diff options
| -rw-r--r-- | blackbody_new.tex | 39 |
1 files changed, 20 insertions, 19 deletions
diff --git a/blackbody_new.tex b/blackbody_new.tex index fd5aceb..d6124de 100644 --- a/blackbody_new.tex +++ b/blackbody_new.tex @@ -77,10 +77,10 @@ Imagine a metal wire connected to a cold reservoir at one end and a hot reservoi \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 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 $2mm\times2mm$: +\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[W/m^2]=\frac{10^{-3}[V/mV]}{22 [V/W]}\cdot \frac{1}{4\cdot -10^{-6}[m^2]} \cdot \Delta V[mV] +S_{det}[W/m^2]=\frac{V_S [V]}{22 [V/W]}\cdot \frac{1}{4\cdot +10^{-6}[m^2]} \end{displaymath} @@ -146,28 +146,29 @@ Before starting actual experiment take some time to have fun with the thermal ra % "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.} +\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 + 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 $\epsilon=0.95$). However, emissivity of a tungsten wire varies from $\epsilon=0.032$ (at $30^{\circ} C$) to $\epsilon=0.35$ (at $3300^{\circ}C$). +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 one needs to know the surface area of the -filament and the precise distance from it to the detector. However, we -should be able to qualitatively verify the validity of Eq.~(\ref{SBlmod}). From -fitting experimental data, find the proportionality coefficient between the -emitted radiation and the filament temperature $T^4$ and its uncertainty. -Estimate the quality of the linear fit (a linear fit would assume that the -emissivity does not depend on temperature). +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 $T^4$. - -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 residuals 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? +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? @@ -183,16 +184,16 @@ power supply, meter stick. \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, so that 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 $S_0$ and the distance to the point source $r$ to be: +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(r)=\frac{S_0}{2\pi r^2} +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 10 V. {\bf Do not exceed 13 V!} Adjust the distance + 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 @@ -201,7 +202,7 @@ S(r)=\frac{S_0}{2\pi r^2} \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}. + 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)? |