Solving the energy crisis, one giant construction project at a time

One famous idea to power an advanced civilization is that of the Dyson sphere, invented by Freeman Dyson (a very good physicist who was heavily involved in the development of quantum electrodynamics). The Dyson sphere consists of an artificial shell constructed around a star to capture a significant fraction of the star's energy.

Let's assume the sun radiates as a perfect black body emitter with a radius of \(7 \times 10^8~\mbox{m}\) and a surface temperature of \(6000 ~\mbox{K}\). What fraction of the energy emitted by the sun must a Dyson sphere capture to satisfy the current energy needs of the human population of the earth, which on average is consumed at about \(15~\mbox{terawatts}\)?

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