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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. |
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Answer» <P> `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` |
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