1.14: Relations between Flux, Intensity, Exitance, Irradiance
- Page ID
- 8000
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In this section I am going to ask, and answer, three questions.
i. (See figure I.3 )
\(\text{FIGURE I.3}\)
A point source of light has an intensity that varies with direction as \(I(\theta , \phi\)). What is the radiant flux radiated into the hemisphere \(\theta < \pi /2\)? This is easy; we already answered it for a complete sphere in equation 1.6.3. For a hemisphere, the answer is
\[\phi = \int_0^{2\pi} \int_0^{\pi/2} I(\theta, \phi) \sin \theta d \theta d \phi. \tag{1.14.1} \label{1.14.1}\]
ii. At a certain point on an extended plane radiating surface, the radiance is \(L(\theta ,\phi\)). What is the emergent exitance \(M\) at that point?
\(\text{FIGURE I.4}\)
Consider an elemental area \(\delta A\) (see figure I.4). The intensity \(I( \theta , \phi\)) radiated in the direction \((\theta ,\phi )\) is the radiance times the projected area \(\cos \theta \ \delta A\). Therefore the radiant power or flux radiated by the element into the hemisphere is
\[\delta \phi = \int_0^{2\pi} \int_0^{\pi/2} L(\theta, \phi) \cos \theta \sin \theta d\theta d \phi \delta A, \tag{1.14.2} \label{1.14.2}\]
and therefore the exitance is
\[M=\int_0^{2\pi} \int_0^{\pi/2} L(\theta, \phi) \cos \theta \sin \theta d\theta d\phi \tag{1.14.3} \label{1.14.3}\]
iii. A point \(O\) is at the centre of the base of a hollow radiating hemisphere whose radiance in the direction \((\theta , \phi )\) is \(L(\theta , \phi)\). What is the irradiance at that point \(O\) ? (See figure I.5.)
\(\text{FIGURE I.5}\)
Consider an elemental area \(a^2 \ \sin \theta \ \delta \theta \ \delta\phi\) on the inside of the hemisphere at a point where the radiance is \(L(\theta ,\phi )\) (figure I.5). The intensity radiated towards \(O\) is the radiance times the area:
\[\delta I (\theta, \phi) = L(\theta, \phi) a^2 \sin \theta \delta \theta \delta \phi \tag{1.14.4} \label{1.14.4}\]
The irradiance at \(O\) from this elemental area is (see equation (1.10.1)
\[\delta E = \frac{\delta I (\theta,\phi) \cos \theta}{a^2} = L (\theta, \phi) \cos \theta \sin \theta \delta \theta \delta \phi, \tag{1.14.5} \label{1.14.5}\]
and so the irradiance at O from the entire hemisphere is
\[E = \int_0^{2\pi} \int_0^{\pi/2} L(\theta, \phi) \cos \theta \sin \theta \delta \theta \delta \phi. \tag{1.14.6} \label{1.14.6}\]
The same would apply for any shape of inverted bowl - or even an infinite plane radiating surface (see figure I.6.)
\(\text{FIGURE I.6}\)