In special relativity, the electromagnetic fields can be derived from a 4-vector potential $A^{\mu}=(\phi / c, \mathbf{A})$. Using the Minkowski metric tensor $\eta_{\mu \nu}$ and its inverse $\eta^{\mu \nu}$, state how the electromagnetic tensor $F_{\mu \nu}$ is related to the 4-potential, and write out explicitly the components of both $F_{\mu \nu}$ and $F^{\mu \nu}$ in terms of those of $\mathbf{E}$ and $\mathbf{B}$.

If $x^{\prime \mu}=\Lambda_{\nu}^{\mu} x^{\nu}$ is a Lorentz transformation of the spacetime coordinates from one inertial frame $\mathcal{S}$ to another inertial frame $\mathcal{S}^{\prime}$, state how $F^{\prime \mu \nu}$ is related to $F^{\mu \nu}$.

Write down the Lorentz transformation matrix for a boost in standard configuration, such that frame $\mathcal{S}^{\prime}$ moves relative to frame $\mathcal{S}$ with speed $v$ in the $+x$ direction. Deduce the transformation laws

$$\begin{aligned} E_{x}^{\prime} &=E_{x} \\ E_{y}^{\prime} &=\gamma\left(E_{y}-v B_{z}\right) \\ E_{z}^{\prime} &=\gamma\left(E_{z}+v B_{y}\right) \\ B_{x}^{\prime} &=B_{x} \\ B_{y}^{\prime} &=\gamma\left(B_{y}+\frac{v}{c^{2}} E_{z}\right) \\ B_{z}^{\prime} &=\gamma\left(B_{z}-\frac{v}{c^{2}} E_{y}\right) \end{aligned}$$

where $\gamma=\left(1-\frac{v^{2}}{c^{2}}\right)^{-1 / 2}$

In frame $\mathcal{S}$, an infinitely long wire of negligible thickness lies along the $x$ axis. The wire carries $n$ positive charges $+q$ per unit length, which travel at speed $u$ in the $+x$ direction, and $n$ negative charges $-q$ per unit length, which travel at speed $u$ in the $-x$ direction. There are no other sources of the electromagnetic field. Write down the electric and magnetic fields in $\mathcal{S}$ in terms of Cartesian coordinates. Calculate the electric field in frame $\mathcal{S}^{\prime}$, which is related to $\mathcal{S}$ by a boost by speed $v$ as described above. Give an explanation of the physical origin of your expression.