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Why is it that the incoming radiation is spread over an area of a disc = πr^2 rather than half the surface area of the earth?

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The Earth intercepts an amount of solar radiation equivalent to that falling on a disc with its same radius, facing the Sun, but the Earth itself is (roughly) spherical, so that radiation will spread over half its surface area.

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The shadow of the Earth indicates the light area that it absorbs. The shadow is a disk. Remember that light that strikes at an angle (like sunlight that strikes the Earth at dawn or dusk) is less intense.

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  • $\begingroup$ If that's the case then πr^2 must be an approximation. Even if the shadow is a circular disc, that does not mean that the light is being distributed on one. $\endgroup$
    – Arth Gupta
    Nov 3 '16 at 21:22
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First assume the light source is far enough away that all light rays are parallel. Then the shadow would be the disk you refer to. This means that that much light struck the Earth. However, that light is not distributed evenly over the lit half of the Earth. (That is why we have seasons).

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