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Let's say it's night, and I look at a lamp 20 meters away from me. I'm obviously in the darkness, but I can see the light from the lamp. If the photons need to stimulate my eyes in order for me to see, why isn't there light all the way to me, why so few of the photons get to me? But I can still see clearly what's going on near the lamp so why do some of the photons reach my eyes but some don't?

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There is light all around the lamp, and all around you (except of course for the points that light cannot reach because of walls or whatever). You just only perceive the light that comes into your eyes. The light that comes directly from the lamp is generally more intense than that reflected by the objects around you, so that you tend to ignore the latter.

You can "prove" this using a simple mirror. If you place a mirror anywhere around you, you can still see the light from the lamp reflected into it, thus "proving" that light is also present where you placed the mirror.

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intensity is defined as power/area, here, more energy that reaches your eyes, more light you observe,

By symmetry if we consider a isotropic (symmetrical) point source with total power being emitted P, intesity at any d(A) {small area} at the surface of imaginary sphere will be P/4π(r)^2 so, Intesity falls by second power of radial distance

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