Finding motion/position of a particle travelling through static magnetic field of a bar magnet

Question

I am trying to figure out an expression for the position of a charged particle travelling downwards in the z-direction at some constant velocity through the static magnetic field generated by a bar magnet.

Following this post I have the expression for the magnetic field

$\mathbf{B}=\frac{\mathrm{B_r} V}{4 \pi \sqrt{x^2+y^2+z^2}^5} \left(3zx,3zy,2z^2-x^2-y^2\right)$

Where $B_r$ is the magnetic flux density and $V$ is the volume of the magnet. I then plug this into the relativistic Lorentz force equation $\frac{dp}{dt}$ = $\frac{e}{c}v\times B$, from which I should be able to get a function describing the position of the particle. However I am unsure how. If I take one of the components, e.g. the x component

m$\ddot{x}$ = $\frac{e}{c}$($-v_zB_y$)

and integrate by time twice to get a function of x, then I still have B as part of the function, which is a function of the space coordinates. I would like a function that has t as an argument alone. I assume I am missing something, but I am not sure what. Any help is appreciated.

Answer 1

In general, the equation(s) of motion will be position dependent. There may or may not be an explicit time dependence. Take the 1D simple harmonic oscillator with force $F = -kx$. Newton's second law reads $$m\ddot x=-kx.$$ Time doesn't explicitly appear in this equation, yet it's a differential equation that's easily solved (partly because of the lack of time dependence).

The equation you have isn't easy (if not impossible) to solve analytically, except for certain initial conditions. You would normally either employ certain simplifications such as approximating the magnet's field as being uniform, or solve the problem numerically. See for example Runge-Kutta or Verlet methods.