A Point Charge in an Electric Field — Question 4

Question 4

A linear restoring field produces harmonic motion for only one charge sign

Along a constrained frictionless x axis, the prescribed field is $E_x=-\kappa x$, with $\kappa>0$. A particle of mass m and charge q starts at $x=A>0$ from rest. The constraint prevents transverse motion and does no work.

1. For $q>0$, derive the angular frequency, motion and maximum speed.

2. For $q<0$, solve the initial-value equation and classify the equilibrium.

Question 4 - Solution

A linear restoring field produces harmonic motion for only one charge sign

(a) Method and calculation

$$m\ddot x=-q\kappa x,\quad\omega=\sqrt{q\kappa/m}.$$$$x(t)=A\cos\omega t,\quad v(t)=-A\omega\sin\omega t,\quad v_{\max}=A\omega.$$

(b) Interpretation and further calculation

$$q<0:\quad\ddot x=\Omega^2x,\quad\Omega=\sqrt{|q|\kappa/m}.$$$$x(t)=A\cosh\Omega t,\quad v(t)=A\Omega\sinh\Omega t.$$Positive charge has a restoring force in this constrained direction; negative charge is driven away from the origin.

Physical check

This one-dimensional model makes no claim of stable trapping in unconstrained three-dimensional space.