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'The Lorentz Force Law' printed from https://nrich.maths.org/
In Cartesian coordinates, the position ${\bf r}$ of the charged
particle at time $t$ is determined by three functions$x(t), y(t),
z(t)$, so that $${\bf r} = (x(t), y(t), z(t))$$
In component form, the Lorentz force law becomes
$$\left(\frac{m}{q} \ddot{x}, \frac{m}{q} \ddot{y}, \frac{m}{q}
\ddot{z}\right)
=
\left( E_x+\dot{y}B_z-\dot{z}B_y, E_y+\dot{z}B_x-\dot{x}B_z,
E_z+\dot{x}B_y-\dot{y}B_x \right)
$$
You can then equate components to make three separate
(simultaneous) differential equations.
Try differentiating these to solve for particular cases of ${\bf
E}$ and ${\bf B}$.
You can make life easier for yourself by supposing the the electric
and magnetic fields do not change over time.