How does an electron undergoing uniform circular motion exactly lose energy? 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? 
 A: 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.
A: 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
A: 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".
