Potential Due to a Point Charge — Question 6

Question 6

An attractive source gives a finite escape threshold from a finite radius

A positive particle q of mass m begins at radius R from a fixed negative point charge $-Q$, where Q>0. It is launched radially outward.

1. Find the minimum launch speed for escape to infinity.

2. For launch speed $v_0$ below that threshold, find the outer turning radius.

Question 6 - Solution

An attractive source gives a finite escape threshold from a finite radius

(a) Method and calculation

$$\mathcal E=\tfrac12mv_0^2-\frac{kQq}{R},\qquad\mathcal E_{\mathrm{escape}}\ge0.$$$$v_{\mathrm{esc}}=\sqrt{\frac{2kQq}{mR}}.$$

(b) Interpretation and further calculation

$$-\frac{kQq}{r_t}=\tfrac12mv_0^2-\frac{kQq}{R}.$$$$r_t=\frac{R}{1-v_0^2/v_{\mathrm{esc}}^2}\quad(0\le v_0<v_{\mathrm{esc}}).$$

Physical check

The turning radius diverges as the launch energy approaches zero from below; escape at threshold has zero asymptotic speed.