A ray crossing 2 media of different indices and energy conservation

[Paulpaulpa]

The $I_i$ are the intensity of the rays, in other words energy per surface units per radians by seconds.

The d$\Omega$ are the solid angles

The equation p75 isis what I don't understand. I suppose that each side represent the energy going and out of the surface dS but I don't understand why it has this form. It looks like the projection of the intensities along the x-axis, as if energy was a a vector quantity (like the momentum, for which the three components are conserved separately). But only the total energy going through dS should matter and I don't understand why it has this form.

If we say that $I_1$ is independent of the spacetime coordinates and is large enough to wrap around dS, the energy going in should be $I_1 d\Omega dS$ since the surface is totally exposed to the ray. I am probably wrong but I don't know why.

 

[tech99]

Some energy is reflected.