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?
Answer
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.
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