Physics

Problem

A star with surface area

5.0\times10^{12}\,\text{m}^{2}

normally radiates uniformly at a temperature of

5.0\times10^{3}\,\text{K}

. During a brief flare, the surface temperature jumps to

6.0\times10^{3}\,\text{K}

and remains at that value for

15\,\text{min}

before returning to normal. Assuming the surface area stays unchanged and using

\sigma = 5.67\times10^{-8}\,\text{W}\,\text{m}^{-2}\,\text{K}^{-4}

, what is the additional energy radiated by the star because of the flare (compared with the energy it would have emitted had the flare not occurred)? a)

1.7\times10^{22}\,\text{J}

3.3\times10^{23}\,\text{J}

1.7\times10^{23}\,\text{J}

1.3\times10^{23}\,\text{J}

Intuitive explanation

The key is the Stefan–Boltzmann law: power radiated depends on $T^4$ , so even a modest temperature increase produces a dramatic boost in energy output. Calculate the power at each temperature using $P = \sigma A T^4$ , find the difference in power, then multiply by the flare duration to get the extra energy released. The $T^4$ dependence means temperature changes matter far more than you might initially expect.

Intuitive explanation

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