Use Planck’s formula for blackbody radiation R(f , T) in terms of frequency to approximate the location of the power peak in R(f , T) as a function of T (i.e., for fixed T, find the f that maximizes power). Show that this is roughly linear in T. (This is known as Wien’s displacement law.) To do so, fix T, and define a variable u = f /T, and write R(f , T) as a function S(u) = T3 u3 D(u) , where D(u) = e(h/k)u − 1, which depends only on the single variable u. Under what condition on u is S(u) a maximum? (You may assume that e(h/k)u >> 1.)
For T near 10,000◦K (roughly the temperature of the surface of the sun), the peak is at about frequency 5 × 1014 Hz, which is just about in the middle of the range of visible light (roughly 4 × 1014 Hz to 7 × 1014 Hz); this means that the human visual system is most sensitive to the energy that’s most common (sunlight).
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