Stefan-Boltzmann Law

The Stefan-Boltzmann law describes the power radiated from a black body radiator.  The Watts radiated = s (T₁⁴ - T₂⁴) , where s = (2p ⁵ k⁴ )/(15c² h³) — a constant, and T₁ is the (black body) temperature (in K) and T₂ is the ambient temperature. For this exercise, T₁ was  3100K, and T₂ was 293K (20C) at 12W radiated power.  The first term represents radiation of the black body to ambient. The second term represents radiation of the ambient to the black body.  (T₂ is so small as to have essentially no effect for this discussion.  It is included for completeness)  A tungsten filament may be approximated by a black body.

Given the above equation for 12W radiated, and T₁ = 3100K, then s can be determined.  Given this, one can determine T₁ for other radiated powers.  In this exercise, 9W yields a T₁ of  2884K, and 6W yields a T₁ of 2607K.

Efficacy (lumens/watt) measures the visual efficiency of light -- how well we see it.   The efficacy changes dramatically for filaments of varying temperature.  Shown here is a curve of efficacy versus ideal radiator temperature.   Note that the highest practical filament temperature is about 3200K.  The sun radiates at 5785K - -near the peak of the curve.  A filament radiating at 3100K is about 22 LpW.  A filament radiating at 2600K is only about 8 LpW.