body yielded spectral distribution graphs thatcould not

be accurately derived from theory in the 19th century.

An example of a failed theory for the shape of the spectral distribution graph is theRayleigh-

Jeans formula that relies on classical physics theory. The formula somewhatapproaches the


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experimental data for long wavelengths of emitted radiation, but deviates badly forradiation

with shorter wavelengths. The experimental data basically showed that the maximum

irradiated lights wavelength tended to decrease as the temperature of the object was

increased, which Wiens displacement law describes. The second trend noted in the

experimental results is that the total power of radiation increased with increasing

temperature. This observation was accurately quantified by considering the powerradiated

per unit area as a function of object TemperatureThis is the Stefan-Boltzmann Law

of

radiation : ([2], p 96-100)

_ = ____......................(1)

Planck further investigated functions that could describe the spectral distribution ofradiation

due to temperature, assuming Maxwells electromagnetic wave theory. He came to the

conclusion that the electromagnetic thermal radiation particles can only have discrete

energies. Therefore, the heat energy must be radiated by EM waves, which we mostly

encounter as Infrared radiation.

In order to physically measure the thermal radiation given off by a Tungsten filament,it is

necessary to manipulate the equations of theory, whereafter the 4th power relation

between

Irradiance per unit area and Temperature can be determined. This proportionality isvalid for

the Non-Blackbody radiator filament since it has an absorption coefficient of less thanone.


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Plancks Law of radiation:

__, __ = ______

__ __

______

.....................(2)

By integrating the above equation over all wavelengths, we obtain the Stefan-Boltzmann

Law:

____ = ___

__ . __

__ . __............................(3),

where =

___

__ . __

__ = 5.67 x 10-8 W.m-2.K-4

The thermal energy flux which reaches a detector a fixed distance from the heatedsource is

proportional to P(T). As this thermal energy reaches a detector, which in our case

is a

series of thermocouples whose potential difference reading indicates temperature

of

the detector absorbing the thermal radiation, the potential difference, Utherm is

proportional to the 4th power of the temperature of the heat source.


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For the thermocouple pile detector at room temperature, a correction must beincorporated.

Finally, taking logarithms on both sides of the proportionality:

!"__#$ = %&'()*'). __............................(4)

Thus, a straight line trend with a gradient of 4 will arise in a Logarithmic-Logarithmicplot if

the thermal radiation obeys the Law in (3):

+&,!"__#$ = +&,%&'()*') + 4+&,_............................(5)

The Tungsten filament used in the incandescent lamp in this experiments temperature

could

be determined by measuring its current at an applied potential difference after whichits

Resistance could be calculated using Ohms Law. The resistance of the filament at

0oC is

given by: ([3], p 143)

/0 = 1_"2_

_34."235."2_

...................(6),

where tR denotes the room temperature in oC, and R(tR) is the filament resistance atroom

temperature.

Finally, the temperature of the filament related to its measured resistance at any

Temperature T is :

_ = 273 + _

_5 . 9:;_ + 4< =1_>_


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1?

1B D....................................(7)

Method:

In order to finally utilize equation (5), which is the double-logarithmic plot, we willfirstly need

to measure the thermocouple stacks potential difference (Utherm) at its exposure to

the

heated filament at various intensitiesor equivalently powers of radiation. Thebrighter the

filament glows, by Wiens displacement law the hotter it is and thus the greater willbe the

power with which it emits thermal radiation which the thermocouple stack will absorbto

increase its own temperature and thus Utherm values.

These voltages are very small, and in order to record data that may be quantitatively

manipulated, an amplifier was connected between the thermopile and the voltmeter.Output

voltage readings of the order of 0.1 mV could be expected.

Before the bulb was placed nearby the sensor, The reading of the Utherm voltmeterwas

zeroed by means of adjusting the amplifier.

The thermopile sensor was placed approximately 29.5 cm away from the lamp, andlevel to

the light bulb. A DC power supply to the lamp was connected, along with an ammeterto

monitor applied current and a voltmeter to record Potential difference across thefilament in


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order to compute Resistance via Ohms Law.

To compute the filaments resistance at 0oC via Equation (6), Room temperature was

recorded and the resistance of the filament at room temperature was determined by

connecting a 100 resistor in series between the power supply and the filament, and

measuring the voltage across it at several supplied currents between 100mA and200mA to

prevent heating of the filament which would of course change its resistance.

When this was complete, the resistor was removed, and the light bulb was suppliedwith

electricity at Voltages between 0.5 V and 7.0 V, several measurements of the current

through the bulb and the thermocouple stack Potential difference being made.

The corresponding Filament Temperature at each recorded step could now becomputed

using equation (7).

Results:

Below follows a table containing several measured voltages and current of the Wfilament at

room temperature, as well as calculated resistances and an average resistance at room

temperature.

Another table containing the measured filament current, filament voltage andThermopile

Potentials follows that, along with computed Reistance of the filament andcorresponding

Temperatures of the Filament.


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Using the applicable data, a Log-Log plot of Thermopile potential as a function ofFilament

temperature is shown.

Table 1 : Measured values of current and Voltage over the Tungsten filament atroom

Temperature 26oC

Applied Current I (mA)

1 mA

Voltage over Filament (V)

0.001 V

Computed Resistance*

() 0.01

100 0.023 0.23

125 0.029 0.23

150 0.034 0.23

175 0.040 0.23

200 0.046 0.23

The Resistance was computed via / = E

F

Average Filament Resistance at 26oC :

R (tR) = 0.23 0.01

Thus, the filaments resistance at 0oC by equation (6):

/0 =


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0.23H

1 + ;. 26 +


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Utherm (mV)

0.1 mv

1.257 1.857 0.68 700.5 0.1

2.504 2.702 0.93 960.6 0.2

3.109 3.027 1.03 1047.0 0.3

3.538 3.289 1.08 1088.2 0.4

4.039 3.557 1.14 1138.4 0.5

4.427 3.738 1.18 1179.0 0.6

4.849 3.920 1.24 1222.3 0.8

5.460 4.192 1.30 1275.7 0.9

5.95 4.430 1.34 1308.4 1.1

6.53 4.631 1.41 1361.9 1.3

7.08 4.843 1.46 1403.0 1.5

Resistance was calculated by Ohms Law.

The Temperature of the Filament was calculated using equation (7). For the first line

in Table 2:

_ = 273 +

1

2

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