A solid aluminium sphere and a solid copper sphere of twice the radius are heated to the same temperature and are allowed to cool under identical surrounding temperatures. Assume that the emisssivity of both the spheres is the same. Find ratio of (a) the rate of heat loss from the aluminium sphere to the rate of heat loss from the copper sphere and (b) the rate of fall of temperature of the aluminium sphere to the rate of fall of temperature of copper sphere. The specific heat capacity of aluminium `=900Jkg^(-1)C^(-1)` . and that of copper `=390Jkg^(-1)C^(-1)` . The density of copper `=3.4` times the density of aluminium.

Answer

(a) `1:4`
(b) `2.9 :1`

Explanation

`C `-cooling rate `H-` heat loss
`(H_(Al))/(H_(Cu)) = (sigmaA_(AI)(T_(AI^(4)-T_(0)^(4))))/(sigmaeA_(Cu)(T_(Cu^(4)-T_(0)^(4))))`
at `t = 0 T_(Al) = T_(Cu) =T =(A_(Al))/(A_(Cu)) = (R^(2))/((2R)^(2)) = (1)/(4)`
(b) `C-` cooling rate
`(C_(Al))/(C_(Cu)) = ((A_(Al))/(m_(Al)S_(Al)))((m_(Cu)S_(Cu))/(A_(Cu)))`
`= ((A_(Al))/(A_(Cu)))((m_(Cu))/(m_(Al)))((S_(Cu))/(S_(Al)))`
`= ((R )/(2R))^(2) xx {((2R)/(R))^(3)xx3.4}xx(390)/(900)=2.9`