Electric Potential Energy — Question 7

Question 7

A double-well electric energy distinguishes local stability from barrier crossing

A constrained charged particle has prescribed electric energy $U(x)=U_0[(x/a)^2-1]^2$ with $U_0,a>0$. Its mass is m and no other forces do work.

1. Find all equilibria and classify them using the second derivative of U.

2. Find the small-oscillation frequency about x=a and the threshold launch speed from x=-a for crossing toward the other well.

Question 7 - Solution

A double-well electric energy distinguishes local stability from barrier crossing

(a) Method and calculation

$$U\prime=\frac{4U_0x}{a^2}\left(\frac{x^2}{a^2}-1\right)=0\Rightarrow x=0,\ \pm a.$$$$U\prime\prime=\frac{4U_0}{a^2}\left(\frac{3x^2}{a^2}-1\right):\quad U\prime\prime(0)<0,\quad U\prime\prime(\pm a)=8U_0/a^2>0.$$

(b) Interpretation and further calculation

$$\omega=\sqrt{U\prime\prime(a)/m}=\sqrt{8U_0/(ma^2)}.$$$$\tfrac12mv_{\mathrm{crit}}^2=U(0)-U(-a)=U_0\Rightarrow v_{\mathrm{crit}}=\sqrt{2U_0/m}.$$Finite-time crossing requires a speed strictly above the threshold.

Physical check

Stable small oscillations about one minimum do not rule out crossing to another minimum when enough energy is supplied.