Energy flow

The power radiated by a source of light is described by the flow of light away from it, into a particular direction, and through a defined surface. The intensity $I$ is the energy emitted per unit time, per unit area of the source, per unit frequency interval, per unit solid angle into a chosen direction. The flux $F$ is the energy flowing per unit time onto a detector, through a unit area of its surface, per unit frequency interval. So if a point source emits energy at a rate $P$ joule/sec uniformly into all directions and summed over all frequencies, then the flux through a sphere around it of radius $r$ is simply

$$F = P / (4 \pi r^2)$$ (11)

This is the inverse square law describing the decline of flux with increasing distance from the source.

The energy of light is carried by quanta, each with energy $h\nu$. Since these quanta travel at the speed of light $c$, those in a shell of thickness $dr$ will pass through the outer surface of the shell in the time $dr/c$. Since the area of the shell is $4 \pi r^2$, if there are $n$ photons per unit volume in the shell, the number passing through the surface per second will be

$$\left(n 4 \pi r^2 dr\right)/\left(dr/c\right) = 4 \pi r^2 n c$$ (12)

each with energy $h\nu$. With this quantum view, the flux must be this number times the energy per photon, divided by the surface area, or

$$F = h \nu \left(4 \pi r^2 n c\right) / \left(4 \pi r^2 \right)$$ (13)

$$F = h \nu n c$$ (14)

The equations for flux must measure the same thing. It follows that the density of photons depends on distance from the source too according to

$$h \nu n c = P / (4 \pi r^2)$$ (15)

$$n = P / (4 \pi r^2 h \nu c)$$ (16)

The photon density also decreases as the $2^{nd}$ power of $r$. Either a detector counting photons or one measuring energy will show an inverse square law with distance from the source.