This is a concluding note on my ‘series’ on light. The ‘series’ gave you an overview of the ‘classical’ theory: light as an electromagnetic wave. It was very complete, including relativistic effects (see my previous post). I could have added more – there’s an equivalent for four-vectors, for example, when we’re dealing with frequencies and wave numbers: quantities that transform like space and time under the Lorentz transformations – but you got the essence.

One point we never ever touched upon, was that magnetic field vector though. It is there. It is tiny because of that 1/c factor, but it’s there. We wrote it as

All symbols in bold are vectors, of course. The force is another vector vector cross-product: **F** = q**v**×**B**, and you need to apply the usual right-hand screw rule to find the direction of the force. As it turns out, ** that force – as tiny as it is – is actually oriented in the direction of propagation, **and it is what is responsible for the so-called

**radiation pressure**.

So, yes, there is a ‘pushing momentum’. How strong is it? What power can it deliver? Can it indeed make space ships sail? Well… The magnitude of the unit vector **e**_{r’ }is obviously one, so it’s the values of the other vectors that we need to consider. If we substitute and average **F**, the thing we need to find is:

〈F〉 = q〈vE〉/c

But the charge q times the field is the electric force, and the force on the charge times the velocity is the work dW/dt being done on the charge. So that should equal the *energy absorbed that is being absorbed from the light per second. *Now, I didn’t look at that much. It’s actually one of the very few things I left – but I’ll refer you to Feynman’s *Lectures *if you want to find out more: there’s a fine section on light scattering, introducing the notion of the *Thompson scattering cross section*, but – as said – I think you had enough as for now. Just note that 〈F〉 = [dW/dt]/c and, hence, that the *momentum *that light delivers is equal to *the energy that is absorbed* (dW/dt) *divided by c*.

So the momentum carried is 1/c times the energy. Now, you may remember that Planck solved the ‘problem’ of black-body radiation – an anomaly that physicists couldn’t explain at the end of the 19th century – by re-introducing a corpuscular theory of light: he said light consisted of photons. We all know that photons are the kind of ‘particles’ that the Greek and medieval corpuscular theories of light envisaged but, well… They have a particle-like character – just as much as they have a wave-like character. They are actually neither, and they are physically and mathematically being described by these wave functions – which, in turn, are functions describing *probability amplitudes*. But I won’t entertain you with that here, because I’ve written about that in other posts. Let’s just go along with the ‘corpuscular’ theory of photons for a while.

Photons also have energy (which we’ll write as W instead of E, just to be consistent with the symbols above) and momentum (p), and Planck’s Law says how much:

W = h*f *and p = W/c

So that’s good: we find the same multiplier 1/c here for the momentum of a* photon*. In fact, this is more than just a coincidence of course: the “wave theory” of light and Planck’s “corpuscular theory” must of course link up, because they are both supposed to help us understand real-life phenomena.

There’s even more nice surprises. We spoke about polarized light, and we showed how the end of the electric field vector describes a circular or elliptical motion as the wave travels to space. It turns out that we can actually relate that to some kind of *angular *momentum of the wave (I won’t go into the details though – because I really think the previous posts have really been too heavy on equations and complicated mathematical arguments) *and that we could also relate it to a model of photons carrying angular momentum*, “like spinning rifle bullets” – as Feynman puts it.

However, he also adds: “But this ‘bullet’ picture is as incomplete as the ‘wave’ picture.” And so that’s true and that should be it. And it will be it. I will really end this ‘series’ now. It was quite a journey for me, as I am making my way through all of these complicated models and explanations of how things are supposed to work. But a fascinating one. And it sure gives me a much better feel for the ‘concepts’ that are hastily explained in all of these ‘popular’ books dealing with science and physics, hopefully preparing me better for what I should be doing, and that’s to read Penrose’s advanced mathematical theories.