Saved Bookmarks
| 1. |
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)` `=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))` |
|