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

Options

A. `72/7(r rho c)/(sigma)`
B. `7/72(r rho c)/(sigma)`
C. `27/7(r rho c)/(sigma)`
D. `7/27(r rho c)/(sigma)`

Answer

B

Explanation

`(dT)/(dt) = (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`].
`(100)/(dt) = (sigma 4 pi r^(2))/(4/3 pi r^(3)rho cJ) (200^(4)-0^(4))`
`rArr dt = (r rho c J)/(48 sigma) xx 10^(-6) s = (r rho c)/(sigma)*(4.2)/(48) xx 10^(-6)`
`=7/80 (r rhoc)/(sigma) mu s ~= (7)/(72) (r rho c)/(sigma) mu s [ "As" J = 4.2]`.