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. The total radiated power is For a single point charge When deriving the Larmor formula by explicitly considering the velocity of the moving particle, one arrives at Lienart-Larmor expression: The g6 factor means that when g is very close to one (i.e. b < < 1) the radiation emitted by the particle is likely to be negligible. However when g is greater than one, the radiation explodes as the particle tries to lose its energy in the form of EM waves. It's also interesting that when the acceleration and velocity are orthogonal the power is reduced. The faster the motion becomes the greater this reduction gets. In fact it seems to imply that as b tends to one the power radiated tends to zero (for orthogonal motion). This would suggest that a charge moving at the speed of light, in instantaneously circular motion, emits no radiation. However, it would be impossible to accelerate a charge to this speed because the g6 would explode to , meaning that the particle would radiate a gigantic amount of energy which would require you to put more and more energy in to keep accelerating it. This would imply that there is a cosmic speed limit, namely c. Such a connection wasn't made until 1905 when Einstein published his paper on Special Relativity. Larmor formula

Synchrotron Radiation Synchrotron radiation is electromagnetic radiation generated by a synchrotron. It is similar to cyclotron radiation, but generated by the acceleration of ultrarelativistic (i.e., moving near the speed of light) charged particles through magnetic fields. This may be achieved artificially in synchrotrons or storage rings, or naturally by fast electrons moving through magnetic fields in space. The radiation produced may range over the entire electromagnetic spectrum, from radio waves to infrared light, visible light, ultraviolet light, X-rays, and gamma rays. It is distinguished by its characteristic polarization and spectrum. Angular distribution of the radiation; z-axis points along v Supernova remnants - the remaining material after the explosion of a high mass star - are sources of synchrotron emission. When a particle is accelerated in a magnetic field, the particle release high energy photons:

Issues and implications Radiation reaction The radiation from a charged particle carries energy and momentum. In order to satisfy energy and momentum conservation, the charged particle must experience a recoil at the time of emission. The radiation must exert an additional force on the charged particle. This force is known as the Abraham-Lorentz force in the nonrelativistic limit and the Abraham-Lorentz-Dirac force in the relativistic limit. Atomic physics A classical electron orbiting a nucleus experiences acceleration and should radiate. Consequently the electron loses energy and the electron should eventually spiral into the nucleus. Atoms, according to classical mechanics, are consequently unstable. This classical prediction is violated by the observation of stable electron orbits. The problem is resolved with a quantum mechanical or stochastic electrodynamic description of atomic physics. Example: An a particle with energy 1 MeV collides head-on with a uranium nucleus. How much of the initial kinetic energy is converted to energy of radiated electromagnetic waves? We shall ignore the recoil of U (Z=92, A=238). Let the U nucleus be at rest at the origin and alpha particle (Z=2) travels along the x-axis. It starts at infinity with v=-v0, deaccelerates, reverses direction at x=x0 (distance of the closest approach,) and accelerates again. =16/15

Radiation reaction Since accelerating electric charges launch electromagnetic waves that move away from the source charges at the speed of light carrying energy and momentum with them, the charges must experience a recoil force, that is, a force of "radiation reaction". Normally, electromagnetic radiation reaction forces are ridiculously small. For example, if you rigged up a radio antenna to put out a kilowatt of power in one direction, the reaction force on the antenna would correspond to the weight of several fleas. So in almost all circumstances you can just ignore radiative reaction effects, pretend that they don't exist. But not always. In high energy elementary particle accelerators (like the ones at Fermilab or CERN) radiation reaction is an obvious fact of life. As the particles traveling at nearly the speed of light are bent into their circular paths by magnets, they are accelerated. And they radiate. The reaction force produced by the radiation slows the particles down, unless power is applied to replace the radiated energy and momentum. Let us calculate average energy loss of particle between times t1 and t2:, assuming that at t2 the particle returns to its initial state at t1: The Abraham-Lorentz force or the radiation reaction force The force is proportional to the square of the object's charge, times the so-called "jerk" (rate of change of acceleration) that it is experiencing. The force points in the direction of the jerk. For example, in a cyclotron, where the jerk points opposite to the velocity, the radiation reaction is directed opposite to the velocity of the particle, providing a braking action.

Consider a charged particle moving in the absence of external forces. If the external force is present The integral extends from the present to infinitely far in the future. Thus future values of the force affect the acceleration of the particle in the present. Therefore, signals from an interval approximately t into the future affect the acceleration in the present. For an electron, this time is approximately 10−24 sec, which is the time it takes for a light wave to travel across the "size" of an electron. This conceptual problems created by self-fields are highlighted in Jackson: “The difficulties presented by this problem touch one of the most fundamental aspects of physics, the nature of the elementary particle. Although partial solutions, workable within limited areas, can be given, the basic problem remains unsolved. One might hope that the transition from classical to quantum-mechanical treatments would remove the difficulties. While there is still hope that this may eventually occur, the present quantum-mechanical discussions are beset with even more elaborate troubles than the classical ones. It is one of the triumphs of comparatively recent years (~ 1948 - 1950) that the concepts of Lorentz covariance and gauge invariance were exploited sufficiently cleverly to circumvent these difficulties in quantum electrodynamics and so allow the calculation of very small radiative effects to extremely high precision, in full agreement with experiment. From a fundamental point of view, however, the difficulties remain.” The Abraham-Lorentz force is the result of the most fundamental calculation of the effect of self-generated fields. It arises from the observation that accelerating charges emit radiation. The Abraham-Lorentz force is the average force that an accelerating charged particle feels in the recoil from the emission of radiation. The introduction of quantum effects leads one to quantum electrodynamics. The self-fields in quantum electrodynamics generate a finite number of infinities in the calculations that can be removed by the process of renormalization. This has led to a theory that is able to make the most accurate predictions that humans have made to date. for an electron