Potential Due to a Point Charge — Question 9

Question 9

A circular Coulomb orbit partitions kinetic and potential energy

A particle of charge $-q$ and mass m moves in a circular orbit of radius R about fixed $+Q$, with q,Q positive. Ignore radiation and use nonrelativistic mechanics.

1. Use centripetal balance to find speed and kinetic energy.

2. Find total mechanical energy and the additional energy required to reach infinity with zero final speed.

Question 9 - Solution

A circular Coulomb orbit partitions kinetic and potential energy

(a) Method and calculation

$$\frac{mv^2}{R}=\frac{kQq}{R^2}\Rightarrow v=\sqrt{\frac{kQq}{mR}}.$$$$K=\frac{kQq}{2R},\quad U=-\frac{kQq}{R}.$$

(b) Interpretation and further calculation

$$\mathcal E=K+U=-\frac{kQq}{2R}.$$$$\Delta\mathcal E_{\mathrm{escape}}=0-\mathcal E=\frac{kQq}{2R}.$$

Physical check

Escape energy from an orbit is half the magnitude of the interaction energy because the particle already has kinetic energy.