Transmission Of Heat Question 127
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
[IIT-JEE 1991]
Options:
A) $ \frac{72}{7}\frac{r\rho c}{\sigma } $
B) $ \frac{7}{72}\frac{r\rho c}{\sigma } $
C) $ \frac{27}{7}\frac{r\rho c}{\sigma } $
D) $ \frac{7}{27}\frac{r\rho c}{\sigma } $
Show Answer
Answer:
Correct Answer: B
Solution:
$ \frac{dT}{dt}=\frac{\sigma ,A}{mc},,(T^{4}-T _{0}^{4}) $ [In the given problem fall in temperature of body $ dT=(200-100)=100K $ , temp. of surrounding T0 = 0K, Initial temperature of body $ T=200K]. $
$ \frac{100}{dt}=\frac{\sigma 4\pi r^{2}}{\frac{4}{3}\pi r^{3}\rho ,c,J}(200^{4}-0^{4}) $
Therefore $ dt=\frac{r\rho ,c,J}{48\sigma }\times {{10}^{-6}}s=\frac{r\rho ,c}{\sigma }.\frac{4.2}{48}\times {{10}^{-6}} $
$ =\frac{7}{80}\frac{r\rho ,c}{\sigma }\mu ,s\tilde{}\frac{7}{72}\frac{r\rho ,c}{\sigma }\mu ,s $ [As J = 4.2]
BETA
