Electric Potential Energy of a System of Point Charges — Question 8

Question 8

A constrained middle charge minimizes a three-body energy along its track

Two positive charges q are fixed at x=0 and x=L. A third positive charge q of mass m moves on a frictionless straight guide within $0<x<L$. Include all three pair energies.

1. Find U(x), its equilibrium position and minimum value.

2. Find the curvature at equilibrium and the small-oscillation angular frequency along the guide.

Question 8 - Solution

A constrained middle charge minimizes a three-body energy along its track

(a) Method and calculation

$$U(x)=kq^2\left(\frac1L+\frac1x+\frac1{L-x}\right).$$$$U\prime=kq^2[-x^{-2}+(L-x)^{-2}]=0\Rightarrow x=L/2,\quad U_{\min}=5kq^2/L.$$

(b) Interpretation and further calculation

$$U\prime\prime=kq^2\left[\frac2{x^3}+\frac2{(L-x)^3}\right]\Rightarrow U\prime\prime(L/2)=32kq^2/L^3.$$$$\omega=\sqrt{\frac{32kq^2}{mL^3}}.$$

Physical check

The fixed endpoint pair adds a constant to U, affecting the minimum value but not the equilibrium or frequency.