Fog visibility

Vision is enabled by light scattering off of an object and taking a straight line path directly to your eye. On a clear day, nearly all of the light from an object makes it to your eye (assuming no large objects block the path).

But on a foggy day, the atmosphere is filled with droplets of water that redirect light in random directions if it hits them, blurring the image. As the fog gets thicker and thicker, faraway images become badly scrambled.

Suppose that on a foggy day there are roughly \(N_\text{droplet} = 200\) water droplets per cubic centimeter of atmosphere, and that each droplet is about \(d_\text{droplet} = \SI{20}{\micro\meter}\) in diameter. About how far away can an object be located before \(90\%\) of its image is scrambled by the fog?

Assume that when light scatters off of a water droplet, it goes off in a random direction.

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