Question

A solid copper sphere (density $\rho$ and specific heat capacity (c) of radius r at an initial temperature 200K is suspended inside a chamber whose walls are at almost 0K. The time required (in $\mu$ s) for the temperature of the sphere to drop to 100 K is

• A. $\frac{80}{7}\frac{r\rho c}{\sigma }$
• B. $\frac{7}{80}\frac{r\rho c}{\sigma }$
• C. $\frac{27}{7}\frac{r\rho c}{\sigma }$
• D. $\frac{7}{27}\frac{r\rho c}{\sigma }$

Answer 1

The correct option is B $\frac{7}{80}\frac{r\rho c}{\sigma }$

$\frac{dT}{dt}=\frac{\sigma A}{mcJ}(T^4-T_0^4)$ [In the given problem fall in temperature of body dT=(200-100)=100K , temp. of surrounding $T_0=0K$, Initial temperature of body T=200k].
$\frac{100}{dt}=\frac{\sigma 4\pi r^2}{\frac{4}{3}\pi r^3\rho cJ}({200}^4-0^4)$
$\Rightarrow dt=\frac{r\rho cJ}{48\sigma }\times {10}^{-6}s=\frac{r\rho c}{\sigma }.\frac{4.2}{48}\times {10}^{-6}$ [As J = 4.2]
$\frac{7}{80}\frac{r\rho c}{\sigma }\mu s$