Magnetic field energy of electron

• I
I've got a question regarding the magnetic field of an electron, and wheter or not it has some form of "self inductance" or resistance to be put in motion.

The magnetic energy per volume is equal to u= 1/2*μ*H2.

Say I've got an electron at rest, then the energy of the magnetic field is zero. OK.
If I give it a push so it reaches an arbitrary velocity, then it will set up a magnetic field in space, which requires some energy to create according to the formula above.

Will the electron require some extra energy other than the kinetic energy to be put in motion? If so, why isnt it mentioned anywhere and is it significant? And if not, where does it get the energy from? The electric field? Or is the answer in neither category?

I tried asking my lecturers about this, but they didnt know the answer. I also tried to find the total field energy of ∫u dV over space, but I got a diverging integral since I treated the electron as a point particle... It was however dependent on the square of velocity, so it could somehow be "merged" into 1/2*m*v2. But this whole thing strikes me as odd and a bit out of place, so I would be happy if someone could clear it up.

OmegaKV

Drakkith
Staff Emeritus
I don't know the mathematical details of this, but I don't believe any extra energy would be required. Consider an observer who is already in motion relative to the electron. They see the EM field from the electron as partly electric and partly magnetic. If I instead start with an observer at rest relative to the electron and then accelerate that observer, would that somehow imply that work was performed on the electron since it would now have magnetic energy from the perspective of the accelerated observer?

I believe that when you put the electron in motion you simply "transfer" some energy from the electric component of the field to the magnetic. I put transfer in quotes because I don't think it's the right term. That seems to imply that extra work had to be done, which I don't think is true.

Hopefully someone more knowledgeable in this area can chime in.

Well, it is more energy consuming to accelerate two electrons when they are close to each other (justification: potential energy has mass in special relativity), and there is more energy in the magnetic field created by the two electrons when they are close to each other (justification: I checked some magnetic field energy formulas).

But the above does not really matter because:

Electron's mass is 9.10938356 × 10-31 kg, so electron behaves like a normal classical object with that mass. But only in vacuum, and when not accelerated too quickly.

We have decided that electron's mass is the mass it seems to have in vacuum, we might even say with tongue in cheek that electron's mass is its effective mass in vacuum.

https://en.wikipedia.org/wiki/Effective_mass_(solid-state_physics)

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Drakkith
Staff Emeritus
It's certainly true that a system of two electrons close together have more mass than two electrons far apart thanks to the increase in potential energy, but I don't quite see how that, or the rest of your post, is related to the OP's question.

I've got a question regarding the magnetic field of an electron, and wheter or not it has some form of "self inductance" or resistance to be put in motion.

I forgot to mention that charged particles undergoing acceleration will radiate energy away as EM radiation. But I think this is different from what you were asking about. The energy required to create the EM radiation is not stored in the particle's magnetic field, but is radiated away. It cannot be "retrieved" if you slow the particle down.

It's certainly true that a system of two electrons close together have more mass than two electrons far apart thanks to the increase in potential energy, but I don't quite see how that, or the rest of your post, is related to the OP's question.

OP was interested about the kinetic energy and the magnetic energy of a small charged particle. I tried to point out that in relativity squeezing a moving electron pair increases the kinetic energy of that system by increasing the rest mass of the system, while in classical physics the mass does not change, but a more energetic magnetic field appears around the more compact pair.

Classically we can have a microscopic particle with charge 1 Coulomb and mass 10-99 kg. Classically it would be a very good approximation to say that the mass of that particle is zero, and any work done by accelerating that particle becomes the energy of the magnetic field around the particle, right?

Now the question is, how wrong is classical physics in that extreme case I invented, and how wrong is it in less extreme cases? I mean how wrong is classical physics about the energy needed to accelerate small charged things.

Drakkith
Staff Emeritus
Classically it would be a very good approximation to say that the mass of that particle is zero, and any work done by accelerating that particle becomes the energy of the magnetic field around the particle, right?

That I don't know.

It is a good point you are raising about the accelerating observer, Drakkith. I havent thought about it from that perspective.
I am also a bit curious as to how one could express the magnetic/electric field near an electron. In wires and conductors its generally pretty easy to use the current density. Are there any expression for the charge distribution in an electron? And how would quantum mechanics such as the wavelike nature of particles and the uncertainty principle affect this?

Also, thanks to those who have replied!

Drakkith
Staff Emeritus