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Part of the first answer to the question: Is there a travelling speed of for electric field? If yes, what is it? says:

...any change in an established field (say, due to shifting the position of the charge creating the field) won't be felt by a distant object until enough time has passed for a photon from the source to make it to the observation point.

My question is: if we keep in the classical field framework, can we say: "...until enough time has passed for an EM wave...?"

Suppose a charge and an observation point in the same frame of reference. If the charge is suddenly moved (and stopped after that) radially to a place closer to the observation point, the electric field there must increase after a delay. But the EM wave, sent by the charge while being accelerated, doesn't have a component of E in the radial direction, only perpendicular to it.

So I don't understand how it can increase the radial component of E.

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But the EM wave, sent by the charge while being accelerated, doesn't have a component of E in the radial direction, only perpendicular to it.

This is not entirely correct. The total field can be decomposed into a monopole and a dipole source. The dipole term has no radial component along this line, but the monopole term does. The monopole term falls off as $1/r^2$, so it is not considered radiation, but that doesn’t mean it is not allowed to change.

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Static force fields are fields, such as a simple electric, magnetic or gravitational fields, that exist without excitations. The most common approximation method that physicists use for scattering calculations can be interpreted as static forces arising from the interactions between two bodies mediated by virtual particles, particles that exist for only a short time determined by the uncertainty principle.1 The virtual particles, also known as force carriers, are bosons, with different bosons associated with each force

The quote you give assumes the above description, i.e. the underlying quantum mechanical picture, which has the velocity of light limit.

The wave created in you example by the accelerating and descellerating charge does not carry information of the the charge, in classical electrodynamics. There will be a light pulse, which will depend on the acceleration of the charge, and the two cannot be unscrambled without further data on the motion. There will be the energy the light pulse carries, the direction, and possibly polarization. In any case the light pulse will arrive with the velocity of light and its E and B will depend on the acceleration, not only on the original field.

So you question becomes, suppose the light pulse detectors are on the moon. If we had an accurate electric field detector on the moon, would we measure the difference in the electric field before the light arrived?

In the quantum framework the answer is , no, because nothing goes faster than light , the photon wave functions making up light are Lorenz invariant and all the quantum mechanical formulations . It must be no in the classical since the classical emerges from the quantum framework.

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