# How exactly do accelerated charges produce electromagnetic waves

So I know that accelerated charges produce changing magnetic fields which in turn produce an electric field, according to Faraday's law. (Of course keeping in mind that the electric and magnetic fields are perpendicular to each other)

What after that? How is the electromagnetic wave created?

• "changing magnetic fields [...] produce an electric field" - This is a very old urban legend. Charges produce fields. Fields don't produce fields. Oct 27 '19 at 3:44
• It is not an urban legend. There are vacuum solutions to Maxwell’s equations which contain fields without charges. Since there are charges in reality it is fair to say that those solutions are unphysical, but to say that it is an urban legend is going too far.
– Dale
Oct 27 '19 at 11:52
• An "electromagnetic wave" is identical to any electromagnetic field, because by Maxwells equations, all EM fields obey the Helmholtz wave equation. Radiation is a name given to the approximate field solutions for charges at a distance far from the charge, which can be characterized by the 1/r falloff we know and expect from radiation. The fact is, the EM field cannot propagate instantaneously, and so waves will always be generated when you "slosh a charge around" Oct 27 '19 at 14:04

Charges affect each other at a distance after a time delay, whether they are moving or accelerating or not.

When both are stationary we consider the force an electric field.

When the target is stationary and the source is moving directly toward or away from the target at constant velocity, it's still an electric field but the intensity of the force is different.

If the target is stationary and the source is moving sideways, it's still an electric field and the intensity is something else.

If the source and the target are both moving, and the source has some motion sideways, while the target has some motion in the same plane as the direction between them and the source velocity, then there is also a magnetic force. In this particular case, the equations for the electric force do not add up in different frames. In a frame where one charge is stationary magnetic force on the target is zero. In a frame where they are both moving, the equations give a different wrong result and magnetic force is needed to make up the error.

You get radiation only when the source charge is accelerating sideways.

The radiation will be a wave pattern when the motion of the source charge fits a wave.

So I know that accelerated charges produce changing magnetic fields which in turn produce an electric field, according to Faraday's law.

It is the "in turn" that is not correct, the law cannot apply to light as light has no charges. Look at this animation according to the mathematics of Maxwell equations:

Electromagnetic waves can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. This 3D animation shows a plane linearly polarized wave propagating from left to right. The electric and magnetic fields in such a wave are in-phase with each other, reaching minima and maxima together

The solutions of Maxwell's equations describe the behavior of classical electromagnetic waves. Charges have electric fields, moving charges create magnetic fields , and these observations become the laws, entering Maxwell's theory axiomatically . It is a full semester course to understand classical electrodynamics in all its complications. Yes, accelerated charges produce electromagnetic waves, as the one shown in the annimation.

To understand how the classical wave is created from the underlying quantum mechanical framework of nature requires studying both quantum mechanics and quantum electrodynamics, QED. More than semester courses.

It can be shown that the classical electromagnetic wave emerges from a confluence of photons, the photon is quantum of light, not in an additive way, but in a superposition of the quantum mechanical wavefunctions of the photons.

• Anna, would you mind clarifying why it takes acceleration of a charge to produce radiation? Oct 27 '19 at 6:04
• If there is no acceleration , classically the particle keeps in a straight line and does not lose energy. If it does not lose energy no radiation will come out, because the radiations carries energy. en.wikipedia.org/wiki/Poynting_vector#Plane_waves . Quantum mechanically if there is no dp/dt in an interaction to give to the photon energy and momentum there can be no photon (dp/dt is instantaneous acceeration) Oct 27 '19 at 8:32
• What about when the speed increases? For example when the electron accelerates in an atom while falling down from a higher state to a lower state Oct 27 '19 at 8:38
• @SwaroopJoshi In quantum mechanics, the whole atom is in an energy level, if it is in a higher level, there is a probability for the atom to emit a photon and go to a lower energy level . The steps are discreet, that is why there are spectral lines. The model of an electron around a nucleus is a semi classical model, and the photon has to be postulated in the Born model. In QED it happens because of the mathematics of quantum electrodynamic field theory. Oct 27 '19 at 11:15
• Okay, anyway, so those anti-nodes in the magnetic field are caused due to the instantaneous velocity of the electrons being 0 when they reach a crest/trough? Oct 30 '19 at 3:25