Charged particle oscillation about the origin

In summary, the conversation discusses the behavior of an electric field along the z-axis, with a positively charged ion at the origin undergoing oscillations. The question is whether the amplitude of the field that the ion sees is constant or not, assuming the ion is fixed on the z axis and moving at non-relativistic velocities. The answer is that the field value varies with time and its offset from the origin/ion, but the approximation to only consider static fields is valid in the near-field zone. The dipole approximation is a simple solution for this problem.
  • #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
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?
 
  • #3
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).
 
  • #4
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.
 
  • #5
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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1. What is charged particle oscillation about the origin?

Charged particle oscillation about the origin refers to the movement of a charged particle in a circular or elliptical path around the origin point, due to the influence of an electric field.

2. What causes charged particles to oscillate about the origin?

The oscillation of charged particles about the origin is caused by the interaction between the charged particle and an external electric field. This interaction causes the particle to experience a force that changes its direction of motion, resulting in an oscillatory movement.

3. How is the frequency of charged particle oscillation determined?

The frequency of charged particle oscillation is determined by the strength of the external electric field and the mass and charge of the particle. The more powerful the electric field, the higher the frequency of oscillation.

4. What is the significance of studying charged particle oscillation about the origin?

Studying charged particle oscillation about the origin is important in understanding the behavior of charged particles in electric fields. This phenomenon is also utilized in various technologies such as particle accelerators and mass spectrometers.

5. How is the motion of a charged particle affected by the direction of the electric field?

The direction of the electric field affects the direction of the force exerted on the charged particle, which in turn affects the direction of its oscillatory motion. If the electric field changes direction, the charged particle will also change its direction of oscillation.

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