A spherical black body of radius r radiates p | vedclass

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(D) For a black body,the power radiated $P$ is given by the Stefan-Boltzmann law: $P = \sigma A T^4$,where $\sigma$ is the Stefan-Boltzmann constant,$

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$(ii), (iv)$

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(D) For a black body,the power radiated $P$ is given by the Stefan-Boltzmann law: $P = \sigma A T^4$,where $\sigma$ is the Stefan-Boltzmann constant,$A$ is the surface area,and $T$ is the temperature.Since the body is spherical,$A = 4\pi r^2$,therefore $P \propto r^2$. Thus,statement $(ii)$ is correct.The rate of cooling $R$ is defined as $R = -\frac{dT}{dt} = \frac{P_{net}}{ms}$,where $m$ is the mass and $s$ is the specific heat capacity.$m = \rho V = \rho (\frac{4}{3}\pi r^3)$,where $\rho$ is the density.Substituting these into the expression for $R$: $R = \frac{\sigma (4\pi r^2) (T^4 - T_s^4)}{(\rho \frac{4}{3}\pi r^3) s} \propto \frac{r^2}{r^3} \propto \frac{1}{r}$.Thus,statement $(iv)$ is correct.Therefore,the correct relations are $(ii)$ and $(iv)$.

$A$ spherical black body of radius $r$ radiates power $P$ and its rate of cooling is $R$. Which of the following relations are correct?
$(i) \ P \propto r$
$(ii) \ P \propto r^2$
$(iii) \ R \propto r^2$
$(iv) \ R \propto \frac{1}{r}$

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$A$ spherical black body of radius $r$ radiates power $P$ and its rate of cooling is $R$. Which of the following relations are correct?$(i) \ P \propto r$$(ii) \ P \propto r^2$$(iii) \ R \propto r^2$$(iv) \ R \propto \frac{1}{r}$