A) \[\text{S}\propto {{\text{T}}^{\text{4}}}\]
B) \[\text{S}\propto {{\text{T}}^{2}}\]
C) \[\text{S}\propto {{\theta }^{2}}\]
D) \[\text{S}\propto \theta \]
Correct Answer: A
Solution :
Let radius of sun = R Distance of earth from the sun = d Power radiated from the sun \[=4(\pi {{R}^{2}})\sigma {{T}^{4}}=P\] Energy received/area/second \[=S=\frac{P}{4\pi {{d}^{2}}}\] \[=4\pi {{R}^{2}}\sigma \frac{{{T}^{4}}}{4\pi {{d}^{2}}}\] \[=\sigma {{T}^{4}}\frac{{{R}^{2}}}{{{d}^{2}}}\] \[=\frac{1}{4}\sigma {{T}^{4}}{{\left( \frac{2R}{d} \right)}^{2}}\] Angle subtended by sun at earth \[\alpha =\frac{2R}{d}\] S = constant \[\times {{T}^{4}}\times {{\alpha }^{2}}\] \[S\propto {{T}^{4}}\].
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