Problems are listed in approximate order of difficulty. A single dot (•) indicates straigh...

Problems are listed in approximate order of difficulty. A single dot (•) indicates straightforward problems involving just one main concept and sometimes requiring no more than substitution of numbers in the appropriate formula. Two dots (••) identify problems that are slightly more challenging and usually involve more than one concept. Three dots (•••) indicate problems that are distinctly more challenging, either because they are intrinsically difficult or involve lengthy calculations. Needless to say, these distinctions are hard to draw and are only approximate.

• The total intensity I(T) radiated from a blackbody (at all wavelengths λ) is equal to the integral over all wavelengths, 0 < λ < ∞, of the Planck distribution (4.28) (Problem 1). (a) By changing variables to x = hc/λkBT, show that I(T) has the form

where σ is a constant independent of temperature. This result is called Stefan’s fourth-power law, after the Austrian physicist Josef Stefan, (b) Given that , show that the Stefan-Boltzmann constant σ is σ = 2π5kB4/15h3c2. (c) Evaluate σ numerically, and find the total power radiated from a red-hot (T = 1000 K) steel ball of radius 1 cm. (Such a ball is well approximated as a blackbody.)

Problem 1

The intensity distribution function I(λ, T) for a radiating body at absolute temperature T is defined so that the intensity of radiation between wavelengths λ and λ + dλ is

This is the power radiated per unit area of the body with wavelengths between λ and λ + dλ. The Planck distribution function for blackbody radiation is

where h is Planck’s constant, c is the speed of light, and kB is Boltzmann’s constant. Sketch the behavior of this function for a fixed temperature for 0 < λ < ∞. Explain clearly how you figured the trends of your graph. [Hint: You should probably think about the two factors separately.]