The wavelength at which the blackbody emissive power reaches its maximum value at \(300 \mathrm{~K}\) is (a) \(5.1 \mathrm{\mu m}\) (b) \(9.7 \mu \mathrm{m}\) (c) \(15.5 \mu \mathrm{m}\) (d) \(38.0 \mu \mathrm{m}\) (e) \(73.1 \mu \mathrm{m}\)

Short Answer

Expert verified
Answer: (b) 9.7 μm

Step by step solution

01

Recall Wien's Displacement Law

Wien's Displacement Law states that the wavelength at which the blackbody emissive power reaches its maximum value is inversely proportional to the temperature. Mathematically, it can be written as: \(\lambda_{max}T = b\) where \(\lambda_{max}\) is the wavelength at which the emissive power is maximum, \(T\) is the temperature in Kelvin, and \(b\) is Wien's constant, which is equal to \(2.898\times10^{-3}\mathrm{~m.K}\).
02

Calculate the wavelength at maximum emissive power

To find the wavelength at maximum emissive power, we will use Wien's Displacement Law with the given temperature: \(\lambda_{max} = \frac{b}{T}\) Plug in the known values: \(\lambda_{max} = \frac{2.898 \times 10^{-3} \mathrm{~m.K} }{300 \mathrm{~K}}\) Calculate the result: \(\lambda_{max} = 9.66 \times 10^{-6} \mathrm{~m}\)
03

Convert the result to micrometers and compare it with the given options

Since the options are given in micrometers (\(\mu\)m), we need to convert our result to micrometers: \(\lambda_{max} = 9.66 \times 10^{-6} \mathrm{~m} \times \frac{10^6 \mathrm{~\mu m}}{1 \mathrm{~m}} = 9.66 \mathrm{~\mu m}\) Approximate the result and compare it with the given options: \(\lambda_{max} \approx 9.7 \mathrm{~\mu m}\) The closest value to our calculated result is option (b) \(9.7 \mu \mathrm{m}\). Therefore, the correct answer is: (b) \(9.7 \mu \mathrm{m}\)

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Most popular questions from this chapter

Consider a black spherical ball, with a diameter of \(25 \mathrm{~cm}\), is being suspended in air. Determine the surface temperature of the ball that should be maintained in order to heat \(10 \mathrm{~kg}\) of air from 20 to \(30^{\circ} \mathrm{C}\) in the duration of 5 minutes.

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

At a wavelength of \(0.7 \mu \mathrm{m}\), the black body emissive power is equal to \(10^{8} \mathrm{~W} / \mathrm{m}^{3}\). Determine \((a)\) the temperature of the blackbody and \((b)\) the total emissive power at this temperature.

The variation of the spectral absorptivity of a surface is as given in Fig. P12-78. Determine the average absorptivity and reflectivity of the surface for radiation that originates from a source at \(T=2500 \mathrm{~K}\). Also, determine the average emissivity of this surface at \(3000 \mathrm{~K}\).

An opaque horizontal plate is well insulated on the edges and the lower surface. The irradiation on the plate is \(3000 \mathrm{~W} / \mathrm{m}^{2}\), of which \(500 \mathrm{~W} / \mathrm{m}^{2}\) is reflected. The plate has a uniform temperature of \(700 \mathrm{~K}\) and has an emissive power of \(5000 \mathrm{~W} / \mathrm{m}^{2}\). Determine the total emissivity and absorptivity of the plate.

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