From 015e2355f4577bbf82a0997c694f5b547a633414 Mon Sep 17 00:00:00 2001
From: "Eugeniy E. Mikhailov" <evgmik@gmail.com>
Date: Fri, 4 Sep 2020 10:03:11 -0400
Subject: Irina's edits

---
 blackbody_new.tex | 87 ++++++++++++++++++++++++++++++-------------------------
 1 file changed, 48 insertions(+), 39 deletions(-)

(limited to 'blackbody_new.tex')

diff --git a/blackbody_new.tex b/blackbody_new.tex
index 9144318..4230be0 100644
--- a/blackbody_new.tex
+++ b/blackbody_new.tex
@@ -8,7 +8,6 @@
  \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.
-\vspace{0.25in}
 
 \section*{Theory}
 
@@ -38,36 +37,36 @@ where $S$ is the radiated power per unit area ($W/m^2$), $T$ is the temperature
 \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{displaymath}
+\begin{equation}
  u(\lambda,T) = \frac{8\pi kT}{\lambda^4}
-\end{displaymath}
+\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 black body temperature as described by {\bf Wien's law:}
-\begin{displaymath}
+\begin{equation}
  \lambda_{max}T = 2.898\times 10^{-3} m\cdot K
-\end{displaymath}
+\end{equation}
 and 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{displaymath}
+\begin{equation}
 u(\lambda,T)=\frac{8\pi hc\lambda^{-5}}{e^{hc/\lambda kT}-1}
-\end{displaymath}
+\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!} \\
+%\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}.''
 %
-``\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}
+%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}
 
@@ -77,16 +76,13 @@ Imagine a metal wire connected to a cold reservoir at one end and a hot reservoi
 \\
 \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 for radiated power per unit area $S$ in the correct units ($W/m^2$), you need to use the voltage-to-power conversion factor $22~mV/mW$, and the area of the sensor $2mm\times2mm$:
+\textbf{Sensor calibration}: To obtain the radiation sensor readings 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[W/m^2]=\frac{S[mV]}{22 [mV/mW]}\cdot 10^{-3}\cdot \frac{1}{4\cdot
+S[W/m^2]=\frac{S[mV]\cdot 10^{-3}[V/mV]}{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
@@ -122,41 +118,50 @@ Before starting actual experiment take some time to have fun with the thermal ra
 
 \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 one
+\item  Turn on the lamp power supply. Set the voltage, $V$, in steps of 1-2
 	volt from 1-12 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 Fig.
-	\ref{w_res:fig}.
+	\ref{w_res:fig}. Don't forget to convert the measured temperatures to Kelvin scale: $T[K]=T[^oC]+273$.
+	
+\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}
 
-\subsection*{Safety}
-The Stefan lamp will get very hot - be careful!!!
-
 \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.}
 \end{figure}
 
 
-In the lab report plot the reading from the radiation sensor (convert to $W/m^2$) (on the y axis) versus the temperature $T$ on the x axis. According to the Stefan-Boltzmann Law, the data should show a $S\propto T^4$ behavior. Do a fit to $S=C T^n$ where $C$ and $n$ are free parameters. Report the value of each parameter and its uncertainty. Does $n=4$, within uncertainty? How does it compare to the accepted value of Stefan's constant?  
 
-Examine the fit.  How many points are within 1 uncertainty ($\sigma$) of the line?  How many are between 1 and 2 $\sigma$? For a good fit, about 2/3 of the points should be within $1\sigma$ of the line, and the other 1/3 between 1-$2\sigma$, with perhaps one point between 2 and $3\sigma$, if we have about 20 points. That latter number scales up with the number of points. Unless you have a few 100 datapoints, a good fit should not have any points further than $3\sigma$. Based on this criteria, was is your fit good?  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?
+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 black body 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{SBl}
+ 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 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 $\epsilon=0.95$). However, emissivity of a tungsten wire varies from $\epsilon=0.032$ (at $30^{\circ} C$) to $\epsilon=0.35$ (at $3300^{\circ}C$). Using the result of your fit, and assuming we know the Stephan-Boltzman constant $\sigma$ by some other means, what is $\epsilon$ and what is the uncertainty on it?  Is it consistent with tungsten? What else could be affecting this measurement?
+
+Let us examine the quality of the fit more carefully. For that it is convenient to 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 experimental points to be randomly distributed around zero. 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? 
+
+
+
 
-The parameter $C$ is equal to $A\sigma$. For an ideal black body $A=1$, but in your case it will be very different. $A<1$ means the object does not absorb (or emit) all the radiation incident on it (this object only radiates a fraction of the radiation of a true blackbody). The material lampblack has $A=0.95$ while tungsten wire has $A=0.032$ (at $30^{\circ} C$) to 0.35 (at $3300^{\circ}C$). Using the result of your fit, and assuming we know the Stephan-Boltzman constant $\sigma$ by some other means, what is $A$ and what is the uncertainty on it?  Is it consistent with tungsten? What else could be affecting this measurement?
 
 
 \section*{Test of the inverse-square law}
 \textbf{Equipment needed}: Radiation sensor, Stefan-Boltzmann lamp, multimeter,
 power supply, meter stick.
-\begin{figure}
+\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. You will determine if this
-is true for a lamp.
+the inverse square of the distance from that source, so that the intensity at the certain distance from the black body 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 $P$ and the distance to the point source $r$ to be:
+\begin{equation} \label{invlaw_theory}
+S(r)=\frac{P}{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.
@@ -169,10 +174,14 @@ is true for a lamp.
 	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 $S= B + C x^n$ where $B$, $C$ and $n$ are free parameters. Based on the criteria in the previous section, how good is the fit? 
+ \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= 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 What are the values of $B$, $C$ and $n$ (and, of course, their uncertainties)?  How do they agree with what you expect from the inverse square law?
+\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?
 
-- 
cgit v1.2.3

