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# 1.6: Relation between Flux and Intensity

• • Contributed by Jeremy Tatum
• Emeritus Professor (Physics & Astronomy) at University of Victoria

$\Phi=4\pi I. \label{1.6.1}$

$\Phi = \int Id\omega, \label{1.6.2}$

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.

Example:

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

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