Rate of interaction of free electron with photons from sunlight How many photons does a free electron (in sunlight say) interact with per second? 
I did a rough calculation assuming the electron interacts with any photon that enters through an area the size of the Thomson cross-section, and that the light is monochromatic with angular frequency 2x10^14 Hz and electric field of 1000 Vm^-1. And the power through this area is the Poynting vector times the area. Then the rate of interactions is the power divided by the energy of one photon:
$$ \text{rate} = \frac{\epsilon_0 c E_0^2}{2 \hbar \omega} \frac{8 \pi}{3} r_e^2 \approx 4\times10^{-6}s^{-1}$$
where $r_e$ is the classical electron radius.
But this means it takes 3 days to interact once which doesn't sound right. I may have made some bad assumption or something here. 
I'd like to know if this approach is valid.
If not, where, conceptually, does this approach fail and how should it be improved?
A possible avenue to answer the question could be to describe a known calculation of the interaction rate between free electrons and light and point out how to connect it to this problem.
 A: The electron doesn't absorb the photons, it scatters them. The energy absorbed by an interaction depends on the scattering angle - this can be determined using the Compton formula for the wavelength of the scattered photon. And the electron tends to scatter more at different angles (proportional to the Thomson differential cross section). From this I found the rate of interactions to be more like once every 30 seconds.
A: The scattered power from an electron is just $N \sigma_T$, where $N$ is the incident power per unit area and $\sigma_T$ is the Thomson scattering cross-section of $6.6\times 10^{-29}$ m$^2$. For visible light there is no need to consider quantum effects and Compton scattering.
If each photon has an energy of $h\nu$ or $hc/\lambda$, then the rate of scattering interactions is
$$R = \frac{N\sigma_T \lambda}{hc}\ . $$
For sunlight at the Earth, $N\sim 1300$ W m$^{-2}$ and of $\lambda = 500$ nm, then $R = 2\times 10^{-7}$ s$^{-1}$ or once every 50 days.
I disagree with you only in the sense that your assumed frequency is not correct for visible light.
