path: root/blackbody_new.tex

\documentclass[./manual.tex]{subfiles}
\begin{document}

\chapter*{Blackbody Radiation}

%\date {}
%\maketitle \noindent
 \textbf{Experiment objectives}: explore radiation from objects at certain temperatures,
    commonly known as ``blackbody radiation''; make measurements testing the Stefan-Boltzmann
    law; measure the inverse-square law for thermal radiation.

\section*{Theory}

    A familiar observation to us is that dark-colored objects
    absorb more thermal radiation (from the sun, for example) than
    light-colored objects. You may have also observed that a good
    absorber of radiation is also a good emitter (like
    dark-colored seats in an automobile). Although we observe
    thermal radiation (``heat'') mostly through our sense of touch,
    the range of energies at which the radiation is emitted can
    span the visible spectrum (thus we speak of high-temperature
    objects being ``red hot'' or ``white hot''). For temperatures
    below about $600^{\circ}C$, however, the radiation is emitted in the
    infrared, and we cannot see it with our eyes, although there
    are special detectors (like the one you will use in this lab)
    that can measure it.

    An object which absorbs all radiation incident on it is known
as an ``ideal blackbody''.  In 1879 Josef Stefan found an empirical
relationship between the power per unit area radiated by a blackbody and the
temperature, which Ludwig Boltzmann derived theoretically a few years later.
This relationship is the {\bf Stefan-Boltzmann law:}
\begin{equation}\label{SBl}
 S =\sigma T^4
\end{equation}
where $S$ is the radiated power per unit area ($W/m^2$), $T$ is the temperature (in Kelvins), and $\sigma=5.6703
\times 10^{-8} W/m^2K^4$ is the Stefan's constant.

Most hot, opaque objects can be approximated as blackbody emitters, but the most ideal blackbody is a closed volume (a cavity) with a very small hole in it. Any radiation entering the cavity is absorbed by the walls, and then is re-emitted out. Physicists first tried to calculate the spectral distribution of the radiation emitted from the ideal blackbody using {\it classical thermodynamics}. This method involved finding the number of modes of oscillation of the electromagnetic field in the cavity, with the energy per mode of oscillation given by $kT$. The classical theory gives the {\bf Rayleigh-Jeans law:}
\begin{equation}
 u(\lambda,T) = \frac{8\pi kT}{\lambda^4}
\end{equation}
where $u(\lambda)(J/m^4)$ is the spectral radiance -- energy radiated per unit area at a single wavelength $\lambda$. This law agrees with the experiment for radiation at long wavelengths (infrared), but predicts that $u(\lambda)$ should increase infinitely at short wavelengths. This is not observed experimentally (Thank heaven, or we would all be constantly bathed in ultraviolet light - a true ultraviolet catastrophe!). In reality, the peak of radiation distribution as a function of its wavelength depends on the blackbody temperature as described by {\bf Wien's law:}
\begin{equation} \label{weins}
 \lambda_{max}T = 2.898\times 10^{-3} m\cdot K
\end{equation}
and the spectral radiance approaches zero for short  wavelengths.

    The breakthrough came when Planck assumed that the energy of the oscillation modes can only take on discrete values rather than a continuous distribution of values, as in classical physics. With this assumption, Planck's law was derived:
\begin{equation}
u(\lambda,T)=\frac{8\pi hc\lambda^{-5}}{e^{hc/\lambda kT}-1}
\end{equation}
where $c$ is the speed of light and $h=6.626076\times 10^{-34} J\cdot s$ is the Planck's constant. This proved to be the correct description.

%\begin{boxedminipage}{\linewidth}
%\textbf{Sometimes physicists have to have crazy ideas!} \\
%%
%``\emph{The problem of radiation-thermodynamics was solved by Max Planck, who
%was a 100 percent classical physicist (for which he cannot be blamed). It was
%he who originated what is now known as {\it modern physics}. At the turn of the
%century, at the December 14, 1900 meeting of the German Physical Society,
%Planck presented his ideas on the subject, which were so unusual and so
%grotesque that he himself could hardly believe them, even though they caused
%intense excitement in the audience and the entire world of physics}.''
%
%From George Gamow, {\it ``Thirty Years that Shook Physics, The Story of Quantum Physics''}, Dover Publications,
%New York, 1966.
%\end{boxedminipage}


\section*{Radiation sensor operation principle}

Imagine a metal wire connected to a cold reservoir at one end and a hot reservoir at the other.  Heat will flow between the ends of the wire, carried by the electrons in the conductor, which will tend to diffuse from the hot end to the cold end.  Vibrations in the conductor's atomic lattice can also aid this process. This diffusion causes a potential difference between the two ends of the wire. The size of the potential difference depends on the temperature gradient and on details of the conductive material, but is typically in the few 10s of $\mu V/ K$. A thermocouple, shown on the left, consists of two different conductive materials joined together at one end and connected to a voltmeter at the other end. The potential is, of course, the same at the joint, but the difference in material properties causes $\Delta V=V_1 - V_2 \neq 0$ between the separated ends. This $\Delta V$ is measured by the voltmeter and is proportional to $\Delta T$.  Your radiation sensor is a thermopile, simply a ``pile'' of thermocouples connected in series, as shown at the right. This is done to make the potential difference generated by the temperature gradient easier to detect.
\begin{figure}
\includegraphics[height=1.5in]{./pdf_figs/thermopile}
\caption{\label{sensor}\emph{Left}: thermocouple construction; \emph{right}: thermopile - an array of thermocouples connected in series.}
\end{figure} \\
\textbf{Important}: When using the thermal radiation sensor, make each reading quickly to keep the sensor from heating up. Use sheets of white isolating foam (with the silvered surface facing the lamp) to block the sensor between measurements.
\\
\textbf{Sensor calibration}: To obtain the radiation sensor readings $V_S$ for radiated power per unit area $S$ in the correct units ($W/m^2$), you need to use the voltage-to-power conversion factor $22~V/W$, and the area of the sensor $2mm\times2mm$:
\begin{displaymath}
S_{det}[W/m^2]=\frac{V_S [V]}{22 [V/W]}\cdot \frac{1}{4\cdot
10^{-6}[m^2]}
\end{displaymath}


\section*{Test of the Stefan-Boltzmann Law}

\textbf{Equipment needed}: Radiation sensor, multimeters, Stefan-Boltzmann
Lamp, Power supply.

Before starting actual experiment take some time to have fun with the thermal radiation sensor. Can you detect your lab partner? What about people across the room? Point the sensor in different directions and see what objects affect the readings. \textbf{These exercises are fun, but you will also gain important intuition about various factors which may affect the accuracy of the measurements!}

\begin{enumerate}
\item  \textbf{Before turning on the lamp}, measure the resistance of the filament of the Stefan-Boltzmann lamp at room temperature. Record the room temperature, visible on the wall thermostat.

%
%\begin{tabular}{lr}
%  T (room temperature)=&$\underline{\hskip .7in}K$\\
% Resistance of filament (room temperature)=&$\underline{\hskip .7in}$
%\end{tabular}

\item To indirectly measure the temperature of the filament, we will use the known dependence of its resistance on the temperature, given in table shown in Table.~\ref{w_res:table}. To ensure the accurate measurement, we will again use the four-point probe method (review the video on the course web site, if you need a refresher) by measuring the voltage drop across the lamp. VERY IMPORTANT:
	make all connections to the lamp when the power is off, and ask the instructor to check your connections before proceeding.

\item Place the thermal sensor at the same height as the filament, with the front face of the sensor approximately 5~cm away from the filament and fix their relative position. Make sure no other objects are viewed by the sensor other than the lamp.
%
\item  Turn on the lamp power supply. Set the voltage, $V$, in steps of 1-2
	volt from 1-6 volts. At each $V$, record the current running through the lamp and the voltage from the radiation sensor.
	Calculate the resistance of the lamp using Ohm's Law and determine
	the temperature $T$ of the lamp from the table shown in Table
	\ref{w_res:table}. Don't forget to use Kelvin scale for the temperatures (conversion equation is $T[K]=T[^oC]+273$).
	
\item Calculate the values of $T^4$ - these are going to be the $x$-values for the graph. Are they more or less equally distributed? If not (which is probably the case), estimate the big gaps, and measure additional points to fill them in.
	

\end{enumerate}

% \begin{figure}[h]
% \includegraphics[width=\columnwidth]{./pdf_figs/w_res}
% \caption{\label{w_res:fig}Table of tungsten's resistance as a function of temperature. Here, $R_{300K}$ is the resistance of tungsten at the temperature of 300~K.}
% \end{figure}

\begin{table}[h]
	\centering
\begin{tabular}{|ccc|ccc|ccc|ccc|}
	\hline
	$\mathrm{R/R_{300K}}$ & \multicolumn{1}{p{0.3in}}{\centering Temp\\ K} & \multicolumn{1}{p{0.7in}|}{\centering Resistivity\\ $\mu\Omega \mathrm{cm}$}
& $\mathrm{R/R_{300K}}$ & \multicolumn{1}{p{0.3in}}{\centering Temp\\ K} & \multicolumn{1}{p{0.7in}|}{\centering Resistivity\\ $\mu\Omega \mathrm{cm}$}
& $\mathrm{R/R_{300K}}$ & \multicolumn{1}{p{0.3in}}{\centering Temp\\ K} & \multicolumn{1}{p{0.7in}|}{\centering Resistivity\\ $\mu\Omega \mathrm{cm}$} \\
	\hline
	\hline
1.0  & 300  & 5.65  & 5.48  & 1200 & 30.98 & 16.29 & 3000 & 92.04 \\
1.43 & 400  & 8.06  & 6.03  & 1300 & 34.08 & 16.95 & 3100 & 95.76 \\
1.87 & 500  & 10.56 & 6.58  & 1400 & 37.19 & 17.62 & 3200 & 99.54 \\
2.34 & 600  & 13.23 & 7.14  & 1500 & 40.36 & 18.28 & 3300 & 103.3 \\
2.85 & 700  & 16.09 & 7.71  & 1600 & 43.55 & 18.97 & 3400 & 107.2 \\
3.36 & 800  & 19.00 & 8.28  & 1700 & 46.78 & 19.66 & 3500 & 111.1 \\
3.88 & 900  & 21.94 & 8.86  & 1800 & 50.05 & 26.35 & 3600 & 115.0 \\
4.41 & 1000 & 24.93 & 9.44  & 1900 & 53.35 &       &      & \\
4.95 & 1100 & 27.94 & 10.03 & 2000 & 56.67 &       &      & \\
	\hline
\end{tabular}
% this table is lifted from Pasco manual
% numbers somewhat disagree with literature for example
% "Analytical expressions for thermophysical properties of solid and liquid
% tungsten relevant for fusion applications"
% https://doi.org/10.1016/j.nme.2017.08.002
\caption{\label{w_res:table}Table of tungsten's resistance and resisitivity as a function of temperature. Here, $\mathrm{R_{300K}}$ is the resistance of tungsten at the temperature of 300~K. This dependence can be approximated 
by the following relationship between the filament temperature $T$ (in Kelvin)s and the relative resistivity $\mathrm{R/R_{300K}}$: $T=292\cdot\left(\mathrm{R/R_{300K}}\right)^{5/6}$.}

\end{table}


In the lab report plot the reading from the radiation sensor (convert to $W/m^2$) (on the y axis) versus the temperature $T^4$ on the x axis. According to the Stefan-Boltzmann Law, the data should show a linear dependence, since  according to Eq.(\ref{SBl}) $S\propto T^4$. Fit the experimental data using a linear fit and its uncertainty. For an ideal blackbody we expect the slope to be equal to the Stephen constant $\sigma=5.6703 \times 10^{-8} W/m^2K^4$. However, there exists no ideal black bodies. For real objects the Eq.(\ref{SBl}) is modified, and written as:
\begin{equation}\label{SBlmod}
 S =\epsilon\sigma T^4,
\end{equation}
where the coefficient $\epsilon$ is called \emph{emissivity} and is defined as the ratio of the energy radiated from a material's surface to that radiated by a perfect blackbody at the same temperature. The values of $\epsilon$ vary from 0 to 1, with one corresponding to an ideal blackbody. All real materials have $\epsilon<1$, although some come quite close to the ideal (for example, carbon black has $\epsilon=0.95$). The emissivity of a tungsten wire varies from $\epsilon=0.032$ (at $30^{\circ} C$) to $\epsilon=0.35$ (at $3300^{\circ}C$).

Unfortunately, it is impossible to measure the exact value of emissivity
from the experimental data, as the Stephan-Boltzman law describes the amount of radiation \emph{emitted}
by the object per unit area. To relate $S$ to the amount of \emph{detected} radiation $S_{det}$ one needs to know the surface area of the
filament - something we cannot measure without breaking the bulb (please don't!).  
All we can say is that the emitted and detected radiation intensity are proportional to one another. 
As a result, in this lab we are going to only verify the validity of functional dependence 
described by Eq.~(\ref{SBlmod}) by
testing the linear dependence of the detected radiation on the filament temperature $T^4$. 

To do that, fit the experimental data using the linear fit, find the proportionality coefficient and its uncertainty.
To examine the quality of the fit more carefully, make a separate plot of the \emph{residual} - the difference between the experimental points and the fit values. For a proper fit function, we expect the residuals to be randomly distributed around zero within the experimental measurement uncertainties. Analyze your results. Do the points seem to systematically differ from the fit line in a particular region?  Can you think of a reason why that would be? 






\section*{Test of the inverse-square law}
\textbf{Equipment needed}: Radiation sensor, Stefan-Boltzmann lamp, multimeter,
power supply, meter stick.
\begin{figure}[h]
\includegraphics[height=2.5in]{./pdf_figs/bb31}
\caption{\label{bb31}Inverse square law setup}
\end{figure}
A point source of radiation emits that radiation according to an inverse square
law: that is, the intensity of the radiation in $(W/m^2)$ is proportional to
the inverse square of the distance from that source. This way, the intensity at the certain distance from the blackbody integrated over surface of the sphere of such radius is always constant. Mathematically, we expect the following relationship between the total power of the radiation source $P_0$ and the distance to the point source $r$ to be:
\begin{equation} \label{invlaw_theory}
S_{det}(r)=\frac{P_0}{2\pi r^2}
\end{equation}

\begin{enumerate}
\item  Set up the equipment as shown in Fig. \ref{bb31}. Tape the meter stick to the table. Place the Stefan-Boltzmann lamp at one end, and the radiation sensor in direct line on the other side. The zero-point of the meter stick should align with the lamp filament (or, should it?). Adjust the height of the radiation sensor so it is equal to the height of the lamp. Align the system so that when you slide the sensor along the meter stick the sensor still aligns with the axis of the lamp. Connect the multimeter (reading millivolts) to the sensor and the lamp to the power supply.
\item  With the {\bf lamp off}, slide the sensor along the meter stick. Record the reading of the voltmeter at 10 cm intervals. Average these values to determine the ambient level of thermal radiation. You will need to subtract this average value from your measurements with the lamp on.
\item  Turn on the power supply to the lamp. Set the voltage to
	approximately 5-7 V. {\bf Do not exceed 13 V!} Adjust the distance
	between the sensor and lamp from 2.5-100 cm and record the sensor
	reading. \textbf{Before the actual experiment think carefully about
	at what distances you want to take the measurements. Is taking them
	at constant intervals the optimal approach? At what distances would you expect
	the sensor reading change more rapidly?}

 \item  Make a plot of the corrected radiation measured from the lamp versus the distance from the lamp to the sensor $x$.  Fit the data to 
 \begin{equation} \label{invlaw_fit}
 S_{det}= S_{0} + \frac{C}{(x-x_0)^2}.
 \end{equation}

\item What are the values of $S_0$, $C$ and $x_0$ (and, of course, their uncertainties)?  

\item Compare Eqs.(\ref{invlaw_theory}) and (\ref{invlaw_fit}). What are the physical meanings of the parameters $S_0$, $C$ and $x_0$.  Do their values make sense, considering your experimental arrangements and measurements? 

\item  Can the lamp be considered a point source? If not, how could this affect your measurements?

\end{enumerate}

\newpage

\section*{Universal thermometer}
Blackbody radiation gives us an ability to measure the temperature of remote objects. Have you ever asked yourself how do astronomer know the temperature of stars or other objects many light years away? The answer - by measuring the light they emit and analyzing its spectrum composition using the expressions for the blackbody radiation spectrum. Wein's law Eq.(\ref{weins}) links the wavelength at which the most radiation is emitted to the inverse of the object's temperature, thus the colder stars emit predominantly in red (hence the name ``red giants''), while emission pick for hot young stars  is shifted to the blue, making them emit in all visible spectrum.
\begin{figure}[h]
	\centering
	\includegraphics[height=2.5in]{./pdf_figs/blackbody_radn_curves} 
	\caption{Black body radiation spectrum for objects with different temperatures.}%
	\label{fig:bbStars}
\end{figure}

The human bodies, of course, are much cooler than stars and emit in infrared range. This radiation is invisible to a human eye, but using proper detection methods it is possible to create thermal maps of the surroundings with accuracy better than $1/10$th of a degree. Forward-looking infrared (FLIR) cameras have wide range of applications, from surveillance and military operations to building inspection and repairs, night-time navigation and hunting.  As I write this in Fall 2020, in the middle of COVID19 pandemic, more and more locations use such infrared sensors to measure visitors' temperature at the building entrances or the check points in airports.  

\end{document}

\hrule

{\huge Your instructor may have told you to do a ``brief writeup'' of this experiment. If so, please see the instructions starting on the next page (page~\pageref{pag:briefwriteupBB}).}

\hrule

\newpage

\section*{Instructions for a brief writeup}\label{pag:briefwriteupBB}

This lab is reasonably straightforward. There is much data to take and several fits, but describing the setup and providing an introduction seems less necessary than for some of the others. Therefore, just answer the following questions/bullet points in your report. {\bf Please restate each one so it's easy for us to follow along.} {\it Refer the relevant section earlier in the manual for additional discussion. This is just a bullet point summary of the deliverables.}

\subsection*{Thermal radiation rates from different surfaces}
First, tabulate your data and include it (with label and caption).  Then use your data to address the following questions:
\begin{enumerate}
\item Is it true that good absorbers of radiation are good emitters? \textit{Hint:} use what you know about absorption of visible light for these surfaces
\item Is emission from the black and white surfaces similar?
\item Do objects at the same temperature emit different amounts of radiation?
\item Is glass effective in blocking thermal radiation? What about other objects that you tried? 
\end{enumerate}

\subsection*{Tests of the Stephan-Boltzmann Law}


\begin{enumerate}
\item Tabulate your data as in Tab 4.1 and include it.
\item Estimate and report uncertainties on the voltage, current, and radiation sensor.
\item Plot the readings from the radiation sensor (y-axis) vs $T$ (x-axis) with uncertainties and fit to $S=C T^n$. Include the plot in your report.
\item Is it a good fit? 
\item Does the Stephan-Boltzman equation seem to hold? Does $n=4$, within uncertainties?
\item What value (and uncertainty) do your data imply for $A$? Is it consistent with tungsten? What else could be affecting your measurement? 


\end{enumerate}

\subsection*{Test of the inverse square law}

\begin{enumerate}
\item Tabulate your data and include it in the report. Be sure to include the lamp-off data, used to correct the lamp-on data for ambient radiation.
\item Estimate and report uncertainties on the distance between the sensor and lamp,  and on the readings from the radiation sensor.
\item Make a plot of the corrected radiation (y-axis, with uncertainty) vs distance (x-axis) and fit it to $S= B + C x^n$. Include it in your report. 
\item How good is the fit?
\item Does $n=-2$ within uncertainty? Do the data confirm the inverse square law?
\item What are $B$ and $C$?  What did you expect?  Comment on any discrepancies. 
\end{enumerate}

%
%\item The Blackbody Spectrum Equipment: Spectrometer, computer, high
%temperature source.
%\begin{itemize}
%
%\item  There are two spectra at the end of this lab.
%From Wien's law, estimate the temperature of the sources.
%\begin{displaymath}
%T_{ estimate}=\underline{\hskip .75in}K
%\end{displaymath}
%\begin{displaymath}
%T_{ estimate}=\underline{\hskip .75in}K
%\end{displaymath}
%
%
%\item  On a separate graph, plot the expected spectrum
%from the Rayleigh-Jeans law and Planck's law. Which law best
%represents the spectrum acquired above?
%\end{itemize}
%
%Addendum to Blackbody Radiation Handout
%\section*{Thermal Radiation rates from Different Surfaces}
%
%\begin{itemize}
%\item Make sure the shutter on the Sensor is pushed all the
%way open!
%\item Make sure the temperature is stable when you begin readings from
%surfaces of Cube (you may have to wait at least 5 minutes between
%temperature changes).
%\item Make the readings quickly!
%\end{itemize}
%
%
%
%\section*{Inverse Square Law}
%
%Make sure that you keep the Sensor in line with the filament as you
%slide it (and you do not introduce an angle).
%




%\subsection*{The Blackbody Spectrum}
%
%In Fig. \ref{Sun} is an approximate spectrum of the sun. Determine
%the approximate temperature of the sun from this spectrum using Wien's
%law. $T=\underline{\hskip .75in}$.
%In Fig. \ref{Cobe} is the spectrum for the microwave background
%assumed to arise from the time when the photons decoupled from the
%charged particles, i.e., when most free electrons became bound.
%Determine the temperature of the microwave background. Note that what
%is plotted is waves/cm, not cm. If the present size of the visible
%universe is $13\cdot 10^9$ light years. How large was the visible
%universe when the decoupling took place. Hint: $R_u$ has experienced the
%same expansion as the wavelength and the temperature at decoupling must
%correspond to about 10 eV.
%\begin{itemize}
%\item Extra credit: what is the accepted value
%for the
%temperature of the surface of the sun?
%\item How does your extracted value
%compare?
%\item Include a copy of this spectrum in your lab report.
%On a
%separate graph, plot Planck's Law and the Rayleigh-Jeans Law for this
%same temperature.
%\item Which law does the solar spectrum appear to behave?
%\end{itemize}


%
%
%
%\begin{figure}
%\includegraphics[height=2.5in]{LinearSp.eps}
%\caption{\label{Sun}Approximate Sun Spectrum}
%\end{figure}
%\begin{figure}
%\includegraphics[height=2.5in]{cobespc.eps}
%\caption{\label{Cobe}Cobe: Cosmic blackbody Spectrum}
%\end{figure}
%\begin{figure}
%
%\newpage

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