The nonrelativistic trajectory of a point charge *q* in the magnetic field of a steady line current *I* is characterized by the three components of its initial velocity. The motion is periodic in the cylindrical coordinates *s*, $ \varphi \u0307 $, and $ z \u0307 $, describing, in the generic case, a kind of “double helix,” with one helix serving as a guide while the other winds around it. A positive charge “drifts” in the direction of the current (a negative charge goes the other way). The inclusion of a uniform line charge *λ* (coinciding with the current) does not alter the motion qualitatively, but it does change the drift velocity, and can even reduce it to zero, collapsing the trajectory to the surface of a toroid. The relativistic treatment modifies and illuminates these results.

## REFERENCES

**53**, 262 (2015)) the authors check that kinetic energy is conserved in this problem.

*Introduction to Electrodynamics*

*α*is negative? Rotating the apparatus by 180° would reverse the sign of the current and render

*α*positive, while carrying one possible trajectory into another, so there is no loss of generality in assuming

*α*is positive.

*I*is the modified Bessel function of order

_{n}*n*, and

*v*is the (constant) speed of the charge. These results were obtained by Essén and Nordmark (Ref. 5), but with unfortunate typos (their Eqs. (73) and (74)); they were derived in a different way by Asadi-Zeydabadi and Zaidins (Ref. 6).

*Electricity and Magnetism*

*Classical Electrodynamics*

*λ*= 0,

*I*= 0, for which the trajectory is of course a straight line, with constant velocity.

*American Journal of Physics*and

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