Liénard-Wiechert Potential for Spiraling Charge

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The graphics show different views of the radiation patterns produced by a point charge moving in an inward spiral trajectory with angular velocity . Since a charge moving in a circular orbit is continually undergoing centripetal acceleration , it radiates away energy in accordance with Larmor's formula (in Gaussian units). As energy is lost, the radius decreases as the charge spirals inward toward the attracting center. Neglecting any contribution from the radiation field, the Liénard–Wiechert scalar potential produced by the moving charge is computed. This treatment is valid for , and thus excludes the ultrarelativistic domain.

Contributed by: Enrique Zeleny (August 2012)
Open content licensed under CC BY-NC-SA



The Liénard–Wiechert scalar potential is given by where is the electron charge, is the velocity vector, and is the distance from the current position to the original position at the retarded time . The last equation is solved iteratively, updating the direction of the field from the charge's instantaneous position. As mentioned in the Caption, the computation is valid provided that .


[1] M. Trott, The Mathematica Guidebook for Graphics, New York: Springer-Verlag, 2004.

[2] K. Kokkotas. "Radiation by Moving Charges." (May 9, 2010)

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