A small-blackened solid copper sphere of radius 2.5 cm is placed … - Vidyadip

Question

A small-blackened solid copper sphere of radius $2.5\ cm$ is placed in an evacuated chamber. The temperature of the chamber is maintained at $100^{\circ} C$. At what rate energy must be supplied to the copper sphere to maintain its temperature at $110^{\circ} C$ ? (Take Stefan's constant $\sigma$ to be $5.670 \times 10^{-8} J s ^{-1} m^{-2} K^{-4}, \pi=3.1416$ and treat the sphere as a blackbody.)

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Answer

Data $: r=2.5 cm =2.5 \times 10^{-2} m , T _0=273+100=373 K , T =273+110=383 K _t$
$\sigma=5.67 \times 10^{-8} J s ^{-1} m ^{-2} k ^{-4}$
The rate at which energy must be supplied
$ \sigma A\left(T^4-T_0^4\right)=\sigma 4 \pi r^2\left(T^4-T_0^4\right)$
$=\left(5.67 \times 10^{-8}\right)(4)(3.142)\left(2.5 \times 10^{-2}\right)^2\left(383^4-373^4\right)$
$=(5.67)(4)(3.142)(6.25)\left(3.83^4-3.73^4\right) \times 10^{-4}$
$=0.9624 W $

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