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The Feynman Critique of Radiation by Accelerating Charges |
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Richard Feynman raised doubts as to the general validity of the hallowed proposition that accellerating or decelerating charges radiate electromagnetic waves. His critique, given in full below, is found in the work Feynman Lectures on Gravitation which was prepared from notes on the lectures he gave at Caltech during the academic year of 1962-63. The material is from pages 123 and 124 of that work. The symbols v and a represent the vectors of velocity and acceleration.
MODIFICATIONS OF ELECTRODYNAMICS REQUIRED BY THE PRINCIPLE OF EQUIVALENCE
The Principle of Equivalence postulates that an acceleration shall be indistinguishable from any experiment whatsoever. In particular, it cannot be distinguished by observing electromagnetic radiation. There is evidently some trouble here, since we have inherited a prejudice that an accelerating charge should radiate, whereas we do not expect a charge lying in a gravitational field to radiate. This is, however, not due to a mistake in our statement of equivalence, but to the fact that the rule of the power radiated by an accelerating charge,
(dW/dt) = (2/3)(e²/c³)a² (9.1.1)
has led us astray. This is usually derived from calculating the flow from Poynting's Theorem far away, and it is only valid for cyclic motions, or at least motions which do not grow forever in time (as a constant acceleration does). It does not suffice to tell us "when" the energy is radiated. This can only be determined by finding the force of radiation resistance, which is (2/3)(e²/c³)(da/dt). For it is work against this force which represents energy loss. For constant acceleration this force is zero. Generally the work done against it can be written
(dW/dt) = −(2/3)(e²/c³)v·(da/dt)
= (2/3)(e²/c³)a·a −(2/3)(e²/c³)(v·a), (9.1.2)giving a correct expression for (dW/dt). For cyclic or limited motions, the average contribution of the last term over the long run is small or zero (over one cycle since (v·a) is restored to its original value, its contribution vanishes) and the simpler eq.(9.1.1) suffices.
Of course, in a gravitational field the electrodynamic laws of Maxwell need to be modified, just as ordinary mechanics needed to be modified to satisfy the principle of relativity. After all, the Maxwell equations predict that light should travel in a straight line--and it is found to fall towards a star. Clearly, some interaction between gravity and electrodynamics must be included in a better statement of the laws of electricity, to make them consistent with the principle of equivalence.
The Principle of Equivalence is usually stated in terms of an ever accelerating elevator. A more manageable image would be in terms of a space station generating artificial gravity by rapid rotation.
Feynman seems to leave valid the radiation by charges moving in a circular orbit, but the rate at which work is being done against a force is F·v, or equivalently ma·v, and in a circular orbit the centrifugal force and acceleration are perpendicular to the tangential velocity so the work being done is zero.
This might leave open the case of a particle moving in a non-circular elliptical orbit. But we have the cases of the planets in the solar system demonstrating that there is no work being done against the centrifugal force (and centripetal acceleration). Otherwise the solar system would have collapsed long ago.
The explanation for why physical charges do not radiate under accelerartion is utterly simple. The Larmor proposition is for point particles and the energy of radiation is proportional to the square of the charge. Consider a charge ofthat is spread over some space. If that charge is distributed over M pieces then each of the M points radiate an amountof energy proportional to (Q/M)². The M pieces altogether have a total radiation of M(Q/M)²=Q²/M. If M can go to infinity, as it would for a spatially distributed charge no matter how small the region of distribution, then the radiation goes to zero. Real charged particles are spatially distributed and so they do not radiate under acceeleration or deceleration.
(To be continued.)
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