Collective Oscillation and Audit

Displacement Field Pulls Electrons Back

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