Question

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

• A. $\frac{7r\rho C}{72\times {10}^6\sigma }$
• B. $\frac{r\rho C}{6\sigma }$
• C. $\frac{r\rho C}{\sigma }$
• D. none of the above.

Answer 1

The correct option is A $\frac{7r\rho C}{72\times {10}^6\sigma }$

In this question, we will equate the heat loss from the sphere to heat loss due to radiation.

$mC\frac{dT}{dt}=\sigma A{T}^4$

$\left(\frac{4}{3}\pi r^3\right)\left(\rho \right)C\frac{dT}{dt}=\sigma \left(4\pi r^2\right)T^4$

$\frac{r\rho C}{3\sigma }{\int }_{100}^{200}\frac{dT}{T^4}={\int }_0^{t}dt$

$\frac{rC\rho }{9\sigma X{10}^6}(1-\frac{1}{8})=t$

$t=\frac{7r\rho C}{72\times {10}^6\sigma }$