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A metal ball of diameter 14 cm and mass ...

A metal ball of diameter 14 cm and mass 10kg is heated to a temperature of `227^(@)`C and suspended in a box whose waals are at a temperautre of `27^(@)`C. What is the maximum rate at which its temperature will fall ? Stefan's constant = `5.67xx10^(-8)Wm^(-2)K^(-4)`, specific heat capacity of the metal = `420 J kg^(-1)K^(-1)`.

Text Solution

Verified by Experts

The metal ball will emit maximum energy when its relative emittance e = 1 , i.e. when it is treated as a black body.
Maximum energy emitted by the ball per second is E (Fig. 12.10)
`E =eAsigma(T^(4)-T_(0)^(4))`
`e = 1, A = 4pir^(2)` where r = 0.07 m
`T = 227div273 = 500 K, T_(0) =27+273=300K`
`E =4xx3.14xx(0.07)^(2)xx5.67xx10^(-8)(500^(4)-300^(4))`
= 189.83 J/s
Let the fall in temperature of the ball be dT in one second. Then heat lost by the ball per second is
mc dT = 189.83
m = 10kg, c= 420 J `kg^(-1) K^(-1)`
`dT = 189.83/(10xx420)` = 0.045 K / second
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