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1.15: A= \(\pi\) B

There are several occasions in radiation theory in which one quantity is equal to \(\pi\) times another, the two quantities being related by an equation of the form \(A = \pi B\). I can think of three, and they are all related to the three questions asked and answered in section 1.14.

If the source in question i of Section 1.14 is an element of a lambertian surface, then \(I(\theta ,\phi )\) is given by Equation 1.13.1, and in that case Equation 1.14.1 becomes

\[\phi = \pi I (0) \tag{1.15.1} \label{1.15.1}\]

If the element \(\delta \ A\) in question ii is lambertian, \(L\) is independent of \(\theta\) and f , and equation 1.14.3 becomes

\[M = \pi L \tag{1.15.2} \label{1.15.2}\]

This, then is the very important relation between the exitance and the radiance of a lambertian surface. It is easy to remember which way round it is if you think of the units in which \(M\) and \(L\) are expressed and think of \(\pi\) as a solid angle.

If the hemisphere of question iii is uniformly lambertian (for example, if the sky is uniformly dull and cloudy) then \(L\) is the same everywhere in the sky, and the irradiance is

\[E = \pi L \tag{1.15.3} \label{1.15.3}\]