Question #43469

A nuclear battery contains $N_0$ radioactive nuclei at $t=0$, which decay with a decay constant $\lambda$ (in $\text{s}^{-1}$). Each decay event emits exactly one photon of wavelength $\lambda_p$ (in $\text{m}$). All emitted photons are incident on a semiconductor photovoltaic cell with a band-gap energy $E_g$ (in $\text{J}$), where $E_g < \frac{hc}{\lambda_p}$ ($h$ is Planck's constant, $c$ is the speed of light). Due to thermalization, the excess energy of each photon is lost as heat, and the maximum electrical energy generated per absorbed photon is $E_g$. The instantaneous electrical power output of the cell is given by $P(t) = \eta(t) \Phi(t) E_g$, where $\Phi(t)$ is the instantaneous incident photon flux (photons per second). The dimensionless efficiency factor $\eta(t)$ is directly proportional to the incident photon flux $\Phi(t)$. At $t=0$, this factor is $\eta_0$. What is the total electrical energy generated by the cell from $t=0$ to $t \to \infty$?