Chapter 33, Problem 74P

Consider a line of positive charge with line charge density λ 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 = λ /27π∈₀r and that there's no magnetic field (no relativity needed here). Now consider the situation in a frame S' moving at speed v parallel to the line of charge, (b) Show that the charge density measured in S′ is given by λ′ = γλ, where $\gamma =1/\sqrt{1-v^2/c^2}$. (c) Use the result of (b) to find the electric field in S′. Since the charge is moving with respect to S′, there’s a current in S′. (d) Find an expression for this current and (e) for the magnetic field it produces. Determine the values of the quantities (f) $\stackrel{\to }{E}\cdot \stackrel{\to }{B}$ and (g) E² − c²B² 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 $\stackrel{\to }{E}\cdot \stackrel{\to }{B}$ and E² − c²B² are always invariant.