When your book says energy it should say radiant intensity. I didn't read Lambert's Photometria myself, but multiple sources say that this is how Lambert defined his law. A lambertian surface follows Lambert's cosine law, so for this surface we have:
$$I_\theta=I_n \cos\theta$$

Radiance's definition can be written as:
$$L=\frac{\partial I}{\partial A\cos\theta}$$

From this definition we can say that for $\theta=0$, then $\cos\theta=1,$, so:
$$L_n=\frac{\partial I_n}{\partial A}$$
The radiance for some random $\theta$ for a Lambertian surface would be:
$$L_\theta=\frac{\partial I_\theta}{\partial A\cos\theta}=\frac{\partial I_n\cos\theta}{\partial A\cos\theta}=\frac{\partial I_n}{\partial A}$$

This last relation only works for lambertian surfaces since I used the Lambert's law in it. As you can see, if you apply Lambert's law, the definition of radiance doesn't depend on $\theta$ anymore.

To answer your second question, every surface that subtends the same solid angle receives the same light (radiant flux) from the radiant area ($\partial A$). This means that every unit of the fortshortened surface of this receiving surface receives the same radiant flux. Therefore it is not about the distance to the source, but about the solid angle it subtends. If you only think about human eyes as sensors, then it has to do with distance, because the size of the eye is kind of the same for everyone and it doesn't depend on distance, so you have to be at the same distance to subtend the same solid angle, but you shouldn't think of radiance as being a function of distance, but of solid angle.

This is one of the places where wave particle dualism gets some people in trouble. Many are taught that it means that light can be a wave and a particle, and that phrasing can lead to some confusion. I find it more intuitive to just rip the bandaid off quickly and say light is *neither* a wave *nor* a particle. It is something which, in some situations, can be well modeled as a wave, and in some situations can be well modeled as a particle, but it is its own thing (which can be well modeled in all known cases using a more complicated concept, a "wavefunction").

You can think of photons getting randomly reflected or transmitted on the boundary, but the truth is that the billiard-ball photon model really isn't very effective at describing what happens at this boundary. This is one of the regions where wave mechanics models the effects very well, while particle models don't do so well. If you use wave mechanics, the idea of a wave getting partially reflected and partially transmitted isn't difficult to believe at all. In fact, it's pretty easy to prove.

Thinking in wave terms at these boundaries also gives correct answers in peculiar situations where the particle model simply falls on its face. Consider the interesting case of an "evanescent wave."

In this setup, the laser and prism are set up at the correct angles to cause "total internal reflection." This means that, by the simple models, 100% of the light should bounce off the side of the prism and into the detector. Indeed, if the prism is in the open air, we do see 100% reflection (well, within the error bars of absorption). However, bring an object close to the prisim (but not touching) and things change. You end up seeing effects from the object, even though 100% of the light was supposed to be reflected!

If you think of light like photons, this is hard to explain. If you look at it as a wave governed by Maxwell's equations, you see that you would violate the law of conservation of energy if there was a "pure" reflection. Instead it creates a reflection and an "evanescent wave" which is outside the prism, and its strength falls off exponentially, which is really hard to explain with particles!

Of course, these too are all simplifications. The *real* answer to your question is that the wavefunction of the light interacts with the electromagnetic fields of the atoms in the prisim, and the result of that interaction leads to reflection, refraction, diffusion, absorption, and eveansecent waves. However, naturally those equations are a bit harder to understand, so we use the older, simpler models from before quantum mechanics. We just have to be sure to use the one which is most applicable in any given situation, because none of them are *quite* right.

## Best Answer

So a single atomic electron that gets excited in this way indeed does not have this property, and indeed a vast number of them can do this in parallel without functioning this way: this is part of the classical explanation for why the sky is blue.

Furthermore it matters that the surface is flat: if you cut little parallel lines in the surface then you get a spectrometer; this is part of the explanation for why you see rainbows in the "data track" of a CD or DVD.

Furthermore you see the same reflection when you analyze things like reflection from a glass, even though there is also a transmitted wave.

Our modern understanding is that a photon is able to sense every path that it could possibly take from A to B. Each path can be thought of as a little arrow rotating in 2D at a constant rate with respect to time (the photon's "frequency"). We add up all of these little arrows by connecting them tip-to-tail and then ask how far away the final point is from the initial point, which is a measure of the probability that the photon goes in that direction.

Now if there is a path, and there are other possible paths "nearby" that one, like on a mirror, we have to consider if they take a different time or not. If they all have a different time then each arrow is tilted a little more relative to the last one, and as you add them together, they describe something very circle-like and it will not get far away from the initial point. But if the path is either the

fastestorslowestpath from point A to point B, then something magical happens: all of the nearby paths take about the same time, so all of the arrows line up, so you get a very large probability that the object moves in that way.This is known as "Fermat's law of refraction," and it says that light seems to magically find the shortest-time path between two points. And a direct consequence is that the angle of incidence equals the angle of reflection.