Equipotential Surfaces — Question 6

Question 6

A circular equipotential requires a constraint to guide a moving particle

A positive source Q creates $V=kQ/r$. A positive particle q of mass m moves at constant speed v around the circle r=R, held there by an ideal radial guide.

1. Find electric work over any circular arc and the electric force.

2. Find the required guide force, specifying inward as positive for that calculation.

Question 6 - Solution

A circular equipotential requires a constraint to guide a moving particle

(a) Method and calculation

$$\Delta V=0\Rightarrow W_e=-q\Delta V=0.$$$$\mathbf F_e=\frac{kQq}{R^2}\hat{\mathbf r}\quad\text{outward}.$$

(b) Interpretation and further calculation

$$F_{\mathrm{guide,in}}-\frac{kQq}{R^2}=\frac{mv^2}{R}.$$$$F_{\mathrm{guide,in}}=\frac{mv^2}{R}+\frac{kQq}{R^2}.$$

Physical check

An equipotential is a geometric locus, not a free trajectory; the electric force here points away from the required centripetal acceleration.