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Radiation from an Accelerated Charge: The Issues of Charge Distribution and Aggregation |
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There is a hallowed result in mathematical physics to the effect that an accelerated or decelerated charge, even in the absence of interaction with any other electromagnetic fields, will radiate electromagnetic waves and thus lose energy. Based upon some previous analysis by Hendrik Lorentz, Joseph Larmor derived the following formula for the rate of energy radiation from a particle with a charge of q being accelerated at a rate α:
This was published in 1895. There are numerous problems, empirical and theoretical, with this result such that Richard Feynman says that "we have inherited a prejudice that an accelerating charge should radiate." Some of the numerous problem with the proposition that an accelerating charge should radiate electromagnetic waves are dealt with elsewhere. Ultimately the proposition rests upon an unjustified interpretation of the Poynting vector and Poynting Theorem. Here another problem is considered.
The problem is in the nonlinear dependence upon the charge and what is the aggregate effect of multiple charges. Suppose there are two charges of opposite sign, say q and −q. If they are undergoing the same acceleration then, according to the Larmor formula, they would be emitting the same radiation. If the two charges are traveling along parallel paths there electromagnetics fields would be, to a degree, canceling out. Yet according to the Larmor formula the pair would be emitting twice the radiation of a single charge. That is unless their separation distance falls below some critical level and the pair functions as a single charge with a charge equal to their net charge of zero. This would have the radiation as function of separation distance going from a positive value to zero discontinuously.
This problem of aggregation of charges occurs not just with chages of opposite signs. Consider two charges of equal charge q traveling on parallel paths. Their joint emission would be twice that of a single particle unless the separation distance is below some critical level at which their combined charge becomes effective. At that point the combined emission would be four times that of a single charge. The transition from two times that of a single to charge to four times would occur discontinuously. Something is wrong with this picture and most likely it is the Larmor formula and the proposition that accelerating charges emit electromagnetic radiation.
Suppose a charge of q is considered as two ½q charges. This would mean that the energy radiation is 2(q²/4)= q²/2.
If the charge of q is considered to be m charges of q/m each then the energy radiated is (1/m) of energy radiation by a charge of q. As m increases without bound the energy radiated goes to zero.
If the charge is not a point particle but instead a charge distributed over space then it might appropriately be considered an infinite number of infinitesimal charges and hence there would be no electromagnetic waves radiated.
(To be continued.)
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