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How does the path of light become visible due to scattering of light?

According to me it only indicates the presence of colloidal particles, but my book gives the example of a cinema hall projector and says that you could see the path of light due to its scattering. Nevertheless, it's possible to see the lights path without the scattering.

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Light can only be seen when it enters your eye, so it somehow has to find a way between the light source and your cornea. Since light travels in more or less straight lines, it somehow has to change direction if you aren't looking into the source of said light directly.

This process is called scattering. It's what makes the path of light visible, because at every step on this path, the light gets scattered. This of course leads to a loss of intensity at the other end of the light path compared to a non-scattering medium.

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enter image description here

There are many types of scattering what you are talking about is generically called Tyndall effect. You can "see" the path of the light because some of the photons from the source are scattered in others direction so so they can reach your eyes. But in fact what you see are the photons that don't travel in straight trajectory, so not the beam itself but the path of the beam due to the photons that change direction due the effects of particles suspended in the medium.

Actually you can't see the light path without these effects...

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