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 temperature. Assume that the emissivity of both the spheres is the same. Find the ratio of (a) the rate of heat from the aluminium sphere to the rate of heat loss from the copper sphere and (b) the rate of fall of temperature of the the aluminium sphere to the rate of fall temperature of teh copper sphere. The specific heat capacity of aluminium = 900 J//kg -^(@)C and that of copper = 390 J//kg-^(@)C. The density of copper is 3.4 times the density of aluminium.

A solid aluminium sphere and a solid copper sphere of twice the radius

1.	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 temperature. Assume that the emissivity of both the spheres is the same. Find the ratio of (a) the rate of heat from the aluminium sphere to the rate of heat loss from the copper sphere and (b) the rate of fall of temperature of the the aluminium sphere to the rate of fall temperature of teh copper sphere. The specific heat capacity of aluminium = 900 J//kg -^(@)C and that of copper = 390 J//kg-^(@)C. The density of copper is 3.4 times the density of aluminium.
Answer» Solution :(a) `(u_(1))/(u_(2)) = (e sigma A_(1) T^(4))/(e sigma A_(2) T^(4)) = (r_(1)^(2))/(r_(2)^(2)) = ((r )/(2R))^(2) = (1)/(4)` (B) Rate of fall of temperature `(dT)/(dt) = (e sigma A)/(ms) (T^(4) - T_(0)^(4))` `= (e sigma 4 pi r^(2))/(RHO (4)/(3) pi r^(3) s) (T^(4) - T_(0)^(4))` `= (3 e sigma)/(rho RS) (T^(4) - T_(0)^(4))` `((dT//dt))/((dT//dt)) = (rho_(2) r_(2) s_(2))/(rho_(1) r_(1) s_(1)) = (3.4) ((2 r)/(r )) ((390)/(900))` ` = (2.95)/(1)`