If there is a beam of light from a source in space a million light years away travelling towards the earth, does gradually start to become brighter as it gets closer and fades away as it moves further?

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If that is true and the source suddenly disappears, does it mean there is an “arrow” like of light travelling in space until it “goes away” ?

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No, the intensity of light decreases over distance from the point of origin.

Think about what happens if you throw a rock into a pond. When the ripple starts to spread, it starts off large around where the rock hits the water, and as the ripple spreads out, the height of it gets lower and lower.

It’s the same with light, as the beam of light shines from the source, the area that the beam covers gets larger and larger, but since the beam has the same amount of light, it gets dimmer and dimmer as the area gets larger and larger.

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