The spectral absorptivity of aluminum coated with a thin layer of silicon dioxide may be approximated as \(\alpha_{\lambda, 1}=\) 0.98 for \(\lambda<\lambda_{c}\) and \(\alpha_{\lambda, 2}=0.05\) for \(\lambda \geq \lambda_{c}\) where the cutoff wavelength is \(\lambda_{c}=0.15 \mu \mathrm{m}\) under normal circumstances.

(a) Determine the equilibrium temperature of a flat piece of the coated aluminum that is exposed to solar irradiation, \(G_{S}=1368 \mathrm{~W} / \mathrm{m}^{2}\), on its upper surface. The opposite surface is insulated.

(b) The cutoff wavelength can be modified by varying the coating thickness. Determine the value of \(\lambda_{c}\) that will maximize the equilibrium temperature of the surface.