Introduction to Electrodynamics · Problem 10.30

Potentials and Fields — David J. Griffiths

Physics · retarded-potentials

Problem

A uniformly charged rod (length LL, charge density λ\lambda) slides out the xx axis at constant speed vv. At time t=0t = 0 the back end passes the origin (so its position as a function of time is x=vtx = vt, while the front end is at x=vt+Lx = vt + L). Find the retarded scalar potential at the origin, as a function of time, for t>0t > 0. [First determine the retarded time t1t_1 for the back end, the retarded time t2t_2 for the front end, and the corresponding retarded positions x1x_1 and x2x_2.] Is your answer consistent with the Li\'{e}nard-Wiechert potential, in the point charge limit (L≪vtL \ll vt, with λL=q\lambda L = q)? Do not assume v≪cv \ll c.

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