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

A, C

Explanation

The energy emitted per second when the temperature of the
copper sphere is T and the surrounding temperature `T_0`
`=sigma(T^4-T_0^4)xxA=sigma.T^4A[:'T_0=0]…..(i)`
We know that `dQ=mcdT rArr (dQ)/(dt)=mc (dT)/(dt)…..(ii)`
From (i) and (ii)
`sigmaT^4A=mc (dT)/(dt)`
`rArrdt=(mcdT)/(sigmaT^4A)=(rhoxx4/3pir^3cdT)/(sigmaT^4xx4pir^2) [:'m=rhoxx4/3pir^3]`
`rArrdt=(rhorc dT)/(3sigmaT^4)`
Integrating both sides
`int_0^tdt=(rhorc)/(3sigma)int_200^100(dT)/(T^4)=(rhorc)/(3sigma)[-1/(3T^3)]_200^100`
`t=-(rhorc)/(9sigma)[1/((100)^3)-1/((200)^3)]`
`t=(7rhorc)/((72xx10^6)sigma)~~(7rhorc)/((72xx10^6)(5.67xx10^-8))=1.71rhorc