Question #43626

A solid, highly thermally conductive sphere of radius $R$ is placed in deep space at an ambient temperature of $0\text{ K}$. A broad, unidirectional monochromatic laser beam of wavelength $\lambda_0$ and uniform photon flux $\Phi$ (number of photons per unit area per unit time) illuminates the sphere. The surface of the sphere has a wavelength-dependent spectral hemispherical absorptivity $\alpha(\lambda)$ given by $\alpha(\lambda) = \alpha_1$ for $\lambda < \lambda_c$ and $\alpha(\lambda) = \alpha_2$ for $\lambda \ge \lambda_c$. Given that $\lambda_0 < \lambda_c$ and the sphere's steady-state temperature $T$ is such that virtually all of its thermal emission occurs at wavelengths $\lambda > \lambda_c$, which of the following expressions gives the steady-state temperature $T$ of the sphere? (Assume $h$ is Planck's constant, $c$ is the speed of light, and $\sigma$ is the Stefan-Boltzmann constant).