Experimenting with light at the 6th European Physics Olympiad

Incandescent light sources emit light with a spectrum approximately following the Planck's law of black body radiation (figure 2). The wavelength at which the spectral density of the light is the highest is decreasing with increasing temperature in accordance with the Wien's displacement law, from red and orange towards the blue part of the spectrum, which is directly observable from the colour of the emitted light. The temperature of black body emissions uniquely determines its colour, so it is a good parameter to quantify it. We use the term colour temperature to describe the colour of a light source, even for non-incandescent light sources, such as fluorescent and LED lights, in order to specify which temperature of black body radiation it most closely resembles.

Quantitative description of light, as seen by the human eye, is encapsulated by photometric units, which are scarcely included in the high school curriculum. Thus, a brief review is needed: the total amount of light, emitted by a source into all directions, the luminous flux, is measured in lumen [lm]. The received light per unit of area is called illuminance, measured in lux [lx = lm/m²]. This is the quantity measured by the light meter, which was provided in the kit. There are other photometric quantities that also relate angular information, but they will not be needed for this task [12–14].

In astronomy, the brightness of stars is measured in magnitude, which is logarithmically related to the light intensity measured by detectors. The colour of stars is quantified by the colour index, the difference of magnitudes measured through two filters, such as the B–V index for the difference between magnitudes through blue (B) and yellow-green (V stands for visible) filters, which is analogous to a quotient of corresponding intensities [15]. With proper calibration, colour index can be used to determine the temperature of the star [16–18].

The same method works for determining the temperature of other objects [19], which was used by the competitors in their first task. They were to determine the temperature of the light bulb filament at different powers. The temperatures are too high to be measured using conventional thermometers. Three filters were provided—red, green, and blue—and a light meter, making it possible to measure the detector illuminance through different filters. They had to prepare a calibration curve based on the provided table, which we are showing in figure 4(a) in graph form.

Zoom In Zoom Out Reset image size

The first task, adapted for stand-alone clarity, was as follows:

The objective was to select only two filters to measure an associated colour index. While it is possible to make an acceptable measurement using any pair of filters, an inspection of the data shows that values through green and blue filters follow similar curves, so their quotient spans across a narrower range, and a small error in the colour index would result in large uncertainty in temperature, as shown in figure 4(b). Using a red filter in combination with one of the other two is thus a more suitable choice. Additionally, the values obtained through the blue filter are much dimmer and thus carry a larger uncertainty; therefore, the red and green pair was the best choice. With the exception of the lowest temperatures, the calibration curve can be approximated with a linear relationship, which simplifies conversion.

Figure 5 depicts the experimental setup. The distance to the light meter should be the same for both filters to avoid the effect of the inverse square law and directional intensity profile. The filter must cover the entire light meter sensor. Careful measurement yields an expected relationship—the filament temperature increases with power, but not linearly [see figures 5(b), (c)]. Avid students might also notice that the trend roughly follows a model

$$\begin{eqnarray}&&T\propto \sqrt[4]{P},\end{eqnarray} \tag{ 1 }$$

which is shown as a dotted line in figure 5(c). This holds under the assumption that all the power is radiated away as black body radiation following Stefan-Boltzmann law, neglecting convection and the radiation received from the surroundings. At the high temperatures reached by the filament, this is a reasonable assumption and can, in fact, be confirmed by using a bolometer to measure the total light flux emitted by the tungsten filament and comparing it to the values of electrical power consumed.

Zoom In Zoom Out Reset image size

One of the notable mistakes made at this task was to place the filter on the light bulb instead of the light meter, despite the warning on the instructions. Some students misunderstood the instructions, assuming they should recreate the given calibration table themselves by measuring the temperature using the IR thermometer.

Luminous efficacy is the ratio between emitted (visible) luminous flux and the emitted power. For example, sunlight has an efficacy of 93 lm W^-1 [20]. For electric light sources, input electrical power is usually used as the denominator, as it is easier to measure and more relevant for power consumption considerations.

The second task asked for the efficacy of two light sources in relation to the input power: an incandescent light bulb and an LED. It is easy to assume that the LED will be more efficient, but the dependence of the efficacy on how they are driven is less intuitive. A quantitative result requires careful measurements and taking geometry into account, as outlined in the task statement:

To estimate the luminous flux, the light meter must be placed perpendicular to the light direction, eliminating the geometric effect of oblique light incidence. Thus, the measured illuminance E is just equal to the luminous flux density [lm/m²] at that position. For a light source that shines equally in all directions, the luminous flux could be obtained by multiplication of the measured illuminance at distance r by the area of the sphere: Φ = 4π r² E. In our case, this does not suffice, as our light sources are not isotropic. Instead, we can imagine enclosing our light source into a sphere and summing contributions to the flux Φ[lm] over small patches we assume have approximately the same flux density, each multiplied by the patch area (see figure 6). Thus, we account for all the rays and get the entire flux. Mathematically speaking, this is an integral over the solid angle Ω:

$$\begin{eqnarray}&&{\rm{\Phi }}=\oint E({\rm{\Omega }}){r}^{2}\,{\rm{d}}{\rm{\Omega }},\end{eqnarray} \tag{ 2 }$$

but for the purposes of this task, it will be evaluated as a discrete sum over measurements, so knowledge of integrals is not required.

Zoom In Zoom Out Reset image size

We notice that the provided incandescent bulb is cylindrical, with the rotational symmetry axis perpendicular to the forward direction. The LED also has rotational symmetry, but pointing along the forward direction instead. Additionally, the LED emits no light in the backward direction [see figure 7(a)]. Rotational symmetry was a strict constraint for the selection of the light sources for the task because the general asymmetric case makes experimental realisation infeasible and complicates the mathematical process of obtaining the total light flux.

Zoom In Zoom Out Reset image size

This insight indicates we only have to measure the illuminance in the plane of the desk, along the circle from the forward direction to the perpendicular direction [measurements in these directions are shown in figure 7(b)], instead of measuring in all spatial directions. The calculation relies on the area of a spherical annulus between ϑ₁ and ϑ₂, which is $2\pi {r}^{2}(\cos {\vartheta }_{1}-\cos {\vartheta }_{2})$ (see figure 6). This equation was provided as a mathematical hint in the task. Care is required in interpreting the angle ϑ, because the light sources have differently oriented symmetry axes. The competitors had to estimate the number of angular divisions that suffice for a reasonably accurate result. Some competitors chose to measure symmetrically in both directions from the forward direction, which improves the accuracy by accounting for departures from the assumed symmetry, such as a poorly aligned light source and the tilt of the tungsten filament.

The angular measurement only has to be performed once for each light source; we can assume that varying the power only varies the intensity, not the angular distribution of light. Therefore, the ratio between the forward illuminance (E) and the total flux (Φ) does not change and can be determined with a single series of measurements. This gives us a nondimensional factor C in the expression of the form Φ = Cr² E, where it replaces the specific value 4π that would hold for an isotropic light. All further data for the power dependence can then be measured only in the forward direction by changing the current on the power supply.

The factor C can also be estimated analytically based on the simplified geometry of the emitter. The LED can be assumed to be a flat Lambertian emitter, so the illuminance measured in each direction is proportional to $\cos \vartheta$, leading to C_LED = π. In our particular choice of the LED, the values obtained from experiments were somewhat smaller but within the range of variation C_LED = 3.0 ± 0.2. Note that this is half the naive factor of 2π, which only accounts for the fact that LED only emits into half of the space. For the incandescent bulb, assuming an ideally thin straight Lambertian filament, the illuminance is proportional to $\sin \vartheta$, with ϑ measured from the cylinder axis. We obtain C_W = π², which matches the measured value of C_W = 10.0 ± 0.3 quite closely. Competitors who attempted this path were graded equally for the part of the grade dedicated to obtaining the correct coefficient C.

Figure 7(b) depicts the angular profile of the luminous flux density for both light sources, with the luminous efficacy graphs shown in panels 7(c), (d). We notice that the efficacy of the incandescent light increases with power. The previous task suggests that increased power also increases temperature, which maximises the amount of power emitted in the visible part of the spectrum. Unfortunately, the temperature is limited by the melting point of tungsten at 3695 K, while the usable range is even lower due to material softening and evaporation of metal atoms into the bulb.

With the LED, the trend is reversed. Increasing the power reduces efficacy, mostly because of overheating, which reduces the quantum efficiency of the LED (a phenomenon known as thermal droop). An interested reader can find more information in the literature [21, 22]. The takeaway lesson is that for good efficacy, LEDs must be driven with moderate currents or at least be adequately cooled, which also helps increase their longevity. As expected, even in the worst case, LED is much more efficient than an incandescent light bulb.

Note that the prominent divergence of efficacy at the lowest powers is an artefact of the small bias current measured to be around 5 mA that flows even at zero readouts on the power supply. Consequently, the powers calculated from Ohm's law are too low, and after division, lead to overestimated efficacies. The measurements at powers below 0.1 W are thus unreliable.

The most common mistake in this task was the failure to realise that the angular profile of the emitted light must be measured or estimated analytically, leading to a very poor estimate of efficacy. Other issues were related to incorrect angle-dependent weights in regard to the rotational symmetry and the provided geometric hint.

Illuminated objects get hotter. When the light falls onto an opaque object, a part of it is scattered away while the rest is absorbed and converted into internal energy. Ideal black bodies absorb all the incident light, while ideal white bodies scatter all the light and absorb none. Realistic cases lie in between these two scenarios, with dark bodies absorbing the majority of incident light, at least in the visible part of the spectrum. An object reaches thermal equilibrium when the losses into the environment due to conduction, convection, and radiation balance the radiant energy received through the illumination. For moderate temperature differences, we can assume that the losses are proportional to the temperature difference, P/A = h(T₁ − T₀), where h is the heat transfer coefficient and includes both convection and radiation. T₁ is the temperature of the surface, and T₀ the temperature of the surroundings. Because the surroundings are at different temperatures, and not necessarily at the temperature of the air, we must understand T₀ as an effective temperature at which the object would equilibrate if no light was present. P/A is the dissipated power per surface area.

Not all the heat is lost from the front plate surface into the environment. A part of the heat is conducted through the plate and dissipated there. A plate with thickness d conducts $P/A=\tfrac{\lambda }{d}({T}_{1}-{T}_{2})$ of power through it, where T₂ is the temperature of the dark side of the plate, and λ is the thermal conductivity.

The third task was about determining these material parameters for the provided black foamed PVC in the shape of a circular disc with thickness d = 3 mm:

This task is not concerned with human visual perception of light but with the actual energy carried by the light, regardless of its wavelength. Thus, we are using radiometric quantities, which are expressed with the conventional units of power. The radiometric analogue of the luminous flux is the radiant flux [W], and the analogue of illuminance is irradiance [W m^–2], which equals the radiant flux density at perpendicular incidence.

The relations involving the unknown coefficients indicate that we need to quantitatively determine the equilibrium temperatures of the plate as well as the incident radiant flux. Figure 8(a) shows an experimental setup. The front and back temperature of the plate can be measured with the provided IR thermometer—both are needed to extract the thermal conductivity λ, which depends on their difference. The power balance for the front and the back surface, accounting for the incident flux, dissipation into the environment, and exchange due to conductivity, yields the following set of equations:

$$\begin{eqnarray}&&j=h({T}_{1}-{T}_{0})+\displaystyle \frac{\lambda }{d}({T}_{1}-{T}_{2})\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&0=h({T}_{2}-{T}_{0})+\displaystyle \frac{\lambda }{d}({T}_{2}-{T}_{1}).\end{eqnarray} \tag{ 4 }$$

We are faced with the problem of how to extract h and λ out of the temperature measurements. Avoiding complex numerical procedures, the easiest way is to reinterpret the problem as finding the slope of a line. There are various different approaches—one is to take the sum and the difference of the above equations:

$$\begin{eqnarray}&&j=h({T}_{1}+{T}_{2}-2{T}_{0})\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&j=\left(h+2\displaystyle \frac{\lambda }{d}\right)({T}_{1}-{T}_{2}).\end{eqnarray} \tag{ 6 }$$

In this form, the coefficient h can be read out directly as the slope of the flux density j with respect to the sum of temperatures of both sides of the plate [see figure 8(b)], whereas λ requires an extra algebraic step [see figure 8(c)]. Forgetting the contribution of h to the temperature difference would lead to an overestimate of the thermal conductivity.

Zoom In Zoom Out Reset image size

The effective external temperature T₀ is difficult to measure, as it is a consequence of radiative and convective exchange with surrounding objects and walls that are not necessarily at the same temperature. Reading h as the slope of the fitted line avoids the need for measuring T₀, although 2T₀ can be read out as the intercept, which is a good sanity check of our measurements.

The incident flux density j relates to the power P with the same geometric relationship that was established in task 2 for photometric quantities, including the nondimensional factor C due to the anisotropic angular distribution of light:

$$\begin{eqnarray}&&j=\displaystyle \frac{P}{{{Cr}}^{2}}.\end{eqnarray} \tag{ 7 }$$

Without using the factor C, the quantitative value of the incident flux is incorrect. The flux density can be varied either by changing the distance r between the target and the light source, or by varying the power P. The light bulb here acts as a heater. The instructions state the idealized assumptions that the conversion from electricity to light is perfect and that the light is fully absorbed by the black plate. In reality, both assumptions hold only approximately. Some energy is spent on heating the air around the light bulb, and most black materials do not reach reflectivity below 10%. These considerations mean that distance variation is more appropriate than power variation for achieving different flux densities. The crux of the problem lies in the fact that the change in the power also changes the temperature (and thus the spectrum) of the light, as explored in the first part of the task. Such a measurement would rely on the assumption that the absorbance is independent of wavelength, which we cannot rely on, especially in the infrared range.

This task was the most sensitive and demanded careful attention due to the many factors that can affect the measurements, such as air currents, variations of the temperature of the surroundings, differences in temperature across the plate, and the intrinsic fluctuations of the thermometer readings. The confidence intervals of the slopes can be estimated graphically and can be improved by performing more measurements. The slope uncertainties have to be converted correctly into the uncertainties of the final parameters to fulfil the requirement of uncertainty analysis.

The subtask of estimating the albedo (reflectivity) of the white plate requires a repetition of the measurements done for the black sample. The main difference is that given albedo a, only the 1 − a proportion of incident flux is absorbed, giving

$$\begin{eqnarray}&&j=(1-a)\displaystyle \frac{P}{{{Cr}}^{2}}.\end{eqnarray} \tag{ 8 }$$

Assuming the previously obtained h and λ remain unchanged, the slope of the graph 8(b) thus gives h/(1 − a) instead of h. The ratio of the slopes leads to an estimate of the albedo, which is close to a ≈ 0.7 ± 0.1. This may sound low for a white surface but it is quite typical for general-purpose white pigments. To achieve a whiter appearance, we need specialised pigments or whitening agents that rely on fluorescence to increase visual whiteness. We must also take into consideration this measurement approximates the albedo in the peak range of our light source, which extends into the infrared range.

Deeper in the infrared range, around 10 μm, detected by the IR thermometer, both plates are good absorbers and emitters, which we also rely on to be able to use the infrared thermometer. If the plate was 'white' in that range, the infrared thermometer would give incorrect readings, but more importantly, the assumption that the heat transfer coefficient h, which includes the radiative exchange of heat around room temperature, is the same for both plates, would no longer hold in this case.

The above solution eliminates the need for measuring the ambient temperature and was given as the official solution because it reduces the impact of various sources of error by sampling a wider range of temperatures and distances. However, many students solved this task in different ways, some of which included a direct measurement of the ambient temperature. One way is to estimate the average ambient temperature of the surroundings by sampling it using the IR thermometer, which is a valid approach, but it deviates systematically from the T₀ intercept as it involves different measurement conditions. The emissivity of the measured materials is not the same, the chosen surface—such as the cardboard cubicle—is not a good representative of the effective temperature, and the heat exchange with warmer air in the cubicle is neglected. Reading out the plate temperature before the light source was turned on is a better way of estimating T₀, but it can also be understood just as part of the series of measurements of j(P) at P = 0. Using an explicit T₀ in equations (5) and (6) only shifts the origin of the plot, which does not affect the readout of the slope, as long as the trend line is not forced to go through the origin. Treating the origin as a privileged point was one of the mistakes we observed, which introduces a bias towards one measurement and skews the result. Conversely, if one wishes to determine T₀, using the intercept of the graph is more accurate than taking a single measurement because all measurements contribute to reducing the uncertainty of the result.

The incident radiative flux can also be varied in alternative, more or less correct ways—instead of changing the distance, one could vary the angle or the current through the bulb. With such tasks, the grader must thoughtfully and critically inspect each non-standard solution individually. Finally, a very common approach by the competitors was to only measure at two different input powers, most often one of them with the light switched off, eliminating the need to find a way to vary the incident radiative flux. This approach is much faster as it does not require graphing; equations (3) and (4) can be solved algebraically. The downside of this approach is that errors that vary with distance (or power) are not detected or compensated and can influence the precision of the result. Such solutions could receive full marks if the measurements at each incident flux were repeated to obtain their standard deviations and increase their precision.