Question #43208

An equiconvex lens with both radii of curvature $R = 20 \text{ cm}$ is made of a photosensitive material. The material has an initial refractive index $n_0 = 1.50$ and a work function $\Phi = 2.5 \text{ eV}$. The lens is uniformly irradiated by two monochromatic light sources simultaneously:\n\nSource A: wavelength $\lambda_A = 310 \text{ nm}$, power $P_A = 6.4 \text{ mW}$\nSource B: wavelength $\lambda_B = 620 \text{ nm}$, power $P_B = 12.8 \text{ mW}$\n\nThe material's refractive index decreases as it emits photoelectrons, governed by the relation $n(t) = n_0 - \alpha N_e(t)$, where $N_e(t)$ is the total number of photoelectrons emitted up to time $t$, and $\alpha = 1.0 \times 10^{-15}$. The quantum efficiency of the material is $\eta = 1.0\%$ (i.e., $1.0\%$ of incident photons with energy greater than the work function eject an electron).\n\nAssuming the lens is kept in a vacuum and emitted electrons are continuously swept away, at what time $t$ will the lens act as a diverging lens with a focal length of $f = -40 \text{ cm}$?\n(Take $hc = 1240 \text{ eV}\cdot\text{nm}$ and $1 \text{ eV} = 1.6 \times 10^{-19} \text{ J}$)