An Integrable Case of Electron Motion in Electric and Magnetic Field

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Abstract
Electron motion is studied in a two‐dimensional electric field of potential V=A+B(x2−y2)/2 and a uniform magnetic field H=(0,0, −H) normal to the electric field, where A, B, −H are constants. The equipotential lines of the electric field in any plane z=const. consist of rectangular hyperbolas with a point of zero field strength at x=0, y=0. The differential equations of motion are integrated, and expressions are given for the electron paths. For this type of field, the electron motion consists of superposition of elliptic and hyperbolic motions, that is, of a simple harmonic motion along an ellipse whose center moves along a hyperbola. The latter hyperbolas intersect the equipotential hyperbolas, so that, unlike the uniform crossed field case, the electron may drift into regions of higher or lower potential.

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