Does the theory of electromagnetism have this fundamental problem?

[francis20520]

This is about possible problems with the equations of EM theory of Faraday and Maxwell.

EM theory states that when an electron moves a magnetic field is generated.

Does this field move with electron in the direction of the electron (if the electron is in motion)?
Or, does it stay still in the same place (it originated)? If so it stays still with respect to whom?

So, is the above a fundamental problem of EM theory? Is it solved or does it remain unsolved?

 

[Ibix]

Fields don't have velocities in any meaningful sense.

The magnetic field of a moving electron will, loosely speaking, be centered on the electron's current position. There are a few caveats around high accelerations (it takes time for the change in the field to propagate outwards), but you will not find the field centered around where the electron started moving. It's not clear what that position would even mean.

 

[vanhees71]

There's only a problem when you want to take into account the back reaction of the particle's own em. field on its motion (the infamous problem with "radiation reaction"). The best approximate solution (not fully self-consistent) is the Landau-Lifshitz solution.

The electromagnetic field of a point particle in given motion is solved for more than 100 years. It's known as the Lienard-Wiechert retarded potential. For details, see

https://itp.uni-frankfurt.de/~hees/pf-faq/srt.pdf

Sect. 3.5 (p. 59). In the following section, I've given the particularly simple solution for a uniformly moving charge, which you can alternatively also get by using a Lorentz transformation of the Coulomb field of a charge at rest.

 

[francis20520]

Fields don't have velocities in any meaningful sense.

The magnetic field of a moving electron will, loosely speaking, be centered on the electron's current position. There are a few caveats around high accelerations (it takes time for the change in the field to propagate outwards), but you will not find the field centered around where the electron started moving. It's not clear what that position would even mean.

Thanks for the quick reply. So r you saying the there is no "Fundamental problem" like which I stated in my post??

So you are saying the magnetic field will be moving (Centered around) the electron??

Is there any literature you can point me to this? I tried Wikipedia but was not successful.

 

[Ibix]

Is there any literature you can point me to this? I tried Wikipedia but was not successful.

See @vanhees71's response, mentioning the Lienard-Wiechert potentials (which can be googled) and with a link to his lecture notes.

 

[PeroK]

Is there any literature you can point me to this? I tried Wikipedia but was not successful.

https://en.wikipedia.org/wiki/Introduction_to_Electrodynamics

 

[vanhees71]

Thanks for the quick reply. So r you saying the there is no "Fundamental problem" like which I stated in my post??

So you are saying the magnetic field will be moving (Centered around) the electron??

Is there any literature you can point me to this? I tried Wikipedia but was not successful.

I don't know what you mean by "the magnetic field will be moving". A magnetic field is a magnetic field, i.e., a function $\vec{B}(t,\vec{x})$ with a vector-valued value at any given point in space at any given time. As I said you find the solution for a uniformly moving point charge in my manuscript (and any textbook on electromagnetism, for sure also at other online sources). For this and also general acclerated motion the solution is given by the retarded Lienard Wiechert potentials and the resulting em. field.

 

[PeroK]

So you are saying the magnetic field will be moving (Centered around) the electron??

A field, by definition, is a quantity at every point in space. The field doesn't move, as such, but the field values at each point in space may change over time.

An electron at rest creates a static (constant in time) electric field. Pick any point in space and the electric field there is constant over time.

An electron in motion creates a time-dependent Electromagnetic field. Pick any point in space and the electric and magnetic fields at that point wil change over time (according to the solution referred to in post #3).

 

[Ibix]

So you are saying the magnetic field will be moving (Centered around) the electron??

To reiterate what's already been said about this, fields do not and cannot have velocities. Assuming that the electron has a constant velocity, the disturbance of the EM field will take the form of $B(\vec x-\vec vt)$ where $\vec v$ is the velocity of the electron, which is to say that you have a disturbance of the EM field that "travels" with the electron, yes. That shouldn't be read as "the field is moving".