A Hundred Years of Larmor Formula

Sir Joseph LARMOR showed in 1897 that an oscillating electric charge emits radiation energy proportional to (acceleration) 2. At first sight,the result appears to be valid for arbitrary accelerations. But, perpetual uniform acceleration has been a case of nagging doubts, as radiation reaction vanishes and the equivalence principle, as also conformal symmetry of Maxwell equations each require nil energy loss. Special hypotheses are devised by some to justify the assumption of radiation loss for both perpetual and non-perpetual (uniform) accelerations which, as in the case of (uniform) velocities, are really different. The problem is here simply resolved by an explicit computation to show absence of radiation for the perpetual case and by illustrating that Larmor formula makes sense only if there is change in acceleration, just as kinetic energy has nontrivial quantitative sense, only when there is change in velocity. 1 Many a breakthrough came to be announced in 1897. Besides J.J.Thompson's initial discovery of the electron and Planck's reluctant recognition of an essential role of Boltzman's viewpoint leading him in 1899 to his radiation law, Sir Joseph Larmor had two papers of particular historical note. One 1 , third in the series 'On Dynamical Theory of the Electric and Luminiferous Medium', introduced the notions of time-dilation and length contraction with complete transformations of space-time (now known after Lorentz) and of electromagnetic fields to appear in his Adams Prize Essay 2. The other 3 proposed an explanation of Zeeman effect (discovered in 1896) different from Lorentz, giving his famous precision effect on a rotating charge in a magnetic field. The same paper also gives the celebrated Larmor formula for the rate at which an oscillator of charge e emits radiation energy, viz. R = 2 3 e 2 c 3 g 2 ; (1) g = ˙ v is the acceleration and c the velocity of light, which at times, apparent from the context, will be taken as 1. Whereas the subject of the first paper above was completely clarified in Einstein's Special Relativity (SR), and the Larmor precision effect subsumed in quantum theory, the significance of Lar-mor formula (1) continues to be a subject of debate. For instance, retarded field 4,5 of a charge, in arbitrary acceleration, gives the above radiation rate for both relativistic and non-relativistic cases. Lorentz showed that energy loss in non-relativistic case is equivalent to a force of 'damping' or 'radiation 2 reaction' : Γ …