Question

I have an issue with the official solution to this problem from BelPhO:

Visibility on the road is 100 m. Assuming that the diameter of a fog droplet is 1 micron, estimate the concentration of fog droplets in the air.

The official solution is:

If a drop of water gets in the way of a photon, the photon will be scattered or absorbed. To estimate the average flight length of a photon, the following reasoning can be carried out: in cylinder of diameter equal to the diameter equal to the diameter of the droplet and length equal to the free span on average there should be one droplet, i.e.

l πd²/ 4n≈1 ⇒ n≈4/ πd²l ≈ 4/ π(10−⁶)²⋅100≈10¹⁰m−³

I cannot see how this physical situation described by the solution can best fit the model of the initial problem. The only information we are given is the visible radius l
and the size of a droplet. Do you not think (as I think) that the solution admits the untrue assumption that the visibility distance is equal to the mean free path? Doesn't it seem more reasonable to you that the fraction of the surface of a sphere of diameter d
which is obscured depends directly on the number of droplets within the visible radius, which is something proportional to the concentration n
of the fog droplet times a volume?

Answer 1

The official solution to the BelPhO problem estimates the concentration of fog droplets using a simplistic model, but the student rightly suggests considering the collective effect of all droplets within the volume of visibility for more accuracy.

Step-by-step explanation:

The concept of visibility on the road, tied with the diameter of fog droplets to estimate their concentration, is a question that blends optics with atmospheric physics. The official solution posited in the BelPhO problem uses a simplistic model associating visibility distance with the mean free path of a photon. The estimate given by n ≈ 4 / (πd2l) suggests the concentration n of the fog droplets is approximately equal to 1010 m-3, where d is the diameter of a droplet and l is the visibility distance.

However, you raised a legitimate concern. This model does not fully address how the volume of air within the visibility radius and the resulting light absorption or scattering by multiple droplets affect visibility. A more realistic approach would indeed consider the proportion of obscured surface on a larger scale, incorporating the number of droplets throughout the observed volume, rather than just considering the interaction with a single droplet.