Potential Due to a Point Charge — Question 8

Question 8

A chord through a spherical equipotential passes through higher potential

A positive point charge Q sits at the center of a sphere of radius R. Endpoints A and B on the sphere subtend angle $2\theta$, where $0<\theta<\pi/2$. A straight guide connects them.

1. Find the nearest distance of the chord to Q and the maximum potential along the chord.

2. Find the minimum initial energy required energetically for a positive charge q to traverse the chord from A, and compare with an arc at r=R.

Question 8 - Solution

A chord through a spherical equipotential passes through higher potential

(a) Method and calculation

$$r_{\min}=R\cos\theta.$$$$V_{\max}=\frac{kQ}{R\cos\theta},\qquad V_A=V_B=\frac{kQ}{R}.$$

(b) Interpretation and further calculation

$$K_{\mathrm{threshold}}=q(V_{\max}-V_A)=\frac{kQq}{R}(\sec\theta-1).$$$$\text{Circular arc:}\quad\Delta V=0\text{ everywhere},\quad\Delta K=0.$$Finite-time chord passage requires energy above the smooth-barrier threshold.

Physical check

Equal endpoint voltage hides the chord barrier; the guide supplies transverse constraint forces but no work.