# 1.6: Relation between Flux and Intensity

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.

*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

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