a) A particle of charge e and rest mass m moves with an arbitrary velocity in the uniform constant electric field E. At the initial instant of time, t = 0, the particle was at the position x = 0, y = 0, z = 0 and had momentum p0. Determine the three dimensional coordinates and the time t of the particle in the laboratory system as functions of its proper time . By eliminating express the three dimensional coordinates as functions of t. Consider in particular ultra-relativistic and non-relativistic limits. Radiation losses are neglected. Hint: start from equations of motion expressed in 4-dimensional form.
b) Find the trajectory of this particle and make the plot. Also discuss the nonrelativistic limit.