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Chapter 12: Problem 116

Daylight and incandescent light may be approximated as a blackbody at the effective surface temperatures of \(5800 \mathrm{~K}\) and \(2800 \mathrm{~K}\), respectively. Determine the wavelength at maximum emission of radiation for each of the lighting sources.

Short Answer

Expert verified
Answer: The wavelength at maximum emission for daylight is 500 nm, and for incandescent light, it is 1040 nm.

Step by step solution

01

Understanding Wien's Displacement Law

Wien's Displacement Law relates the wavelength at maximum emission to the temperature of the blackbody, which is given by: \(\lambda_{max} = \dfrac{b}{T}\) Where \(\lambda_{max}\) is the wavelength at maximum emission, \(b\) is Wien's displacement constant (\(b \approx 2.9 \times 10^{-3} \mathrm{~m K}\)), and \(T\) is the temperature of the blackbody in kelvins. We will apply this formula to both lighting sources to find the wavelengths at maximum emission.
02

Calculate wavelength for daylight source

Given the temperature of the daylight source as \(T = 5800 \mathrm{~K}\). Then using Wien's displacement law: \(\lambda_{max_{daylight}} = \dfrac{b}{T_{daylight}}\) \(\lambda_{max_{daylight}} = \dfrac{2.9 \times 10^{-3} \mathrm{~m K}}{5800 \mathrm{~K}}\) \(\lambda_{max_{daylight}} = 5\times10^{-7} \mathrm{~m} = 500 \mathrm{~nm}\) The wavelength at maximum emission for daylight source is \(500 \mathrm{~nm}\).
03

Calculate wavelength for incandescent light source

Given the temperature of the incandescent light source as \(T = 2800 \mathrm{~K}\). Then using Wien's displacement law: \(\lambda_{max_{incandescent}} = \dfrac{b}{T_{incandescent}}\) \(\lambda_{max_{incandescent}} = \dfrac{2.9 \times 10^{-3} \mathrm{~m K}}{2800 \mathrm{~K}}\) \(\lambda_{max_{incandescent}} = 1.04 \times 10^{-6} \mathrm{~m} = 1040 \mathrm{~nm}\) The wavelength at maximum emission for the incandescent light source is \(1040 \mathrm{~nm}\).
04

Conclusion

In conclusion, the wavelength at maximum emission for daylight is \(500 \mathrm{~nm}\), and for incandescent light, it is \(1040 \mathrm{~nm}\).

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Most popular questions from this chapter

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) \(108.4 \mathrm{~W} / \mathrm{m}^{2} \cdot \mu \mathrm{m}(d) 2750 \mathrm{~W} / \mathrm{m}^{2} \cdot \mu \mathrm{m}(e) 8392 \mathrm{~W} / \mathrm{m}^{2} \cdot \mu \mathrm{m}\)

By what properties is an electromagnetic wave characterized? How are these properties related to each other?

A circular ceramic plate that can be modeled as a blackbody is being heated by an electrical heater. The plate is \(30 \mathrm{~cm}\) in diameter and is situated in a surrounding ambient temperature of \(15^{\circ} \mathrm{C}\) where the natural convection heat transfer coefficient is \(12 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). If the efficiency of the electrical heater to transfer heat to the plate is \(80 \%\), determine the electric power that the heater needs to keep the surface temperature of the plate at \(200^{\circ} \mathrm{C}\).

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) \(0.50\)

Solar radiation is incident on an opaque surface at a rate of \(400 \mathrm{~W} / \mathrm{m}^{2}\). The emissivity of the surface is \(0.65\) and the absorptivity to solar radiation is \(0.85\). The convection coefficient between the surface and the environment at \(25^{\circ} \mathrm{C}\) is \(6 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). If the surface is exposed to atmosphere with an effective sky temperature of \(250 \mathrm{~K}\), the equilibrium temperature of the surface is (a) \(281 \mathrm{~K}\) (b) \(298 \mathrm{~K}\) (c) \(303 \mathrm{~K}\) (d) \(317 \mathrm{~K}\) (e) \(339 \mathrm{~K}\)
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