# $Etimes B$ drift in strongly nonuniform fields

Physics Asked by Timofey Chernyshev on November 30, 2020

Potential is defined as $${phi,, 0,, A,, 0 }$$; fields are static and depend only on the axial coordinate $$x$$: $$E_x=-partial_xphi$$, $$B_z=partial_x A$$. Charged particle moves in the $${x,y}$$-plane, where $$y$$ is a direction of drift-motion.

There are two constants of motion:
$$frac{m}{2} left(v_{x}^{2} + v_{y}^{2} – v_{x0}^{2} – v_{y0}^{2}right) -ephi$$
and
$$m left(v_{y} – v_{y0}right) – frac{A e}{c}$$, where $$v_{x0}$$ and $$v_{y0}$$ are velocities at given initial point $$x_0$$ (where $$phiequiv0$$ and $$Aequiv0$$).

We assume that $$phi$$ and $$A$$ are monotonic. Thus, a solution for velocities can be expressed as
$$v_x = pmsqrt{- frac{A^{2} e^{2}}{c^{2} m^{2}} – frac{2 A e v_{y0}}{c m} + frac{2 e phi}{m} + v_{x0}^{2}}\v_y = frac{A e}{c m} + v_{y0}$$
And it works fine in the case of uniform fields. However, it gives multi-valued loops in some cases of nonuniform fields. Figures below show both the correct and incorrect examples. The only difference here is a selection of $$x_0$$.

So, the question is how to determine the incorrect branch of the solution without step-by-step calculation from $$x_0$$ to $$x$$?

In these examples: $$phi(x) = {rm erf}(exp(10x))-phi_0$$ and $$A(x)=164int (1-x^2)^2dx – A_0$$, $$v_{x0}=v_{y0}=0$$, position of incorrect solution is around $$xapprox-0.335$$.

The phase-loops are shown below for both solutions:

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