# Measures of the intensity of electromagnetic radiation

1. Nov 26, 2012

### Wox

I've always struggled with the commonly used measures of the intensity of electromagnetic radiation and it's catching up to me lately. Suppose $\bar{P}(R,\phi,\theta)$ is the Poynting vector of an electromagnetic field (in spherical coordinates) with norm $I(R,\phi,\theta)=\|\bar{P}(R,\phi,\theta)\|$ in $J.m^{-2}.s^{-1}$ (strictly speaking it should be the ensemble average of the norm). Then the flux $\Phi$ through a surface $S$, for which $\Psi$ is the angle between Poynting vector and surface normal at $(R,\phi,\theta)$, is given by (in $J.s^{-1}$)
\begin{align}\Phi_{S}&=\iint_{S} I(R,\phi,\theta)\cos\Psi\ \text{d}A\\ &=\iint_{S} I(R,\phi,\theta)\cos\Psi\ R^{2}\ \text{d}\Omega\\ &=\iint_{S} I(R,\phi,\theta)\cos\Psi\ R^{2}\sin\theta\ \text{d}\phi\text{d}\theta \end{align}
In radiometry, this is called the radiant flux. From this measure of intensity, several other measures are derived, but I'm a bit puzzled how. Take for example the radiant intensity $I_{e}$: the power per solid angle (in $J.sr^{-1}.s^{-1}$). From the equations above I understand that
$$I_{e}=\frac{\text{d}\Phi_{S}}{\text{d}\Omega}=I(R,\phi,\theta)\cos\Psi\ R^{2}$$
but how is this independent from surface $S$ (which is the point of using other measures than the radiant flux)? Secondly, while you're at it, why not using
$$J=\frac{\text{d}^{2}\Phi_{S}}{\text{d}\phi \text{d}\theta}=I(R,\phi,\theta)\cos\Psi\ R^{2}\sin\theta$$
which I also don't see to be independent of $S$, but at least you don't need to remember the $\sin\theta$ when integrating it. Then there are measures like the radiance in $J.m^{-2}.sr^{-1}.s^{-1}$ which is defined as
$$L_{e}=\frac{\text{d}^{2}\Phi_{S}}{\cos\Psi\text{d}A \text{d}\Omega}$$
I can write that down but I can't say I understand what it means. Can anyone shed some light on these issues?

Last edited: Nov 26, 2012