Charged particle oscillation about the origin

  • Thread starter kelly0303
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  • #1
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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  • #2
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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?
 
  • #3
kelly0303
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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).
 
  • #4
jbriggs444
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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.
 
  • #5
vanhees71
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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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