Time Delay of Photoelectric Effect

A metal plate is positioned 5 m away from a monochromatic light source with an output power $P_0 =10^{-3} W$. Consider that the electrons emitted by the plate (by photoelectric effect) collect their energies from a circular surface having a radius equal to that of 10 atoms ($R_{tot} =10*R_{atom}=10^{-9} m$). The energy necessary to extract one electron from an atom (the work function) is $W =5 eV$. Assuming light is a wave, what is the time necessary for such a metal “target” to absorb this energy from the light source?

The area of the target surface from which one electron is absorbing the energy is

$S_1 =\pi*R^2 =\pi*(10^{-9})^3 \approx 3*10^{-18} m^2$

The surface of the sphere with radius $R=5m$ (distance to plate) having in its center the light source is

$S_2 =4\pi*R^2 =4\pi*5^2 \approx 300 m^2$

By admitting that the source emits light uniformly (in all directions), the power on the target (for one electron) is

$P_e =P_0*(S_1/S_2) =10^{-3}*(3*10^{-18}/300) = 10^{-23} W$

If all the energy on the target is also absorbed by the electron, the time for the electron “to collect” this energy and to be emitted from the atom is

$t = (E/P_e) =(5*1.6*10^{-19}) /10^{-23} \approx 22 hours$