Collective Oscillation and Audit

Displacement Field Pulls Electrons Back

When electrons are displaced from the ions, the resulting charge separation sets up a restoring electric field that pulls them back, the spring at the heart of plasma oscillation. The restoring field is exact only for the stated charge density, electron displacement, and density in the ideal cold, uniform plasma model. Real plasmas add collisions, temperature spread, boundaries, nonuniform density, diagnostics calibration, and measurement/source uncertainty.

A displaced electron slab sees a restoring force from its sign. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Displacement creates a proportional field

Displacement 2 meters times gradient 3 newtons per coulomb per meter gives field 6 newtons per coulomb.

3 N/(C*m)⋅2 m=6 N/C3\ \text{N/(C*m)}\cdot 2\ \text{m}=6\ \text{N/C}
Collective displacement fieldThe field vector direction is checked.electronEFx=2 mgradient=3 N/(C*m)E=6 N/Cq=-1 CF=-6 NK=12 1/s^2density=3massRatio=6omega2=6 1/s^2

Electron charge flips the force direction

Charge -1 coulomb times field 6 newtons per coulomb gives restoring force -6 newtons.

−1 C⋅6 N/C=−6 N-1\ \text{C}\cdot 6\ \text{N/C}=-6\ \text{N}
Restoring electron forceThe force arrow must point back toward balance.electronEFx=2 mgradient=3 N/(C*m)E=6 N/Cq=-1 CF=-6 NK=12 1/s^2density=3massRatio=6omega2=6 1/s^2