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I was expecting the relationship between these two to be linear, but after conducting an experiment my data fits almost exactly on the curve, $y = 2800.2 x^{2.0173}$, where $y$ is the brightness in Lux and $x$ is the power supplied in Watts.

Is this due to some property of filament lamps which I have missed, or is my data simply inaccurate?

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    $\begingroup$ This is probably because your filiment lamps are non-ohmic (meaning that voltage and current are not proportional, the fact used to correlate power and brightness). In general, most lightbulbs are non-ohmic resistors. $\endgroup$ – Cicero May 27 '15 at 23:02
  • $\begingroup$ How did you measure perceived brightness? If you used the visible spectrum, the increasing temperature as the power goes up will shift more of the output into the visible. If you measure total output over the whole spectrum, the relationship should be linear. $\endgroup$ – Ross Millikan May 28 '15 at 4:22
  • $\begingroup$ @RossMillikan Thankyou! I used a photometer, which I believe only measured light emitted in the visible spectrum, so that may well be why my readings turned out non liniear. $\endgroup$ – user3129805 May 28 '15 at 14:01
  • $\begingroup$ @Cicero perhaps I'm missing something, but it appears to me that the non-ohmic nature of the filament would in fact mean that the resistance is increasing as power increases, meaning that my graph's gradient should have been decreasing rather than increasing as power supplied went up $\endgroup$ – user3129805 May 28 '15 at 14:17
  • $\begingroup$ The increasing resistance should not be a problem if you truly measure power in. It takes more voltage to push a given current, but power in should equal light out as long as you measure all the radiation. $\endgroup$ – Ross Millikan May 28 '15 at 15:14
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How did you measure perceived brightness? If you used the visible spectrum, the increasing temperature as the power goes up will shift more of the output into the visible. If you measure total output over the whole spectrum, the relationship should be linear. The increasing resistance of the bulb should not be a problem if you truly measure power in. It takes more voltage to push a given current, but power in should equal light out as long as you measure all the radiation. The filament of a light radiates much like a black body The temperature will rise as the fourth root of the power. The wavelength of maximum output will decrease as the inverse temperature.

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If you are looking at visual brightness, then you have to fold the wavelength dependent sensitivity of the human eye to the approximate black body spectrum of the filament into the calculation. At low power the filament will emit mostly infrared radiation, which is not visible. Even at the max. temperature of practical filaments the color temperature of the radiation is well below the sensitivity maximum of the human eye, which is in the green part of the spectrum, but as the power goes up, ever more of the emitted light falls into the visual part of the spectrum and the brightness increases more than proportionally with the total power.

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  • $\begingroup$ Thank you! This, just like the comment from RossMillikan, actually explains a lot, but is there a specific formula or law which relates power input to the wavelength of the light emitted? This would maybe help to explain the specific, (almost) quadratic equation that I got from my data. $\endgroup$ – user3129805 May 28 '15 at 14:22
  • $\begingroup$ Stefan-Boltzman tells you the total power emitted for a given temperature: en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_law. You can turn this around and estimate the temperature dependence as a function of power. The spectrum at that temperature is then given by Planck's law en.wikipedia.org/wiki/Planck's_law. Plug this into the human eye response en.wikipedia.org/wiki/Luminosity_function and you can estimate a physiological intensity response as a function of power. $\endgroup$ – CuriousOne May 28 '15 at 16:52

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