Solving Electric and Magnetic Fields for Charged Particle Path

Question

I am using the Lorentz Force Equation and the electric-cross-magnetic field velocity equation] to solve for the $E$ and $B$ fields given the known path of a particle moving in 3D.

So with that I have the following equations where a and v are known: Lorentz Form and the E-cross-B Form

My question: Are these equations enough to solve for the $x, y, z$ components of $B$ and $E$?

----------Edit---------------

So this is actually being used as an analogy for the propagation of nano-scale self replicating cracks in 3D. In this analogy, the incoming tensile force is represented by the electric force, and the delamination is represented by the magnetic force.

So I have a parabaloid spiral shaped crack which will represent the motion of a charged particle. Since I know the shape/path I can directly get the position, velocity, and acceleration functions in each direction.

With that said, is there a way to use the two equations linked to find all components of the electric and magnetic fields?

Answer 1

Consider the case where the particle is moving in a straight line with constant velocity. Then the magnetic field in the direction of that velocity has no effect.

Answer 2

One interesting case when a charged particles is at rest initially in the crossed electric and magnetic fields, it moves in a cycloid.

See the video here.

See also a detailed discussion in Chapter 2 of Fundamentals of Plasma Physics by Bittencourt, J. A. here

Answer 3

Well, I'm not sure if they are sufficient, but you could add magnetostatic equations if your magnetic field is constant over time, or instead the general equation $$\nabla \times \overrightarrow{E} = -\frac{\partial \overrightarrow{B}}{\partial t}$$