The spectral distribution of the radiation emitted by a diffuse surface may be approximated as follows. HW 2 Q2
(a) What is the total emissive power?
(b) What is the total intensity of the radiation emitted in the normal direction and at an angle of 30° from the normal?
(c) Determine the fraction of the emissive power leaving the surface in the directions π/4 ≤ θ ≤ π/2.
The Correct Answer and Explanation is :
To solve this problem, we need to analyze the radiation emitted by a diffuse surface based on the spectral distribution provided. Let’s denote the spectral emissive power by ( E(\lambda) ) and assume that it is given in a specific form (e.g., Planck’s law, or a similar distribution).
(a) Total Emissive Power
The total emissive power ( E ) of a surface can be calculated by integrating the spectral emissive power over all wavelengths:
[
E = \int_0^\infty E(\lambda) \, d\lambda
]
For a blackbody, this can be evaluated using Planck’s law, resulting in the Stefan-Boltzmann law:
[
E = \sigma T^4
]
where ( \sigma ) is the Stefan-Boltzmann constant and ( T ) is the absolute temperature of the surface. If the surface is not a perfect blackbody, it is necessary to multiply by the emissivity ( \epsilon ) of the surface:
[
E = \epsilon \sigma T^4
]
(b) Total Intensity of Radiation Emitted
The intensity ( I ) of radiation emitted in a specific direction is given by:
[
I(\theta) = \frac{E}{\pi} \cos(\theta)
]
where ( \theta ) is the angle from the normal. To find the intensity in the normal direction (( \theta = 0^\circ )) and at ( \theta = 30^\circ ):
- For ( \theta = 0^\circ ):
[
I(0) = \frac{E}{\pi} \cos(0) = \frac{E}{\pi}
]
- For ( \theta = 30^\circ ):
[
I(30) = \frac{E}{\pi} \cos(30) = \frac{E}{\pi} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{3} E}{2\pi}
]
(c) Fraction of Emissive Power in Specific Directions
To determine the fraction of emissive power leaving the surface in the range ( \frac{\pi}{4} \leq \theta \leq \frac{\pi}{2} ), we need to integrate the intensity over this angle and divide by the total emissive power:
[
\text{Fraction} = \frac{\int_{\pi/4}^{\pi/2} I(\theta) \, d\theta}{E}
]
Using ( I(\theta) = \frac{E}{\pi} \cos(\theta) ):
[
\text{Fraction} = \frac{1}{E} \int_{\pi/4}^{\pi/2} \frac{E}{\pi} \cos(\theta) \, d\theta = \frac{1}{\pi} \int_{\pi/4}^{\pi/2} \cos(\theta) \, d\theta
]
Calculating the integral:
[
\int \cos(\theta) \, d\theta = \sin(\theta)
]
Thus,
[
\int_{\pi/4}^{\pi/2} \cos(\theta) \, d\theta = \sin\left(\frac{\pi}{2}\right) – \sin\left(\frac{\pi}{4}\right) = 1 – \frac{\sqrt{2}}{2}
]
So the fraction of the emissive power is:
[
\text{Fraction} = \frac{1 – \frac{\sqrt{2}}{2}}{\pi}
]
This represents the fraction of the total emissive power leaving the surface within the specified angle range.