Consider a line of positive charge with line charge density $\lambda$ as measured in a frame $S$ at rest with respect to the charges.
(a) Show that the electric field a distance $r$ from this charged line has magnitude $E=\lambda / 2 \pi \epsilon_{0} r,$ and that there's no magnetic field (no relativity needed here). Now consider the situation in a frame $S^{\prime}$ moving at speed $v$ parallel to the line of charge. (b) Show that the charge density measured in $S^{\prime}$ is given by $\lambda^{\prime}=\gamma \lambda$, where $\gamma=1 / \sqrt{1-v^{2} / c^{2}},$ (c) Use the result of (b) to find the electric field in $S^{\prime}$. since the charge is moving with respect to $S^{\prime}$, there's a current in $S^{\prime}$, (d) Find an expression for this current and (e) for the magnetic field it produces. Determine the values of the quantities ( $f$ ) $\vec{E} \cdot \vec{B}$ and $(g) E^{2}-c^{2} B^{2}$ in both reference frames, and show that these quantities are invariant. Your result gives a hint at how electric and magnetic fields transform, and demonstrates one instance of the fact that $\vec{E} \cdot \vec{B}$ and $E^{2}-c^{2} B^{2}$ are always invariant.