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Basically I wonder what happens to emitted radiation by douubling a light e.g. twice the surface area of the sun will emit how much more radiation? 4 times more? Is there a formula?

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At fixed temperature - and/or fixed chemical composition, even when the radiation is not exactly thermal - the total power radiated is clearly proportional to the area A. – Luboš Motl Dec 30 '12 at 9:42
... so double the area just means double the radiation. See for example However note that a star of double the surface area of the Sun will not emit twice the energy since it will be at a different temperature. – John Rennie Dec 30 '12 at 10:31
up vote 1 down vote accepted

If you assume a radiator (sun, lightbulb) of constant power (e.g. 60 W), you can increase its surface in any way you like (assuming the material only diffracts the radiation and doesn't absorb it), but it will always emit the same power, which is (e.g.) 60W. This is, simply, conservation of energy. It's not a matter of temperature.

If you are, on the other hand, talking about black body radiation, Wikipedia states that the emitted radiation is proportional to the surface. Double the surface, and you will double the radiation.

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