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A solid copper sphere of density rho, sp...

A solid copper sphere of density `rho`, specific heat c and radius r is at temperature `T_1`. It is suspended inside a chamber whose walls are at temperature `0K`. What is the time required for the temperature of sphere to drop to `T_2`? Take the emmissivity of the sphere to be equal to e.

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The rate of loss of energy due to radiation`P=eAsigmaT^4`
This rate must be equal to `mc(dT)/(dt)`.
Hence ,`-mc(dT)/(dt)=eAsigmaT^4`
Negative sign is used as temperature decreases with time. In this equation`m=((4)/(3)pir^3)rho` and `A=4pir^2`
`-(dT)/(dt)=(3esigma)/(rhocr)T^4` or,`-int_0^tdt=(rrhoc)/(3esigma)int_(T^1)^(T_2)(dT)/(T^4)`
solving this, we get `t=(rrhoc)/(9esigma)((1)/(T_2^3)-(1)/(T_1^3))`.
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