Why don't electrons in shells radiate energy?
In classical electrodynamics, a charged particle, when accelerated radiates energy in form of electromagnetic waves. So, one would expect that an electron orbiting a nucleus will radiate energy because atomic nucleus is positively charged and electron is negatively charged. However it does not happen and electrons seem to be orbiting atom without radiating. This was a puzzle which was solved by advent of quantum mechanics.
According to quantum mechanics, electron moves around an atom in quantized orbits. As long as it is in one of the orbits, it does not radiate. When electron jumps from one quantized orbit to another, it radiates energy in a quantum packet of energy called photon. This happens when the energy of first quantized orbit is higher than that of the second orbit. The electron in lowest energy quantum orbit cannot radiate a photon because there is no lower energy quantum orbit available for the electron to jump to. The whole dynamics of quantum orbits as well as emission of photons from electronic quantum orbits is described by quantum electrodynamics.
The quantum physics is applicable to very small particles like atoms, electrons nuclei and photons.