Question

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 …….

Answer 1

Let at any time temperature of body is `T`.
According to Stefan-Boltzmann law, energy radiated per unit time by the body
`(du)/(dt) = sigma A (T^(4) - T_(0)^(4)) = - ms (dT)/(dt)`
`sigma. 4 pi r^(2) (T^(4) - 0) =- rho. (4)/(3) pi r^(3) c (dT)/(dt)`
`int_(200)^(300) (dT)/(T^(4)) = - (3 sigma)/(rho rc) int_(0)^(t) dt`
`(T^(-3))/(-3) = - (3 sigma)/(rho rc)`
`|(1)/(T^(3))|_(200)^(100) = (9 sigma t)/(rho rc)`
(i) `(1)/((100)^(3)) - (1)/((200)^(3)) = (9 sigma t)/(rho rc)`
(ii) `((8 - 1))/((200)^(3)) = (9 sigma t)/(rho rc)`
`t = (7 rho rc)/(9 (200)^(3) sigma) = (7)/(72) (rho rc)/(sigma) xx 10^(-6)`