Explain how one can write Maxwell's equations in relativistic form by introducing an antisymmetric field strength tensor $F_{a b}$.

In an inertial frame $S$, the electric and magnetic fields are $\mathbf{E}$ and $\mathbf{B}$. Suppose that there is a second inertial frame $S^{\prime}$ moving with velocity $v$ along the $x$-axis relative to $S$. Derive the rules for finding the electric and magnetic fields $\mathbf{E}^{\prime}$ and $\mathbf{B}^{\prime}$ in the frame $S^{\prime}$. Show that $|\mathbf{E}|^{2}-|\mathbf{B}|^{2}$ and $\mathbf{E} \cdot \mathbf{B}$ are invariant under Lorentz transformations.

Suppose that $\mathbf{E}=E_{0}(0,1,0)$ and $\mathbf{B}=E_{0}(0, \cos \theta, \sin \theta)$, where $0 \leq \theta<\pi / 2$. At what velocity must an observer be moving in the frame $S$ for the electric and magnetic fields to appear to be parallel?

Comment on the case $\theta=\pi / 2$.