Lorentz transformation of electromagnetic fields

Lorentz boosts mix transverse electric and magnetic field components.

An ideal long solenoid has $B={\mu }_{0}nI$, with axial turn density n. Under an axial boost, turn density becomes $n^{\prime }=\gamma n$. The transverse circulation path is unchanged while a turn's charge-circulation rate, measured relative to its moving material, becomes $I^{\prime }=I/\gamma$. Thus $n^{\prime }{I}^{\prime }=nI$ and the axial magnetic field remains unchanged. Equivalently, the ideal transverse current-sheet density is invariant under this boost. End effects are excluded.

For ideal oppositely charged parallel plates, the normal electric field is $\sigma /{ϵ}_0$. A boost along that normal leaves each transverse plate area unchanged. Invariance of total charge therefore preserves surface charge density and the normal electric field. The plate separation contracts, but the large-plate field does not depend on that separation. Including longitudinal contraction in a voltage argument while holding voltage fixed would impose an unsupported extra assumption.

Two equal opposite line charge densities moving in opposite directions have zero net line charge but a nonzero total current $I$. Under a longitudinal Lorentz boost, the charge-current four-vector gives ${\lambda }^{\prime }=\gamma (\lambda -vI/c^2)=-\gamma vI/c^2$. Equivalently, for lab densities $n$ and speeds $\pm u$, their separate boosted number densities are $\gamma n(1\mp vu/c^2)$. Their imbalance creates the boosted radial electric field, consistently with Lorentz transformation of electromagnetic fields and Gauss's law.

If a plane is stationary in one frame and an observer moves normally to it with velocity $\mathbf{v}$, thenThe second relation accounts for the magnetic-field discontinuity moving through the observer's frame.