# Pardoxical property of electromagnetic radiation from accelerating charges

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?

• 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. – Ian Jan 5 '17 at 22:07
• 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 – Mahin Jan 5 '17 at 22:34
• 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. – user2554 Jan 6 '17 at 7:51
• Small correction, @AlbertAspect $\mathrm{d}W = v \mathrm{d}P$. Try it with special relativistic equations, and you'll see what I mean. – Sean E. Lake Jan 27 '17 at 19:47
• Canonical equations of motion: $$v = \frac{\partial H}{\partial p}.$$ – Sean E. Lake Jan 28 '17 at 0:00