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Let us consider these two Maxwell equations:

$$\frac{\partial \vec{B}}{\partial t}=-\vec{\nabla}\times \vec{E}$$ and

$$\frac{\partial \vec{E}}{\partial t}=\frac{1}{\epsilon_0}\left(-\vec{J}+\frac{1}{\mu_0}\vec{\nabla}\times \vec{B}\right).$$

When we consider faraday's law of induction, we usually assume that the changes are slow, and thus we can neglect radiation by assuming that the left hand side of the second equation is zero. That is, a changing current creates a changing magnetic field, which in turn creates a changing electric field, per the first equation.

I cannot understand this. First, if we can neglect the change in E from the second equation, should not we also neglect the change in B in the first equation? Second, this imply that we can have changing electric and magnetic fields that are not electromagnetic waves. But are not all changing magnetic or electric fields EM waves? or is this approximation equivalent to a charge moving at constant speed, in which the change in E and B are not due to radiation but just to the translation motion of the static field lines

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An electric or magnetic field always obeys a wave equation this can be proven by eliminating one or the other from the two equations that you display. In order to qualify as radiation the wave should transport energy, that is, propagate. Evanescent fields exist only near the current or charge, see https://en.wikipedia.org/wiki/Evanescent_field. These do not transport energy away from the source and are not radiation. For slowly time varying currents and charges the fields are nearly purely evanescent. The faster the time variation, the higher the fraction of propagating fields is.

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Regular induction in a coil satisfies Maxwell’s equation (usually the integral form is used) and it’s usually not considered radiation.

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Yes, you can have changing fields without radiation.

I suggest looking at the electric and magnetic fields derived from the L-W equation, which gives different insights from Maxwell's equations.

Here is one sample derivation.

The electric field of a point charge moving at a constant velocity $v$, as derived from the Liénard-Wiechert equation, reads $$ E = \frac{q}{4\pi\epsilon_0} \frac{(1-\beta^2)}{(1-n\cdot\beta)^3}\frac{n-\beta}{|r-r_s|^2},$$

where $r$ is the location of the stationary target when the force is applied

$r_s$ is the location of the source when the force leaves it

$ n =\overline{r-r_s}$ and

$ \beta = v/c $.

The rest of the electric force equation depends on acceleration, and that part is considered radiation. This part will change when $v \not= 0$.

This force is correct in a frame where the target does not move.

If we choose a frame where source and target both move in a direction unaligned with $r-r_s$, the electric force equation fails, and the magnetic force equation then provides a fudge factor which corrects the error.

I want to point out that the electric and magnetic fields do not create each other. A moving electric charge creates both of them. The electric charge is the creator, and the fields do not affect each other after they are created.

"But are not all changing magnetic or electric fields EM waves?"

No, not all. First, charges that move without acceleration can create electric fields that change but that do not include radiation.

Second, not all radiation is waves as we usually think of them. Apparently any acceleration of a charge creates radiation. You get a wave if the radiation travels in a periodic way.

So if a charge travels up and down in a sine wave, it creates a linearly-polarized wave, polarized up-down. A radio tower does this. Directly above the radio tower the signal is minimized.

If a charge travels in a circle, then if you stand edge-on to the circle the radiation is linearly polarized perpendicular to the axis of rotation. Along the axis there is a wave that is circularly polarized. In between there are combinations of both and the sum is elliptically polarized.

If a charge travels in some erratic pattern it doesn't exactly make a wave. But you can make a fourier transform to convert whatever it does into some combination of sine waves. In that sense, any charge motion whatsoever is a wave of some sort.

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I would rephrase your question.

Is a static electric or a static magnetic field without charges possible?

It is obvious that a static electric field we get by separating electric charges. How about a static magnetic field? We get it by the alignment of the magnetic dipole moments of the involved subatomic particles. This arrangement is static only if the self-inductance is prevailing the thermic movement of the subatomic particles. This is the case for permanent magnets at room temperature or materials very close to zero Kelvin. Trying to find electric or magnetic fields without separated or aligned charges is inconceivable.
Conclusion: Static electric and static magnetic fields without charges are impossible. (Mostly the charge is the electron with its electric charge and its magnetic dipole moment.)

How do we induce an electric and a magnetic static field?

The charge separation as the reason for any electric field is accompanied by an electric current. Faraday induced an current by rotating a conduction disc inside an external magnetic field. The only interaction between the moving electrons and the magnetic field could be the electrons magnetic dipole moment. With the magnetic dipole moment is a spin associated and this together is the reason for the charge deflection and separation. (The inverse process for example are the Lorentz force or the Hall effect.)

The purest process is the one, we observe in free space. An electron - visible in a cloud chamber - gets deflected under the influence of an external magnetic field and moves in a spiral path until its kinetic energy is exhausted. The kinetic energy by this gets emitted as EM radiation.
Conclusion: A charge separation in an inducting process without the emission of EM radiation is impossible.

The same holds for a static magnetic field of aligned magnetic dipoles. The energy for the alignment has to be realized during the disalignment and this again happens in the form of EM radiation.

Electromagnetic propagation

The only case in which the magnetic and an electric field are independent from charges is the emission of photons. A selfinducting portion of energy is emitted from an accelerated charge and this quanta propagates in free space until it gets absorbed by another subatomic particle.

The observation, that a moving charge gain a magnetic field due to its relative motion to the observer, is a conclusion from 200 years ago without the knowledge of the nature of charges: their intrinsic electric field, their intrinsic magnetic dipole and their intrinsic spin with the emission of EM radiation during accelerations.

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