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