1.6: Relation between Flux and Intensity
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For an isotropic radiator,
Φ=4πI.
For an anisotropic radiator
Φ=∫Idω,
the integral to be taken over an entire sphere. Expressed in spherical coordinates, this is
Φ=∫2π0∫π0I(θ,ϕ)sinθdθdϕ.
If the intensity is axially symmetric (i.e. does not depend on the azimuthal coordinate ϕ ) equation ??? becomes
Φ=2π∫π0I(θ)sinθdθ.
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(θ)=0.5I(0)(1+cosθ)
(Note the use of parentheses to mean "at angle θ ".)
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 θ=0 in equation ???, you get I(θ)=I(0).
Show that the total radiant flux is related to the forward intensity by
Φ=2πI(0)
and also that the flux radiated between θ=0 and θ=π/2 is
Φ=32πI(0).