I'm surprised you haven't found this information online because the equation is one of the most basic in all of physics:
$$\mathbf{F} = m\mathbf{a} = m\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t}$$
(bold represents a vector). For an application like yours, it will be more useful to rearrange this in the form of an evolution equation
$$\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t} = \frac{\mathbf{F}(\mathbf{v}, t)}{m}$$
There are two ways to handle this.
- If the magnetic field configuration is known and is given by a simple function, you may be able to solve the evolution equation analytically. In that case, just program that solution into your code and have the program calculate the position of the particle at each time.
In general, that is not practical, so you will have to investigate the field of numerical integration. Here is a primer: as a crude method of solution, you can replace the derivative operator with a finite difference operator, e.g. for the $x$-component:
$$\frac{\mathrm{d}v_x}{\mathrm{d}t} \to \frac{v_x(t + \delta t) - v_x(t)}{\delta t}$$
Plugging this into the evolution equation, you can rearrange it to
$$v_x(t + \delta t) = v_x(t) + \frac{1}{m}F_x(v_x(t), v_y(t), v_z(t), t)\delta t$$
Since you have a formula for calculating $F_x$ as a function of $\mathbf{v}$ and $t$, you can use the $x$-velocity at any given time step to calculate an approximation to the $x$-velocity at the next time step, and similarly for the other components. This is called the Euler method for numerical integration. Although it's easy to understand, it's highly unstable - that is, it becomes very inaccurate very quickly, because calculating the velocities and forces only at the beginning of each time step (as opposed to, say, the beginning and end and taking an average) leads to errors that accumulate at each time step.
There are more sophisticated methods of numerical integration which arrange for the errors to largely cancel out by calculating values of $F_x$ within the time step. Whatever programming language or framework you are using, there is probably a scientific computation library that includes a numerical integrator, which you should look into using. That sort of thing is beyond the scope of this site, however. (Try searching on Stack Overflow)
