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, , and , 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
1.
M.
Müller
and
K.
Dietrich
, “ Classical motion of a charged particle in the magnetic field of a rectilinear current
,” Z. Phys. D
33
, 101
–107
(1995
). This paper allows the particle to carry a magnetic dipole moment in addition to its charge.2.
D.
Yafaev
, “ A particle in a magnetic field of an infinite rectilinear current
,” Math. Phys., Anal. Geom.
6
, 219
–230
(2003
).3.
J.
Aguirre
,
A.
Luque
, and
D.
Peralta-Salas
, “ Motion of charged particles in magnetic fields created by symmetric configurations of wires
,” Physica D
239
, 654
–674
(2010
).4.
A.
Prentice
,
M.
Fatuzzo
, and
T.
Toepker
, “ Charged particle dynamics in the magnetic field of a long straight current-carrying wire
,” Phys. Teach.
53
, 34
–37
(2015
). This paper treats the two-dimensional example (case C). In an “Addendum” (Phys. Teach. 53, 262 (2015)) the authors check that kinetic energy is conserved in this problem.5.
H.
Essén
and
A. B.
Nordmark
, “ Drift velocity of charged particles in magnetic fields and its relation to the direction of the source current
,” Eur. Phys. J. D
70
, 198
–208
(2016
).6.
M.
Asadi-Zeydabadi
and
C. S.
Zaidins
, “ The trajectory of a charged particle in the magnetic field of an infinite current carrying wire in the nonrelativistic limit
,” Results Phys.
12
, 2213
–2217
(2019
). This paper treats the two-dimensional example (Case C).7.
This follows from integrating the Fourier expansion
.
8.
D. J.
Griffiths
, Introduction to Electrodynamics
, 4th ed. ( Cambridge U. P
., Cambridge
, 2017
), Problem 5.39.9.
What if 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.
10.
To draw the figures we used
A.
Chambliss
and
J.
Franklin
, “ A magnetic velocity Verlet method
,” Am. J. Phys.
88
, 1075
–1082
(2020
).11.
See the supplementary material at https://www.scitation.org/doi/suppl/10.1119/5.0077042 for the Mathematica code used to create the figures.
12.
In Case C (
) these integrals can be evaluated in closed form:
where In is the modified Bessel function of order 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).
13.
We might as well use
as the zero for electric potential. You can use a different reference point if you like: the equations will be a little more cumbersome, but the result is the same.
14.
E. M.
Purcell
and
D. J.
Morin
, Electricity and Magnetism
, 3rd ed. ( Cambridge U. P
., Cambridge
, 2013
), Sec. 5.9.15.
J. D.
Jackson
, Classical Electrodynamics
, 3rd ed. ( John Wiley & Sons
, New York
, 1999
), Eq. (12.12).16.
Jackson (Ref. 15), Eq. (12.17).
17.
The “light-like” case (
) is qualitatively similar, with the exception of the trivial limit λ = 0, I = 0, for which the trajectory is of course a straight line, with constant velocity.
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2022
Author(s)
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