The total power radiated from an accelerating charge depends on the magnitude of it's acceleration, and is proportional to the square of the accelaration. But there is a certain problem with the definition of such an effect. Because than one is unable to figure out how the charged particle began to move . Suppose for example that we activate an external electic field on a charged particle at rest. The amount of work per time the field is doing on a particle is qEv, where v is the component of velocity parallel to the field. But at the begining, v = 0, so for any supposed initial acceleration of the particle we get a violation of the principle of conservation of energy since the the field did zero work while the charge radiated a finite amount of energy.

So how to resolve this paradox?

  • $\begingroup$ I think you're referring to one of several related inconsistencies between theory and experiment that motivated quantum mechanics: if you had a "classical electron" orbiting a "classical proton", then the electron would radiate, and eventually fall into the proton (on a time scale humans could measure). But this doesn't happen for real hydrogen atoms. QM resolves the problem. A similar issue is the "ultraviolet catastrophe" that occurs in the classical theory of black body radiation, where the power emitted by a "classical black body" is infinite. $\endgroup$ – Ian Jan 5 '17 at 22:07
  • $\begingroup$ What Ian is trying to tell that motion of charges never ceases its velocity could be zero but speed wont be and Power is a scalar product $\endgroup$ – Mahin Jan 5 '17 at 22:34
  • $\begingroup$ I dont think i confused a single point with a differential. I said just like you that P is the scalar product of F (force) and v (velocity) , and this shows i did understand the difference. But the point is that when the particle is at zero velocity that single point does become the differenetial. So there is a problem with a power that is defined as an increasing function of acceleration. I think that the solution is closer to what lan said: zero velocity never occurs in reality. $\endgroup$ – user2554 Jan 6 '17 at 7:51
  • $\begingroup$ Small correction, @AlbertAspect $\mathrm{d}W = v \mathrm{d}P$. Try it with special relativistic equations, and you'll see what I mean. $\endgroup$ – Sean E. Lake Jan 27 '17 at 19:47
  • $\begingroup$ Canonical equations of motion: $$v = \frac{\partial H}{\partial p}.$$ $\endgroup$ – Sean E. Lake Jan 28 '17 at 0:00

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