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thephysicsman
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Homework Statement
Two black bodies, a sphere and a disc, are situated at the same distance a from the sun. Calculate the ratio of their temperatures.
Homework Equations
Stefan-Boltzmann law:
[tex]U = \sigma T^4[/tex]
The Attempt at a Solution
At a distance a, the irradiance from the sun is
[tex]U = \frac{r_{sun}^2}{a^2}\cdot \sigma T_{sun}^4 [/tex]
The sphere absorbs
[tex]\pi r^2 \frac{r_{sun}^2}{a^2}\sigma T_{sun}^4[/tex]
and emits
[tex]4\pi r^2 \sigma T_{sphere}^4[/tex]
Equating the absorption and the emission, I get
[tex]T_{sphere}=T_{sun}\sqrt{\frac{r_{sun}}{2a}}[/tex]
The disc absorbs the same as the disc, and emits
[tex]2\pi r^2 \sigma T_{sphere}^4[/tex]
Equating the absorption and emission:
[tex]T_{disc}=T_{sun}\sqrt{\frac{r_{sun}}{a}}\cdot \frac{1}{2^{1/4}}[/tex]
[tex]\frac{T_{sphere}}{T_{disc}}=0.84[/tex]
Correct?
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