Applying Gauss' Law: Planar Symmetry — Question 10

Question 10

A neutral three-sheet trap produces oscillations with an amplitude-dependent period

Three infinite, ideal permeable charge sheets at $x=0,d,2d$ carry $+\sigma,-2\sigma,+\sigma$, with $\sigma>0$. A weak particle of charge $q>0$ and mass m starts at rest at $x=d-A$, with $0<A<d$. Ignore collisions with the ideal sheets and all other forces.

1. Find the field in the two interior gaps and show that the particle accelerates toward $x=d$ on either side.

2. Find the oscillation period and its dependence on A. Compare with amplitude-independent harmonic motion.

Question 10 - Solution

A neutral three-sheet trap produces oscillations with an amplitude-dependent period

(a) Method and calculation

$$E_x=+\sigma/\epsilon_0\quad(0<x<d),\qquad E_x=-\sigma/\epsilon_0\quad(d<x<2d).$$$$a_0=\frac{q\sigma}{m\epsilon_0},\quad\ddot x=-a_0\operatorname{sgn}(x-d).$$

(b) Interpretation and further calculation

$$A=\tfrac12a_0t_q^2\Rightarrow t_q=\sqrt{2A/a_0}.$$$$T=4t_q=4\sqrt{\frac{2Am\epsilon_0}{q\sigma}}\propto\sqrt A.$$The speed is continuous when acceleration reverses at the central ideal sheet; the particle turns at $d+A$ and returns symmetrically.

Physical check

The acceleration is piecewise constant rather than proportional to displacement, so this motion is periodic but not simple harmonic.