Larmor formula

In electrodynamics, the Larmor formula is used to calculate the total power radiated by a nonrelativistic point charge as it accelerates. It was first derived by J. J. Larmor in 1897, in the context of the wave theory of light. When any charged particle (such as an electron, a proton, or an ion) accelerates, energy is radiated in the form of electromagnetic waves. For a particle whose velocity is small relative to the speed of light (i.e., nonrelativistic), the total power that the particle radiates (when considered as a point charge) can be calculated by the Larmor formula: where or — is the proper acceleration, — is the charge, and — is the speed of light. A relativistic generalization is given by the Liénard–Wiechert potentials. In either unit system, the power radiated by a single electron can be expressed in terms of the classical electron radius and electron mass as: One implication is that an electron orbiting around a nucleus, as in the Bohr model, should lose energy, fall to the nucleus and the atom should collapse. This puzzle was not solved until quantum theory was introduced. We first need to find the form of the electric and magnetic fields. The fields can be written (for a fuller derivation see Liénard–Wiechert potential) and where is the charge's velocity divided by , is the charge's acceleration divided by c, is a unit vector in the direction, is the magnitude of , is the charge's location, and . The terms on the right are evaluated at the retarded time . The right-hand side is the sum of the electric fields associated with the velocity and the acceleration of the charged particle. The velocity field depends only upon while the acceleration field depends on both and and the angular relationship between the two. Since the velocity field is proportional to , it falls off very quickly with distance. On the other hand, the acceleration field is proportional to , which means that it falls off more slowly with distance. Because of this, the acceleration field is representative of the radiation field and is responsible for carrying most of the energy away from the charge.