Lorentz transformation of electromagnetic fields

Lorentz boosts mix transverse electric and magnetic field components.

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.