Where does the Magnetic Field Energy come from, in case of a charged particle moving with constant velocity?

A charged particle moving with constant velocity creates Magnetic Field around it. If there is Magnetic Field, then there must be Magnetic Field Energy. The question is, where does the Magnetic Field Energy come from, in case of a charged particle moving with constant velocity?

The electric and magnetic fields are components of the electromagnetic field described by the electromagnetic tensor. Components of this tensor change under the Lorentz transformation while obeying the energy conservation law. This means that the energy of the magnetic field increases with the speed in the same proportion as the energy of the electric field simultaneously decreases. The total electromagnetic energy remains the same.

In the contravariant matrix form, the electromagnetic tensor is given by $$F^{\mu\nu} = \begin{bmatrix} 0 & -E_x/c & -E_y/c & -E_z/c \\ E_x/c & 0 & -B_z & B_y \\ E_y/c & B_z & 0 & -B_x \\ E_z/c & -B_y & B_x & 0 \end{bmatrix}.$$

Its determinant is Lorentz invariant:

$$\det \left( F \right) = \frac{1}{c^2} \left( \mathbf{B} \cdot \mathbf{E} \right)^2$$

From here you can see that $\mathbf{B} \cdot \mathbf{E}$ does not depend on the speed. The inner product also is Lorentz invariant (does not depend on the speed)

$$F_{\mu\nu} F^{\mu\nu} = 2 \left( B^2 - \frac{E^2}{c^2} \right)$$

As the magnetic field increases with the speed, the electric field proportionally decreases and the total electromagnetic energy of a moving charge remains the same. The increase of the energy of the magnetic field with the speed comes from the decrease of the energy of the electric field. As a result, charge does not appear in the Newton's laws of the classical dynamics.

• So the original energy came from charging the particle? Sep 1, 2018 at 18:21
• @drake01 Yes, but please keep in mind that this energy does not depend on the speed. In fact, you can see here that (in the classical limit) three quarters of the total mass-energy of the electron come from its charge (the energy of the electromagnetic field): en.m.wikipedia.org/wiki/Electromagnetic_mass Sep 1, 2018 at 18:41

where does the Magnetic Field Energy come from, in case of a charged particle moving with constant velocity?

The energy comes from the same source that sped up the particle to that velocity, which could be heat, electric field, etc.

• This is incorrect. Speed is relative. Instead of moving a charged object, I can move myself. By doing so I would spend the same amount of energy whether the object is charged or not. Sep 1, 2018 at 6:27
• @safesphere I did not say that the energy spent to speed up a charge or a charged body is equal to the magnetic energy needed to speed it up: I said that the energy for both comes from the same source. If your body was charged, you would spend more energy speeding it up, because, besides developing kinetic energy, you would also create magnetic field and spend some additional energy for that.
– V.F.
Sep 1, 2018 at 11:22
• This is 100% incorrect. Speeding up a charged body does not take more energy. The energy of a moving body depends on the mass and speed only. A charge does not appear in Newton's laws of motion. This is obvious because speed is relative. See my earlier comment. To see how the energy balances out, see my answer. Sep 1, 2018 at 15:28

This response has its primary inspiration from the response of V.F. I have further added my own thoughts into it.

A charged particle moving with a constant velocity has a magnetic field energy associated with it. This energy actually comes from the external agent which gave the charged particle the velocity. During the transient period, the velocity of the charged particle was non uniform and so the magnetic field in the space surrounding the charge was not constant with time, which will give rise to the generation of Non Conservative Electric Field, which will take away some energy of external agent, and store it in the magnetic field in the surrounding of charge.

When charge attains a constant velocity, the magnetic field energy in the space behind it will decrease and in the space ahead of it will increase and the net magnetic field energy associated with the charge will remain the same, which was earlier received from the source, which gave the charge, the kinetic energy.

In this way, we can say, source has supplied energy at two places - 1. In the kinetic energy of the charge. 2. In the magnetic field energy around the charge.

• This is incorrect. Classical dynamics does not depend on the charge. Sep 1, 2018 at 15:41