Charged particle oscillation about the origin

[kelly0303]

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?

 

[Dale]

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?

 

[kelly0303]

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).

 

[jbriggs444]

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.

 

[vanhees71]

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