A sphere of surface area $1.25 \mathrm {~m} ^ { 2 }$ and emissivity 1.0 is at a temperature of $100 ^ { \circ } \mathrm { C }$ At what rate does it radiate heat into empty space? $\left( \sigma = 5.67 \times 10 ^ { - 8 } \mathrm {~W} / \mathrm { m } ^ { 2 } \cdot \mathrm { K } ^ { 4 } \right)$

Question

A sphere of surface area $1.25 \mathrm {~m} ^ { 2 }$ and emissivity  1.0  is at a temperature of $100 ^ { \circ } \mathrm { C }$ At what rate does it radiate heat into empty space? $\left( \sigma = 5.67 \times 10 ^ { - 8 } \mathrm {~W} / \mathrm { m } ^ { 2 } \cdot \mathrm { K } ^ { 4 } \right)$

Quizplus · Accepted Answer

The rate of heat radiation from a body can be calculated using the Stefan-Boltzmann law, given by $P = \epsilon \sigma A T^4$, where $P$ is the power (rate of heat radiation), $\epsilon$ is the emissivity, $\sigma$ is the Stefan-Boltzmann constant, $A$ is the surface area, and $T$ is the absolute temperature in Kelvin. Given $\epsilon = 1.0$, $\sigma = 5.67 \times 10^{-8} \mathrm{W/m^2K^4}$, $A = 1.25 \mathrm{m^2}$, and $T = 100^\circ C = 373 K$, substituting these values gives $P = 1.0 \times 5.67 \times 10^{-8} \times 1.25 \times (373)^4 \approx 1400 W$ or $1.4 kW$.

A Sphere of Surface Area 1.25 m21.25 \mathrm {~m} ^ { 2 }1.25 m2

A Sphere of Surface Area $1.25 \mathrm {~m} ^ { 2 }$