Here are some very short notes which corroborate our derivation today.
And, here is something much fussier which ends up with the same result.
Aside from the fact that your textbook says the same thing, of course, but if it is on the internet it must be true.
Both of these are more than you need to know. The point of my posting them is to give some rationale for my writing a whole bunch of my own notes on radiation & scattering. Every treatment I found was either far, far too mathematically fussy to justify the only-approximately-correct result, or too hand-waving to be of much use. This is particularly true if you want to go further and derive black body radiation - the favorite derivation involves densities of states and mode counting, which is just unnecessarily obtuse. (FWIW, the Feynman lectures do a brilliant job, though you'll have to skip across all three volumes to get the details, and it is Purcell's ingenious derivation of the Larmor radiation formula that I followed in class.)
Our approach was to take the middle ground: put in just enough physics to be dangerous, if we really need to be, but recognize that the basic question is "why is the sky blue" and not "how many watts are scattered per steradian per unit wavelength." And, the answer to the simple-sounding question is quite a bit more complicated than you thought, right? Only answering the more complicated-sounding question can obscure the fundamental understanding: we could have derived the scattering cross section in hideous detail, that doesn't tell you why the sky is blue. We'll come back to the complicated question later.
Anyway: the main point is that we'll put in as much physics as we think we need, and not any more. If you want a 10% solution, that is a much different problem than wanting a 1% solution, so pick the battles that matter.
(Recall what we did today: after determining the radiated power for an oscillating charge, we found the radiation reaction (Abraham-Lorentz) force that must be present (damping). Then we presumed small damping, so all we really did was figure out how a bound charge could be approximated as a simple harmonic oscillator with radiation damping, equivalent to an RLC circuit.)
And, here is something much fussier which ends up with the same result.
Aside from the fact that your textbook says the same thing, of course, but if it is on the internet it must be true.
Both of these are more than you need to know. The point of my posting them is to give some rationale for my writing a whole bunch of my own notes on radiation & scattering. Every treatment I found was either far, far too mathematically fussy to justify the only-approximately-correct result, or too hand-waving to be of much use. This is particularly true if you want to go further and derive black body radiation - the favorite derivation involves densities of states and mode counting, which is just unnecessarily obtuse. (FWIW, the Feynman lectures do a brilliant job, though you'll have to skip across all three volumes to get the details, and it is Purcell's ingenious derivation of the Larmor radiation formula that I followed in class.)
Our approach was to take the middle ground: put in just enough physics to be dangerous, if we really need to be, but recognize that the basic question is "why is the sky blue" and not "how many watts are scattered per steradian per unit wavelength." And, the answer to the simple-sounding question is quite a bit more complicated than you thought, right? Only answering the more complicated-sounding question can obscure the fundamental understanding: we could have derived the scattering cross section in hideous detail, that doesn't tell you why the sky is blue. We'll come back to the complicated question later.
Anyway: the main point is that we'll put in as much physics as we think we need, and not any more. If you want a 10% solution, that is a much different problem than wanting a 1% solution, so pick the battles that matter.
(Recall what we did today: after determining the radiated power for an oscillating charge, we found the radiation reaction (Abraham-Lorentz) force that must be present (damping). Then we presumed small damping, so all we really did was figure out how a bound charge could be approximated as a simple harmonic oscillator with radiation damping, equivalent to an RLC circuit.)
No comments:
Post a Comment