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

The energy emitted per second when the temperature of the copper sphere is `T` and the surrounding is temperature `T_(0)`
`=sigma(T^(4)-T_(0)^(4))xxA` (i)
Here `T_(0)=0K`
We know that
`dQ= -mcdt`
`:. (dQ)/(dt)=-mc(dT)/(dt)` (ii)
Here the `-ve` sign shows that the temperature is decreasing with time.
Energy emitted per second from Eqs. (i) and (ii)
`=sigmaT^(4)A=-mc(dT)/(dt)`
`implies dta=-(mcdT)/(sigmaT^(4)A)=-(pxx(4)/(3)pir^(3)cdT)/(sigmaT^(4)xx4pir^(2))`
`[:. m=pxx(4)/(3)pir^(3)]`
`implies dt=-(prc)/(3sigma)(dT)/(T^(4))`
Integrating both sides,
`int_(0)^(t)dt=-(prc)/(3sigma)int_(200)^(100)(dT)/(T^(4))=-(prc)/(3sigma)[-(1)/(3T^(3))]_(200)^(100)`
`t=(prc)/(9sigma)[(1)/((100)^(3))-(1)/((200)^(3))]`
`t=(7prc)/((72xx10^(6)sigma))`