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1.6: Relation between Flux and Intensity

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  • For an isotropic radiator,

    \[\Phi=4\pi I. \label{1.6.1}\]

    For an anisotropic radiator

    \[\Phi = \int Id\omega, \label{1.6.2}\]

    the integral to be taken over an entire sphere. Expressed in spherical coordinates, this is

    \[\Phi = \int_0^{2 \pi} \int_0^\pi I (\theta,\phi) \sin \theta d \theta d\phi. \label{1.6.3}\]

    If the intensity is axially symmetric (i.e. does not depend on the azimuthal coordinate \(\phi\) ) equation \(\ref{1.6.3}\) becomes

    \[\Phi = 2\pi \int_0^\pi I (\theta) \sin \theta d \theta. \label{1.6.4}\]

    These relations apply equally to subscripted flux and intensity and to luminous flux and luminous intensity.


    Suppose that the intensity of a light bulb varies with direction as

    \[I ( \theta) = 0.5 I (0) (1 + \cos \theta) \label{1.6.5}\]

    (Note the use of parentheses to mean "at angle \(\theta\) ".)

    Draw this (preferably accurately by computer - it is a cardioid), and see whether it is reasonable for a light bulb. Note also that, if you put \(\theta = 0\) in equation \(\ref{1.6.5}\), you get \(I(\theta) = I(0)\).

    Show that the total radiant flux is related to the forward intensity by

    \[\Phi = 2\pi I (0) \label{1.6.6}\]

    and also that the flux radiated between \(\theta = 0\) and \(\theta = \pi/2\) is

    \[\Phi = \frac{3}{2} \pi I (0). \label{1.6.7}\]