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One of the main reasons for the failure of the Rutherford model of atomic structure, it is famously stated, is that the electron undergoing circular orbit loses energy since due to its centripetal acceleration. In the same line of thought, an electron undergoing uniform circular motion due to a perpendicular magnetic field must also lose energy, thus collapsing on its center of rotation. However, this is rarely the case (at least textbooks don't mention them). The electron continues the same motion undefinitely.

Can anyone please clarify the reason why energy is lost in the first case and not in the second?

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A charge radiates every time is accelerated. The power radiated is given by the Larmor formula. Putting this into the introductions to the motion of a charge in electromagnetic fields would be a meaningless complication, as much as considering air friction. But yes, a charge in a magnetic field would not spin indefinitely, even in vacuum.

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    $\begingroup$ Please have a look at this link where the energy losses in a magnetic field are shown google.gr/… $\endgroup$
    – anna v
    Oct 3, 2014 at 17:29
  • $\begingroup$ Great link, Anna! $\endgroup$
    – Gaurav
    Oct 5, 2014 at 5:15
  • $\begingroup$ There are non-radiating conditions, its not every acceleration that produces radiation. $\endgroup$
    – Arc
    Jan 15, 2022 at 3:19
  • $\begingroup$ @Arc that only applies to charge distributions, if you have a single charge there is no escape. $\endgroup$
    – DarioP
    Jan 16, 2022 at 8:48
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It's not the case. In the second, the electron will radiate. This is how light species lose energy, and cool in Penning traps, and one of the factors that limit the energies of particles in circular particle accelerators.

For a reference see: http://en.wikipedia.org/wiki/Cyclotron_radiation

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  • $\begingroup$ The electron radiates in the first case too, right ? $\endgroup$
    – Gaurav
    Oct 3, 2014 at 13:05
  • $\begingroup$ Only in classical mechanics. In quantum mechanics, the electron can only emit quanta of light ("photons"), and has a lowest-possible energy state. $\endgroup$
    – Gremlin
    Oct 3, 2014 at 13:31
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    $\begingroup$ @Eoin: the conclusion is right, but the reason is wrong: the electron has a ground state in both of these cases, and has no way to lose energy -- the quantization of the electron's wave function is what prevents radiation, not the quantization of the electromagnetic field -- photons have nothing to do with the phenomenon. $\endgroup$ Oct 3, 2014 at 16:27
  • $\begingroup$ @JerrySchirmer, in theory, the electron will only radiate if it can couple to the modes of the electromagnetic field. But the field modes may be selected, for example, by building a cavity around the electron, the cavity might - in principle - forbid electron radiation. I don't know for electrons, but for atoms there's the electromagnetically induced transparency. $\endgroup$
    – Arc
    Jan 15, 2022 at 3:23
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In every negative acceleration electron loos energy, and this of course in the form of photons. This is not surprising because negative acceleration could be only after positive acceleration, the electron has to move befor he could be stopped or declined. And how the electron can be accelerated? By electric fields where the electron get the kinetic energy from the photons of this field. The electron get "charged" with the photons kinetic energy.

The magnetic field does not give or take away from the electron energy, the magnetic field influential on the electrons magnetic moment only and that way the electron get in circular motion, loos energy and end in the centre of a spiral path.

An electron in an atom don't run around in an orbit. The electron is dislocated in an orbital area and is in balance with the nucleus. The system is stabil. That means that the electric force between the negative electron and the positive proton at atom radius distance is not more pulling nor "pushing".

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