Question #43628

A perfectly black spherical target of radius $R$ ($\text{m}$), density $\rho$ ($\text{kg m}^{-3}$), and specific heat capacity $s$ ($\text{J kg}^{-1}\text{K}^{-1}$) is initially at temperature $T_0$ ($\text{K}$). It is placed in a vacuum chamber with surroundings at $0\text{ K}$ and cools purely by thermal radiation. The instantaneous radiated power $P(t)$ ($\text{W}$) is continuously measured. In a parallel Davisson-Germer diffraction setup, electrons of mass $m_e$ ($\text{kg}$) are accelerated from rest such that their final kinetic energy $K$ ($\text{J}$) at any time $t$ ($\text{s}$) is exactly equal to $P(t) \cdot t_0$, where $t_0$ ($\text{s}$) is a constant time factor. Which of the following expressions correctly represents the de Broglie wavelength $\lambda(t)$ ($\text{m}$) of these electrons as a function of time $t$? (Assume $\sigma$ is the Stefan-Boltzmann constant in $\text{W m}^{-2}\text{K}^{-4}$ and $h$ is Planck's constant in $\text{J s}$).