Electric Potential Energy — Question 5

Question 5

Equal endpoint energies do not guarantee passage over an intermediate barrier

Along a frictionless constrained path $0\le x\le L$, a charged particle has $U(x)=U_b\sin^2(\pi x/L)$ with $U_b>0$. It starts at x=0 toward positive x with kinetic energy $K_0=.800U_b$.

1. Find the first turning point and explain why the endpoint x=L is inaccessible.

2. Find the threshold initial kinetic energy for reaching the barrier top, and distinguish the threshold from passage in finite time.

Question 5 - Solution

Equal endpoint energies do not guarantee passage over an intermediate barrier

(a) Method and calculation

$$K(x)=K_0-U(x)=U_b[.800-\sin^2(\pi x/L)].$$$$x_t=\frac L\pi\sin^{-1}\sqrt{.800}\approx.352416L.$$

(b) Interpretation and further calculation

$$U_{\max}=U_b\quad\text{at }x=L/2,\qquad K_{0,\mathrm{threshold}}=U_b.$$$$K_0>U_b\Rightarrow\text{finite-time passage};\quad K_0=U_b\Rightarrow\text{asymptotic approach to the smooth maximum}.$$At threshold the speed vanishes linearly with distance from the top, so the arrival-time integral diverges.

Physical check

Energy conservation must be checked at every intermediate position, not only at the endpoints.