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In the german edition of "Tipler: Physics for scientists and engineers" there i a small text about the plum pudding model of the atom. i give a direct translation:

"Thomsons plumpudding model of a atom: The negatively charged electrons are embedded in kind of liquid or porridge [...]. For that kind of configuration, one can compute the resonance frequencies of the electron oscillations. According to the laws of classical physics, such an atom should emit light with this frequency, but Thomson couldn't find such a configuration, for which the computed frequencies correspond to the observed specral lines."

My (counterfaactual) question is, what is meant by "configuration" of electrons? And how could a oscillating electron emit descernible lines as the acceleration - the cause of radiation - would be continuos and not discrete or so.

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how could a oscillating electron emit descernible lines as the acceleration - the cause of radiation - would be continuous and not discrete or so.

The membrane of a speaker, playing a pure sinusoidal musical note, moves back and forth with a continuous acceleration. Despite this, the sound it produces has a single frequency.

In a similar manner, the electron that oscillates with a given frequency around an equilibrium position, generates an electromagnetic wave that is continuous in time and space but has a definite frequency $f_0$. The human eye is sensible just to the frequency (color) of light. The eye can not follow the time varying amplitude of the light waves.

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