Question

A spherical black body of radius $r$ radiates power $P$, and its rate of cooling is $R$.

• A. $P\propto r$
• B. $P\propto r^2$
• C. $R\propto r$
• D. $R\propto \frac{1}{r}$

Answer 1

Answer: (B), (D)

$P\propto r^2$
$R\propto \frac{1}{r}$

The power at which the body radiates is directly proportional to area:
$P\propto A$
$P\propto r^2$
$P=mC\frac{dT}{dt}=\frac{4}{3}\pi r^{3}DS\frac{dT}{dt}$
i.e.
$r^{3}DS\frac{dT}{dt}\propto r^2$
$\frac{dT}{dt}\propto \frac{1}{RDS}$
i.e. Rate of fall of temperature is $R$ $\propto \frac{1}{r}$