Why doesn't a charged particle radiate energy in circular motion in a uniform magnetic field? I have studied in my Physics course that one of the drawbacks of Rutherford's atomic model was that when an electron will revolve around the nucleus, it is undergoing acceleration and so it should radiate energy and consequentially fall into the nucleus.
Similarly when a charged particle is projected in the plane perpendicular to a uniform magnetic field it executes uniform circular motion withradius $r=mv/qB$.
My question is why isn't the charged particle radiating energy here? Even in this case the charged particle is accelerating, just as it was in Rutherford's model of the atom. So shouldn't the radius decrease in this case also?
 A: I only add that there is even a natural (not artificial) source of synchrotron radiation: Crab Nebula (remnant of supernova observed in 1054). The reason why we see it (even in small telescopes) is thus quite diferrent from other celestial sources (where we observe light from hot stars or acretion discs, or excited gas).  
A: A charged particle circulating in a magnetic field  does radiate energy, and it is called synchrotron radiation. All circular particle  accelerators have energy losses due to this radiation.
A: This is a really good question and it puzzled me for many years. 
Physics is composed of many different theories, and as history has progressed, physicists have refined them or created new ones. My fav example is Newton's theory of gravity, which is great at explaining gravity for most applications; however, Einstein's are much more accurate and a more thorough explanation.
Electrons orbiting around a nucleus is true for electrodynamics. The question you are posing was a huge issue (like you mentioned above) for many years. It was a big reason why people thought the E&M theory was not complete.
The reason it does not radiate is explained by quantum mechanics. In QM particles are waves and particles. The electron is a waved function. The wave function surrounding a central potential will have bound states. These bound states DO NOT MOVE; the wave function will not change when in a bound state. Because the bound states dont move, the electric field from the electron does not move, so nothing is radiated. This is a subtle and very sweet concept.
Side note: a central potential is something with only radial dependence, and is the kind of potential a nucleus makes.
