The energy of the magnetic field

[Psi-String]

The energy of the magnetic field, by definition, is the energy needed to establish the field. So if we establish the magnetic field by steady current, we need to establish the current. My question is, in the derivation of

$$U = \frac{1}{2 \mu_0} \int B^2 d\tau$$

it seems that the work we done is to against the induced electric field due to the varying current while establishing the current. But how about the kinetic energy of the charge particle?? Don't we need to include this energy or it is small enough that we can neglect it??

 

[Jhero]

Thank you for your question. The equation you have mentioned, U = (1/2μ0)∫B^2dτ, is the energy stored in the magnetic field and is derived from the work done to establish the field. This work is done against the induced electric field, as you have correctly stated.

The kinetic energy of the charge particles is not included in this equation because it is not directly related to the establishment of the magnetic field. The kinetic energy of the charge particles is a result of the interaction between the magnetic field and the charged particles. This energy is not needed to establish the field, but rather it is a consequence of the field's presence.

In most cases, the kinetic energy of the charged particles is much smaller compared to the energy stored in the magnetic field. Therefore, it is typically neglected in the derivation of this equation. However, in some specific cases, such as in particle accelerators, the kinetic energy of the charged particles may need to be taken into account.

I hope this answers your question. If you have any further inquiries, please do not hesitate to ask.