As shown in figure, there is a thick spherical shell with the walls coated with 'lamp black'. A point source which generates thermal energy at a constant rate 'P' is placed at the centre S of the shell. Derive an expression for the temperature T at point M in steady state, where SM = 1.5 R. Your expression would be as follows :
`T = ((P)/(sigma16piR^(2)))^(1//4)+((P)/(3xxpiKR))`
Here K is the coefficient of thermal conductivity of material of shell and `sigma` is Stefan's constant. Find the value of x.

Answer

8

Explanation

`P=sigma4pi(2R)^(2)T_(S)^(4)`
`dR=(dr)/(k4pir^(2))`
`R_(eq)=(1)/(4pik)underset(r)overset(2R)(int)(dr)/(r^(2))," "R_(eq)=-(1

)/(4pik)[(1)/(r)]^(2R)`
`R_(eq)=-(1)/(4pik)[(1)/(2R)-(1)/(r )]=(1)/(4pik)((1)/(r)-(1)/(2R))=((2R-r))/(8pikRr)`
`P=((T-T_(S))8pikRr)/((2R-r))=(T-T_(S))(24Rpik)`
`T=T_(S)+((P)/(24Rpik))=((P)/(16pisigmaR^(2)))^((1)/(4))+((P)/(24Rpik))`
`x=8`