(a) Table 1 contains the illuminances measured through the red, green and blue filter for a standard incandescent light source at known temperatures. Choose suitable light filters and construct a calibration curve that relates the chosen colour index to the temperature.
(b) Measure the relationship between the electrical input power and the tungsten filament temperature. Plot the result over a relevant range.
Theory
The infrared thermometer cannot be used to measure the filament temperature for several reasons - the range of the IR thermometer (stated on the instrument) only goes up to $500^{\circ} \mathrm{C}$. The filament is also too small to be the only thing measured. IR opacity of the glass bulb is also not guaranteed. Therefore, the only way to measure the temperature is indirectly through the colour index, for which the relation to temperature is provided.
Wien's displacement law suggests that at lower temperatures, the light will contain more red component than green and blue, while at higher temperatures, the green and then the blue will increase faster than red, leading to increasing ratios $G / R, B / R$ and $B / G$. We must, however, consider, which pair of filters will be the most suitable choice.
The values measured through different filters depend on the spectral response of each filter, including its overall opacity. It also depends on the sensitivity of the light meter to each wavelength. Instead of theoretical predictions, we are given reference measurements at known temperatures. If we plot the ratios for all three pairs, we observe that $B / G$ is the least suitable, as it changes much less with temperature, compared to the other two. $B / R$ and $G / R$ are comparable, but the blue filter has lower transmittance, which will lead to lower accuracy (higher relative error).
Any pair of filters is a valid choice to proceed with the measurements, but will affect the end accuracy. Averaging the results is also an option, but including $B / G$ combination may still reduce the accuracy of the end result.
To use the plot for converting the colour index to temperature, we need a trend line. A linear trend is enough to cover most of the range, except at lower temperatures, where the relationship tapers off. We can extend the range by combining two trends across the range, or to draw a smooth curve by hand. Zig-zag interpolation is less suitable due to scatter in the reference measurements.
Using the absolute values from the table instead of ratios is not correct, as the intrinsic luminous flux of the light source and the measurement distance are not given.
Experiment
For the measurement of the power dependence of the temperature, we will read out the voltage and current from the power supply. To sample the expected curve of the $T(P)$ relationship, we must sample it sufficiently well, especially at lower powers where the temperature changes more quickly. We suggest sampling at least 8 powers/temperatures to cover the relationship more precisely and distinguish outliers from reliable measurements. For each power setting, we must measure the illuminance through the chosen filters by covering the sensor of the light meter with a filter. Covering the light meter filters all the light, including the light reflected from the walls and the floor, leading to a better measurement. Placing the filter next to the light source also introduces the risk of burning the filter. Planning ahead, we can simultaneously measure the illuminances without a filter, needed in Task 2.
Each colour index is then converted to a temperature by reading out from the calibration graph. We can also estimate the relationship by employing the StefanBoltzman law if we neglect other losses and the contribution of the ambient temperature:
\begin{equation*}
T \propto \sqrt[4]{P} \tag{1}
\end{equation*}According to measurements with multiple light bulbs in different environments, the fit is
\begin{equation*}
T=\left(1220 \mathrm{KW}^{-1 / 4} \pm 20 \mathrm{KW}^{-1 / 4}\right) \sqrt[4]{P}, \tag{2}
\end{equation*}which is used as a baseline for determining the RMS of students' measurements.
The background illuminance must be measured through all filters - it is most likely zero, but a good experimentalist must check, and if significant, it must be subtracted from measurements. This is also a way for us to detect if they left their desk lamp on – if the background differs significantly from the rest of the contestants.
The distance between the light source and the light meter should be short enough to enable accurate measurements at lower powers. Distance can also be different for different power ranges, but care must be taken, as the effect of the finite size of the filament may play a role, as well as the changing reflections from surroundings and from the top of the light stand.
Measure the dependence of luminous efficacy on the electrical input power for both light sources across the range with detectable light output. Plot the results, one plot per light source. State all steps of the calculation procedure and present all the measured data.
Theory
Light sources do not radiate in all directions equally. The angular distribution of luminous flux $\Phi$ (luminous intensity) must be integrated over the solid angle. A light meter at distance $r$ to the light source, oriented so that the light falls on it perpendicularly, measures the illuminance $E$ of a certain part of the imagined integration sphere surrounding the light source:
\begin{equation*}
\Phi=\oint E(\Omega) r^{2} \mathrm{ d} \Omega \tag{3}
\end{equation*}The LED only shines the light into a hemisphere, and has cylindrical symmetry around the direction straight ahead, so we can simplify the expression,
\begin{equation*}
\Phi_{L E D}=2 \pi \int_{0}^{\pi / 2} E(\theta) r^{2} \sin \theta \mathrm{ d} \theta \tag{4}
\end{equation*}and for the incandescent bulb, the symmetry axis is perpendicular to the direction straight ahead, and shines into full solid angle:
\begin{equation*}
\Phi_{W}=4 \pi \int_{0}^{\pi / 2} E(\theta) r^{2} \cos \theta \mathrm{ d} \theta \tag{5}
\end{equation*}The integrals will have to be evaluated numerically it can be done by using the trapezoidal or the Simpson method, or by using the formula for a spherical segment area given in the hint:
\begin{equation*}
\Phi=2 \pi r^{2} \sum_{i} E\left(\theta_{i+1 / 2}\right)\left(\cos \theta_{i}-\cos \theta_{i+1}\right) \tag{6}
\end{equation*}and equivalent (but with $\sin \Longleftrightarrow \cos$) for the incandescent bulb. Here, choosing evaluation points in the middles of intervals is better than choosing one of the edge points. However, the exception are the "edge" measurements, where the measurement point is actually in the middle of the interval - the point straight ahead for the LED is in the middle of the spherical cap. The same goes for the "poles" of the incandescent light bulb.
The ratio between the head-on measured illuminance and the luminous flux, can be expressed as
\begin{equation*}
\Phi=C r^{2} E(0), \tag{7}
\end{equation*}or, more intuitively, as a correction factor to the isotropic source:
\begin{equation*}
\Phi=\{4 \pi, 2 \pi\} \tilde{C} r^{2} E(0) \tag{8}
\end{equation*}
Analytical estimates
One possible pathway is to estimate these factors without measurements, using reasonable assumptions about the light distribution. The LED can be assumed a planar emitter, with a cosine distribution of luminous flux:
\begin{equation*}
\tilde{C}_{L E D}=\frac{\int_{0}^{\pi / 2} \cos \theta \sin \theta \mathrm{ d} \theta}{\int_{0}^{\pi / 2} \sin \theta \mathrm{ d} \theta}=\frac{1}{2} \tag{9}
\end{equation*}which turns out to match the experiment well.
For the incandescent bulb, a similar assumption can be made based on a thin filament model. The different orientation of the symmetry axis leads to a different result:
\begin{equation*}
\tilde{C}_{W}=\frac{\int_{0}^{\pi / 2} \cos ^{2} \theta \mathrm{ d} \theta}{\int_{0}^{\pi / 2} \cos \theta \mathrm{ d} \theta}=\frac{\pi}{4} \approx 0.79 \tag{10}
\end{equation*}
These approximations can be used to a good effect but are not required in the experimental task.
Experiment
To measure the angular dependence, a suitable distance to the light source must be chosen. Too far, and the signal becomes weak and any background could become noticeable. It is advisable to measure the angular dependence at the highest power in order to improve the signal to background ratio. Measurement can also be performed through one of the filters.
For the incandescent bulb, the finite size of the filament becomes an issue if we measure too close to the bulb. This becomes noticeable at distances lower than 10 cm . This was not an issue for the colour index measurement, but it matters for the absolute flux estimation.
To describe the inflection point in the light distribution well, we will need at least 5 measurements in the $\theta \in$ $[0, \pi / 2]$ interval. We can either rotate the light source on the spot, or position the light meter at different angles in relation to the stationary light source.
For the light distribution left-right symmetry can be assumed, or, alternatively, the entire $\theta \in\left[-90^{\circ}, 90^{\circ}\right]$ range can be measured, allowing to take into account asymmetries and an angular offset in the light distribution. The straight ahead measurement is centered in a symmetric band, which needs care so it is not double-counted in case only half of the range is integrated and then doubled.
With the conversion factors known, the luminous efficacy can be determined by measuring the frontal illuminance at powers that cover the entire range from the lowest detectable illuminance to the maximum allowed power. For the incandescent bulb, this measurement can be done simultaneously with Task 1 for better time efficiency.
$C$ $\tilde C$ $W$ 10.01 0.80 LED 2.63 0.42
Table 1: Example values of the conversion factor between the frontally measured illuminance and the luminous flux for both light sources. The values will vary within some wider distribution because of varying light sources and other errors, which is indicated by the brackets in the grading sheet.
It is not required to measure at the same distance as the angular dependence. Multiple distances may also be used.
We have to avoid placing any additional objects near the light source to avoid introducing more reflected or blocked light - such as placing the light source directly on the white paper, or having other obstructions such as the black paper screen or any filters too close to the light bulb.
To plot the efficacy, we divide the $\Phi$ obtained from eq. (3) for each of the light sources, with $P=U I$ read out from the power supply.
The result shows that the efficacy of the incandescent light starts out at zero at low powers and increasing with power, as its temperature increases. The LED has the highest efficacy at lowest powers, then it drops off at higher powers, mostly due to increased temperature of the light emitting junction.
At the lowest settable currents, the readout on the power source is no longer reliable - for example, LED may glow slightly even at $0~A$ . The pole at the origin can be attributed to this source of error.
(a) Determine the heat transfer coefficient $h$ and the thermal conductivity $\lambda$ for the black plastic, and perform error analysis. Assume the material absorbs all received light and the incandescent light bulb emits all power in the form of electromagnetic radiation.
(b) Estimate the albedo (the fraction of the irradiance that is reflected instead of absorbed) of the white plastic and perform error analysis.
Useful relations: An area of a segment of a sphere with radius $r$ between polar angles $\theta_{1}$ and $\theta_{2}$ with $0 \leq$ $\theta_{1} \leq \theta_{2} \leq \pi$ is $\Delta A=2 \pi r^{2}\left(\cos \theta_{1}-\cos \theta_{2}\right)$.
Theory
The plate receives a radiant flux density $j$, determined by the power $P$ of the light source, and the distance $r$ between the target and the light source. The light source does not shine equal amounts of light in all directions, therefore we must use the correction factor $C$, derived in Task 2, to convert from the total radiant flux to forward radiant flux density. \begin{equation*} P=C r^{2} j \rightarrow j=\frac{P}{C r^{2}} . \tag{13} \end{equation*}Not necessary, but also correct, is to (numerically) integrate/average across the entire plate, $j\left(\pi r^{2}\right)=$ $P \int C r^{-2} \cos \theta \mathrm{~d} A$ to take into account spatial variation of $C, r$ and $\theta$ (angle of incidence).
The incident flux density is dissipated to the environment directly, as well as by heat conduction through the plate. Mark by $T_{F}$ the front temperature and $T_{B}$ the back temperature. Conservation of energy gives us the system of equations \begin{align*} j & =h\left(T_{F}-T_{0}\right)+\frac{\lambda}{d}\left(T_{F}-T_{B}\right) \tag{14}\\ 0 & =h\left(T_{B}-T_{0}\right)+\frac{\lambda}{d}\left(T_{B}-T_{F}\right) . \tag{15} \end{align*} This system of equations leads to the following relations: \begin{align*} & j=h\left(T_{F}+T_{B}-2 T_{0}\right) \tag{16}\\ & j=\left(h+2 \frac{\lambda}{d}\right)\left(T_{F}-T_{B}\right) . \tag{17} \end{align*} Any linear combination of equations $(14,15)$ also allows determination of both $h$ and $\lambda$. A particular linear combination that may be used is the isolation of individual temperatures: \begin{align*} & T_{F}-T_{0}=\frac{1}{2}\left(\frac{1}{h}+\frac{1}{h+2 \frac{\lambda}{d}}\right) j \tag{18}\\ & T_{B}-T_{0}=\frac{1}{2}\left(\frac{1}{h}-\frac{1}{h+2 \frac{\lambda}{d}}\right) j . \tag{19} \end{align*} In our system, $2 \frac{\lambda}{d}>h$, but still in the same order of magnitude. Treating the slope of $T_{F}$ as $1 /(2 h)$ or the slope of $T_{F}-T_{B}$ as $(2 \lambda / d)^{-1}$ is a reasonable approximation, but still not theoretically correct.
Error analysis
Errors should be propagated from the slope. For example, if they obtain slopes $k_{1}=1 / h$ and $k_{2}=1 /(h+$ $2 \lambda / d)$, they should propagate the errors. We should allow both straight addition of error contributions of different terms, or adding squared errors (independent errors), e.g. \begin{align*} & h=\frac{1}{k_{1}} \pm \frac{\sigma_{1}}{k_{1}^{2}} \tag{20}\\ & \lambda=\frac{d}{2}\left(\frac{1}{k_{2}}-\frac{1}{k_{1}}\right) \pm \frac{d}{2}\left(\frac{\sigma_{1}}{k_{1}^{2}}+\frac{\sigma_{2}}{k_{2}^{2}}\right) \tag{21} \end{align*} and analogously for other slope definitions.
Albedo
For the white plate, only a part of the incident flux is absorbed, so we replace $j$ by $j(1-a)$ if $a$ is the albedo: \begin{equation*} j=(1-a) \frac{P}{C r^{2}} . \tag{22} \end{equation*}As a consequence, any slope measured for both plates will be in the ratio $(1-a)$ to each other. This can be expressed as a fraction of trend slopes, ratio of temperature differences, or similar.
Experiment
The radiant flux density can be varied in two main ways, or a combination of both: by changing the distance, or by changing the current through the light bulb. Both methods are acceptable, but varying the current also changes the spectrum and the efficiency of the light bulb, so it may produce biased and nonlinear results. The students should know that varying a single parameter is the correct procedure.
The required measurements in this task are the front and back temperature at different powers, for black and white plate. It is essential to wait for equilibration, which includes waiting the back temperature to stabilize. It is advisable to measure starting with the lowest flux density, because it will require the least equilibration time from the initial room temperature of the plate.
The target should not be too close to the light source, not only because of the risk of burning, but also because close to the light bulb, the light is very nonuniformly distributed across the plate. Increased convection rate due to high temperature also starts deviating from the linear regime. Placing the target too far from the light source leads to a negligible heating and thus a very large relative error in temperature differences, especially for the white plate.
In this task, the measurements are subject to many sources of errors: measuring from different distances and at different angles may include different proportions of background or reflected IR radiation from the light source (if the targeted area is still illuminated), if the measurement takes too long, the plate may start cooling down (this is noticeable in a few seconds), air currents may increase convective heat dissipation, and the ambient temperature may also change during the measurement (especially if the light source is placed too close to the wall, or if the power source's fan exhaust is too close to the measurement setup). The errors are most noticeable at low radiant flux and for the white plate, where increases in temperature are the smallest.
For these reasons, it is advisable to take more than one measurement per data point and average the results, and to cover a sufficiently wide range to reduce the slope error. At least 3 points are needed to draw a trend, but 5 is better. With more points, it is easier to spot outliers and utilise the measurements which are least subjected to errors. Back and front temperatures are best measured in pairs one after the other to reduce the error in the temperature difference signal due to changing conditions.
Note: the ambient temperature $T_{0}$ is an effective temperature that combines air temperature and radiative exchange with the surrounding walls, ceiling and other objects. We do not need its value, we only need the slopes of the linear trends. Inexact $T_{0}$ can lead to inaccuracies if used together with an assumption the linear relations go through the origin. $T_{0}$ cannot be reliably determined by measuring surrounding temperatures, but it can be estimated by measuring the equilibrium temperature of the plate in the absence of the light source.
The measurements of the front and back temperature at different radiant fluxes, must be processed and plotted to extract the necessary slopes. For the black plate, two plots will be needed, based on equations $(14,15)$, equations $(16,17)$, or any linearly independent pair. Linear regression gives us the slopes $h$ and $h+2 \frac{\lambda}{d}$ (or their reciprocals). $T_{0}$ is best determined by the $j=0$ intercept of the trend line for (Eq. 16) or any equivalent plot, and should match $T_{0}$ determined by other methods. If measured correctly, the intercept of the trend line for (Eq. 17) should be zero within the error margin.
It is possible to calculate the necessary slopes from a measurement at a single input power (for each plate color), if $T_{0}$ is measured well. This can be done without a graph. However, using multiple measurements decreases the impact of statistical errors and enables us to better estimate the error, so a single measurement will carry a significant error.
Albedo, as defined in the task text through irradiance units, cannot be measured using a light meter, which measures in photometric units. Additionally, light reflected from a white plate introduces additional geometric considerations and angular distribution of reflected light, that cannot easily be taken into account.
The albedo can be estimated as a fraction of the corresponding line slopes between the black and the white plate, taking any of the relations $(14,15,16,17)$. This means that for the white plate, measuring only one side of the plate is enough to determine the albedo, assuming $h$ and $\lambda$ remain the same. The difference slope or the back temperature slope are the least suitable, as they introduce a large relative error to the measurement due to a minimal increase in temperature.
