24.1. Background¶
A body at a certain temperature continuously absorbs and emits electromagneticradiation. If it absorbs more energy that it emits its temperaturewill increase and ifit emits more energy than it absorbs it’s temperature will decrease. Whenit is in a thermal equilibrium with its surroundings then it absorbsthe same amount of power as it emits. An ideal body that absorbs allelectromagnetic radiation that impinges on its surface is called ablack body. The spectrum of the radiation emitted by a black body is entirelydetermined by its temperature and independent of its composition.
In 1900, Max Planck obtained his famous blackbody formula thatdescribes the energy density per unit wavelength interval of theelectromagnetic radiation emitted by a blackbody at a temperature\(T\) :
(24.1)¶\[u(\lambda,T) = \frac{8\pi h c}{\lambda^5(e^{hc/\lambda kT} - 1)}\]
where \(\lambda\) is the wavelength, \(T\) is the temperatureof the body, \(k\) is the Boltzmann constant, \(h\) isPlanck’s constant and \(c\) is the speed of light.
A good realization of a black body is a cavity with a smallhole through which a small amount of radiation can exit (forobservation) without disturbing the thermal equilibrium inside. Ifone assumes for simplicity that the cavity is made out of an electric conductor,one can show that only certain standing transverse electromagnetic waves areallowed. Each electromagnetic wave behaves statistically like aharmonic oscillator of a certain frequency. While a classical oscillator can contain any amountof energy, in order to obtain his result, Planck had to assumethat a harmonic oscillator is quantized i.e. can onlycontain a total amount of energy given by \(E=n\hbar \omega_0\)where \(n = 1,2, ...\). In addition he assumed for a given type ofoscillator (characterized by \(\omega_0\)) that the ratio ofthe number of oscillators in a state \(j\) to the number ofoscillators in another state \(i\) is given by\(\frac{N_i}{N_j} = e^{-\Delta E/kT}\) where \(\Delta E =E_i - E_j\). Using this assumption the average energy per oscillatoris:
(24.2)¶\[ \begin{align}\begin{aligned}<E> = \frac{E_{tot}}{N_{tot}} = \frac{N_0\hbar\omega_0\left( 0 +x + 2x^2 + 3x^3 + ...\right)}{N_0\left(1+x+x^2+x^3 + ...\right)}\\x = e^{-\hbar\omega_0/kT}\end{aligned}\end{align} \]
evaluating the infinite sums gives:
(24.3)¶\[<E> = \frac{\hbar\omega_0}{e^{\hbar\omega_0/kT} - 1}\]
He further assumed that when an oscillator changes from an initial state\(i\) to a final state \(f\) he emits (or absorbs) radiation with afrequency such that \(E_f - E_i = 2\pi \hbar \omega\). Sincethe electromagnetic radiation is confined in a cavity with a volume\(V\) the number of oscillators with a frequency between\(\nu\) and \(\nu + d\nu\) is given by:
(24.4)¶\[dZ = \frac{8\pi V \nu^2}{c^3} d\nu\]
leading to an energy density per unit frequency inside the cavity of
(24.5)¶\[u_{\nu} = \frac{8\pi h \nu^3}{c^3\left( e^{h\nu/kT} -1 \right)}\]
corresponding to (24.1) when \(\nu\) is replaced by\(\lambda\). The surface intensity (per unit solid angle) is obtained as follows:
(24.6)¶\[ \begin{align}\begin{aligned}I(\nu, T) = \frac{c}{4\pi} u_{\nu} = \frac{2 h \nu^3}{c^2\left(e^{h\nu/kT} -1 \right)}\\I(\lambda, T) = \frac{2 h c^2}{\lambda^5\left(e^{hc/\lambda kT} -1 \right)}\end{aligned}\end{align} \]
The total power per unit area and unit solid angle, a \(P\), radiated by a blackbodyis found by integrating \(I(\lambda, T)\) overthe entire wavelength range from zero to infinity. When this is done,we obtain the Stefan-Boltzmann law:
(24.7)¶\[ P = \sigma T^4\]
where the \(\sigma\) is the Stefan-Boltzmann constant. For an incandescentsolid, the ratio of the energy radiated to that from a true blackbody atthe same temperature is called the emissivity, \(e\), a numberwhich is always lessthan one.
The goal of this experiment is to investigate the relationship between\(E\) and \(T\) for the tungsten filament in an ordinary lampto see how close it behaves like a black body radiator. Theemissivity of tungsten is not quite constant but decreases withincrease in wavelength and increases in temperature. The temperatureof the filament will be determined from its change in resistance withtemperature.
24.2. Experimental Equipment¶
The setup for this experiment is shown in Fig. 24.1. Itconsists of a 12V light bulb with a tungsten filament and a radiationdetector which produces an output voltage proportional to the detectedradiation power (22 mV per mW).
24.2.1. Radiation Detector¶
To measure the emitted radiation power a thermopile sensor is usedwhich consists of a series of thermocouples connected alternatively toan active side and a reference side (see Fig. 24.2). The active side typicallyconsists of a very thin layer of material. The output voltage of thisseries is proportional to the temperature difference between theactive layer and the reference layer. The total radiation is absorbedin the active layer where a temperature increase proportional to theabsorbed radiation power is detected and converted to a voltage.
The wavelength range for this detector is indicated on its side.
24.2.2. Wheatstone Bridge¶
For this experiment we will also use a Wheatstone bridge to accuratelydetermine the resistance, \(R_{ref}\), of the lamp filament at roomtemperature (~ 300K) and the resistance of the wires connecting thelamp to the power supply.
The schematic of this instrument is shown in Fig. 24.3
The actual instrument with the various controls and connections isshown in Fig. 24.4
24.3. Experimental Procedure¶
Use a 3V battery pack as the power supply and first determine theresistance of the lamp connecting leads. A dummy lead of the sametotal length is provided for this purpose. Then measure theresistance of the lamp plus connecting leads and, by subtraction,obtain the filament resistance alone. These measurements need to bedone very accurately as a small error will lead to large errors infilament temperatures.
Connect the radiation sensor (thermopile) to a digital multimeter thatshould be set on a 100 or 200-millivolt DC range. Connect the lamp toa DC power supply together with an ammeter and voltmeter to recordfilament current and voltage respectively.
HAVE YOUR CIRCUIT CHECKED BY AN INSTRUCTOR BEFORE SWITCHING ON. NOTEALSO THAT THE LAMP VOLTAGE SHOULD NEVER EXCEED 12 V.
Measure the distance between the lamp filament and the detectoropening. Also record the detector diameter.
For filament voltages of between 1V and 12V in steps of about 1V, record thefilament voltage, current and the sensor millivoltmeter reading. Make thelatter readings quickly and in between readings, place a reflectingheat shield between the lamp and sensor with the reflecting surfacefacing the lamp. This will help to keep the sensor at a relativelyconstant temperature. Note that the millivoltmeter reading,designated “Rad” in Table 1, is proportional to the total powerradiated by the lamp.
Measure the temperature in the room (you can read the thermometer fromthe Franck-Hertz experiment for this) and record it as well.
24.4. Analysis¶
First you need to determine the filament temperature. To do this weuse the known tungsten resistivity as a function of temperature. Youcan downloadW_resistivity.data to get a data filewith the resistivity information. From this file you can get theresistivity (R) as a function of temperature (T).
First we need to know the resistivity of the tungsten at the currentroom temperature. To do this you will firstread the data an fit a 7-th order polynomial to get Ras a function of T:
In [1]: rd = B.get_file('W_resistivity.data')In [2]: R = B.get_data(rd, 'R')In [3]: B.get_data(rd,'T')In [4]: Rf = B.polyfit(T,R, order = 7)
You should get an output like:
chisq/dof = 0.000444541390639parameter [ 0 ] = 0.786467375234 +/- 0.20463740899parameter [ 1 ] = 0.00842119057779 +/- 0.00124878972052parameter [ 2 ] = 3.38132589377e-05 +/- 2.84216959531e-06parameter [ 3 ] = -2.98493892381e-08 +/- 3.20401959016e-09parameter [ 4 ] = 1.58106971254e-11 +/- 1.97223496095e-12parameter [ 5 ] = -4.78596183333e-15 +/- 6.73770637135e-16parameter [ 6 ] = 7.70654530692e-19 +/- 1.19806308507e-19parameter [ 7 ] = -5.11638418846e-23 +/- 8.64022222901e-24
Now you can calculate the resistivity of tungsten at your roomtemperature:
In [5]: R_ref = Rf.poly(295.) # it was assumed that the room is at 295K
We will not be able to directly measure the resistivity of thetungsten filament but its resistance (check your physics book on thedifference between resistivity and resistance). The ratio of thefilament resistance at different temperatures is dominated by thecorresponding ratio of the tungsten resistivity. We therefore need tocalculate the ratio of the resistivity at a give temperature to theone at our current room temperature \(R_{rel}(T) = R(T)/R_{ref}\). Then we fit \(T\) as a function of \(R_{rel}\) again using a7-th order polynomial:
In [6]: R_rel = R/R_refIn [7]: Temp = B.polyfit(R_rel, T, order = 7)chisq/dof = 0.748733057894parameter [ 0 ] = 35.0101136416 +/- 3.42110385654parameter [ 1 ] = 288.88629356 +/- 4.42701538565parameter [ 2 ] = -32.1923642159 +/- 2.00772236211parameter [ 3 ] = 5.39380439371 +/- 0.433502793096parameter [ 4 ] = -0.535908552352 +/- 0.0498179195972parameter [ 5 ] = 0.0300782229751 +/- 0.00312310690913parameter [ 6 ] = -0.00088651572878 +/- 0.000100685923028parameter [ 7 ] = 1.06612957893e-05 +/- 1.30509370423e-06
Now you have a function to get the temperature as a function of theresistivity ratio.
From your measured data read the current and voltages of the lampand determine its resistance.
Correct the resistance for thelead wire resistances and calculate the resistivity ratios for yourmeasured data.
Use the fitted function todetermine the filament temperature.
Fit a line to the logarithm of the measured power (include itsestimated error) as a function of the logarithm of thetemperature. What power to you determine from this ?
Select only those points where the temperature is larger than about1500-2000K. And repeat the previous step
Make a linear fit of the measured power as a function of\(T^4\). Can you estimate the value of \(\sigma\) in theStefan-Boltzmann law ((24.7), remember it is the power perunit area of emitter)
