A) \[\frac{4\pi {{r}^{2}}\sigma \,{{t}^{4}}}{{{R}^{2}}}\]
B) \[\frac{{{r}^{2}}\,\sigma \,{{(t+273)}^{4}}}{4\pi {{R}^{2}}}\]
C) \[\frac{16\,{{\pi }^{2}}\,{{r}^{2}}\,\sigma {{t}^{4}}}{{{R}^{2}}}\]
D) \[\frac{{{r}^{2}}\,\sigma \,{{(t+273)}^{4}}}{{{R}^{2}}}\]
Correct Answer: D
Solution :
From Stefan's law, the rate at which energy is radiated by sun at its surface is |
\[P=\sigma \times 4\pi {{r}^{2}}{{T}^{4}}\] |
[Sun is a perfectly black body as it emits radiations of all wavelengths and so for it \[e=1\]] |
The intensity of this power at earth's surface (under the assumption \[R>>{{r}_{0}}\]) is |
\[I=\frac{P}{4\pi \,{{R}^{2}}}\] |
\[=\frac{\sigma \times 4\pi \,{{r}^{2}}{{T}^{4}}}{4\pi {{R}^{2}}}\] |
\[=\frac{\sigma \,{{r}^{2}}{{T}^{4}}}{{{R}^{2}}}\] |
\[=\frac{\sigma {{r}^{2}}\,{{(t+273)}^{2}}}{{{R}^{2}}}\] |
where \[\sigma \] is the Stefan's constant. |
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