A solid sphere of radius 'R' density `rho` and specific heat 'S' initially at temperature `T_(0)` Kelvin is suspended in a surrounding at temperature 0K. Then the time required to decrease the temperature of the sphere from `T_(0)` to `(T_(0))/(2)` kelvin is (Assume sphere behaves like a black body)

Options

A. `(rhoSR)/(9sigmaT_(0)^(3))`
B. `(5rhoSR)/(9sigmaT_(0)^(3))`
C. `(7rhoSR)/(9sigmaT_(0)^(3))`
D. `(8rhoSR)/(9sigmaT_(0)^(3))`

Answer

C

Explanation

`-mS(dT)/(dt)=sigmaAT^(4)`
`prho(4)/(3)piR^(3)S(dT)/(dt)=sigma4piR^(2)T^(4)`
`underset(T_(0))overset(T_(0)//2)int(dT)/(T^(4))=(3sigma)/(rhoSR)underset(0)overset(t)intdt`
`[(1)/(3T^(3))]_(T_(0))^(T_(0)//2)=(3sigma)/(rhoSR)t`
`((8)/(T_(0)^(3)))-(1)/(T_(0)^(3))=((9sigmat)/(rhoSR))`
`thereforet=(7rhoSR)/(9sigmaT_(0)^(3))`