The method of J. J. Thomson of analyzing the kink in the electric-field lines is used to obtain the expression for the electric field of a charged particle undergoing a parallel acceleration while moving atanyvelocityvless thanc. Thomson originally applied the method to aslowlymoving charge and thereby obtained a result correct to zeroth order inv/c. The technique is rigorously and directly applicable, without approximation, to a charge moving with large velocity. Thus one can obtain the acceleration field of a fast-moving charge without resorting to more elaborate treatments such as the use of the Lienard-Wiechert potentials.