This problem reconstructs the trajectory r₀(t) of a charged particle from the fields produced by the particle at observation points where the magnetic field does not vanish.

(a) Use cB = n̂_ret × E to deduce that

The Li´enard-Wiechert electric field determines the factor q/|q|. 

(b) Use the information in part (a) to show that

(c) Evaluate the formula for cB in part (a) using the Heaviside-Feynman electric field, solve for R_ret, and use the result of part (b) to conclude that

(d) Part (b) and (c) express R = Rn̂ = r − r₀ at the retarded time entirely in terms of the fields at the present time. Show that we also have enough information to express t_ret entirely in terms of present-time quantities.

fret - E E = 9 |q| E2-cB.