Charged particle oscillation about the origin

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Discussion Overview

The discussion revolves around the behavior of a positively charged ion oscillating in an electric field described by ##E = -kz##. Participants explore the nature of the electric field experienced by the ion in different reference frames, particularly focusing on whether the field amplitude remains constant during oscillation and the implications of electromagnetic waves in this context.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant proposes that the electric field seen by the ion in its rest frame is ##-kz_0 cos(\omega t)## and questions if the amplitude of the field is constant.
  • Another participant challenges the notion of the ion being fixed in its own rest frame, seeking clarification on the reference frame being used.
  • A clarification is made that the ion is constrained to move along the z axis in the lab frame, and in its rest frame, it remains at the origin.
  • It is noted that in the lab frame, the electrostatic field varies linearly with position and periodically with time, while the field value at the origin will vary with time alone.
  • One participant argues that considering only static fields is an incomplete picture, as electromagnetic waves must be accounted for, particularly in the near-field zone where the static approximation holds.
  • A reference to the dipole approximation is introduced as a simple solution to the problem, linking to external resources for further exploration.

Areas of Agreement / Disagreement

Participants express differing views on the completeness of the static field approximation and the implications of electromagnetic radiation, indicating that multiple competing perspectives remain in the discussion.

Contextual Notes

Limitations include the dependence on the assumption of non-relativistic velocities and the neglect of magnetic fields, as well as the scope of the discussion being primarily focused on electrostatic fields without resolving the complexities introduced by electromagnetic waves.

kelly0303
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Hello! This is probably something simple but I am getting confused about it. Assume we have an electric field along the z axis given by ##E = -kz##, with ##k>0##, so the field on both sides of the xy-plane points towards the origin. Let's say that we have a positively charged ion at the origin and we give it a kick upwards such that it now oscillates as ##z = z_0cos(\omega t)##. What is the field that the ion sees in its rest frame (assume the ion is fixed on the z axis so we can ignore magnetic fields and it moves at nonrelativistic velocities)? Is it ##-kz_0 cos(\omega t)##? My main confusion is: does the amplitude of the field that the ion sees is constant or not?
 
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kelly0303 said:
assume the ion is fixed on the z axis
I don’t understand this comment. In its own rest frame shouldn’t the ion be fixed at the origin? Can you clarify the reference frame you are describing?
 
Dale said:
I don’t understand this comment. In its own rest frame shouldn’t the ion be fixed at the origin? Can you clarify the reference frame you are describing?
Sorry, I meant the ion is constrained to move on the z axis in the lab frame. Basically this is a 1D problem in the lab frame (in its rest frame the ion is always at the origin).
 
kelly0303 said:
Sorry, I meant the ion is constrained to move on the z axis in the lab frame. Basically this is a 1D problem in the lab frame (in its rest frame the ion is always at the origin).
Seems simple enough. Ignoring magnetism and relativity (i.e. considering only a fixed electrostatic field) and using a coordinate system anchored to the ion then you have an electrostatic field that varies both linearly with position and periodically with time.

The field value locally (right at the origin/right at the ion) will, of course, vary with time alone. It is the field values elsewhere which will vary with time and with their offset from the origin/ion.
 
But that's a pretty incomplete picture since you get in any case electromagnetic waves. The approximation to only consider the static fields is valid in the near-field zone, i.e., at distances close to the particle, where close means at distances much smaller than the wavelength of the radiation, ##\lambda=f/c##, were ##f## is the frequency of the oscillation.

The most simple approximate solution for this problem is the dipole approximation:

https://en.wikipedia.org/wiki/Dipole_antenna#Hertzian_dipole
 
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