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Chapter 12: Problem 22

Consider a surface at a uniform temperature of \(800 \mathrm{~K}\). Determine the maximum rate of thermal radiation that can be emitted by this surface, in \(\mathrm{W} / \mathrm{m}^{2}\).

Short Answer

Expert verified
Answer: The maximum rate of thermal radiation emitted by a surface at a uniform temperature of 800 K is approximately 36,550.4 W/m².

Step by step solution

01

Write down the Stefan-Boltzmann Law formula

The Stefan-Boltzmann Law is given by the equation: $$ P = \sigma T^4 $$ where \(P\) is the power of radiation emitted per unit surface area (in \(\mathrm{W} / \mathrm{m}^{2}\)), \(\sigma\) is the Stefan-Boltzmann constant (\(5.67 \times 10^{-8} \mathrm{W} / \mathrm{m}^{2} \mathrm{K}^{4}\)), and \(T\) is the absolute temperature of the body (in Kelvin).
02

Plug the temperature value into the formula

Given that the surface is at a uniform temperature of 800 K, we can substitute the given value of T into the equation: $$ P = \sigma (800 \mathrm{K})^4 $$
03

Calculate the maximum rate of thermal radiation

Using the value of the Stefan-Boltzmann constant, we can now calculate the maximum rate of thermal radiation: $$ P = (5.67 \times 10^{-8} \mathrm{W} / \mathrm{m}^{2} \mathrm{K}^{4})(800 \mathrm{K})^4 = 36,550.4 \mathrm{W} / \mathrm{m}^{2} $$ So the maximum rate of thermal radiation that can be emitted by this surface is approximately \(36,550.4\, \mathrm{W} / \mathrm{m}^{2}\).

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Most popular questions from this chapter

Consider a radio wave with a wavelength of \(10^{7} \mu \mathrm{m}\) and a \(\gamma\)-ray with a wavelength of \(10^{-7} \mu \mathrm{m}\). Determine the photon energies of the radio wave and the \(\gamma\)-ray, and the photon energy ratio of the \(\gamma\)-ray to the radio wave.

A 1-m-diameter spherical cavity is maintained at a uniform temperature of \(600 \mathrm{~K}\). Now a 5 -mm-diameter hole is drilled. Determine the maximum rate of radiation energy streaming through the hole. What would your answer be if the diameter of the cavity were \(3 \mathrm{~m}\) ?

Consider a surface at \(500 \mathrm{~K}\). The spectral blackbody emissive power at a wavelength of \(50 \mu \mathrm{m}\) is (a) \(1.54 \mathrm{~W} / \mathrm{m}^{2} \cdot \mu \mathrm{m}\) (b) \(26.3 \mathrm{~W} / \mathrm{m}^{2} \cdot \mu \mathrm{m}\) (c) \(108.4 \mathrm{~W} / \mathrm{m}^{2} \cdot \mu \mathrm{m}(d) 2750 \mathrm{~W} / \mathrm{m}^{2} \cdot \mu \mathrm{m}(e) 8392 \mathrm{~W} / \mathrm{m}^{2} \cdot \mu \mathrm{m}\)

A surface at \(300^{\circ} \mathrm{C}\) has an emissivity of \(0.7\) in the wavelength range of \(0-4.4 \mu \mathrm{m}\) and \(0.3\) over the rest of the wavelength range. At a temperature of \(300^{\circ} \mathrm{C}, 19\) percent of the blackbody emissive power is in wavelength range up to \(4.4 \mu \mathrm{m}\). The total emissivity of this surface is (a) \(0.300\) (b) \(0.376\) (c) \(0.624\) (d) \(0.70\) (e) \(0.50\)

A microwave oven is designed to operate at a frequency of \(2.2 \times 10^{9} \mathrm{~Hz}\). Determine the wavelength of these microwaves and the energy of each microwave.
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