A radiation thermometer is a device that responds to a radiant flux within a prescribed spectral interval and is calibrated to indicate the temperature of a blackbody that produces the same flux.

  (a) When viewing a surface at an elevated temperature T_s and emissivity less than unity, the thermometer will indicate an apparent temperature referred to as the brightness or spectral radiance temperature T_λ. Will T_λ be greater than, less than, or equal to T_s?

  (b) Write an expression for the spectral emissive power of the surface in terms of Wien's spectral distribution (see Problem 12.27) and the spectral emissivity of the surface. Write the equivalent expression using the spectral radiance temperature of the surface and show that

                         1  =  1  +  λ   Inɛ_λ

                         T_s    T_λ      C₂

where λ represents the wavelength at which the thermometer operates.

  (c) Consider a radiation thermometer that responds to a spectral flux centered about the wavelength 0.65 μm. What temperature will the thermometer indicate when viewing a surface with ɛ_λ (0.65 μm) = 0.9 and T_s = 1000 K? Verify that Wien's spectral distribution is a reasonable approximation to Planck's law for this situation.