Question

A sphere of radius $R$, density $D$ and specific heat $S$ is heated to temperature $\theta$ and surrounding is at temperature ${\theta }_0$. Its rate of fall of temperature will be proportional to

• A. $DS/R$
• B. $R/DS$
• C. $1/RDS$
• D. $RDS$

Answer 1

The correct option is C $1/RDS$

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 $\propto \frac{1}{RDS}$