Potential Due to a Continuous Charge Distribution — Question 6

Question 6

Removing a central disk changes the center voltage and axial curvature

A uniformly charged annulus has inner radius a, outer radius b>a and surface density $\sigma>0$. Set zero potential at infinity.

1. Find the potential along its symmetry axis by subtracting two disk potentials.

2. Find V(0) and its expansion through z squared near the center; classify the axial extremum.

Question 6 - Solution

Removing a central disk changes the center voltage and axial curvature

(a) Method and calculation

$$V(z)=2\pi k\sigma\left(\sqrt{b^2+z^2}-\sqrt{a^2+z^2}\right).$$$$V(0)=2\pi k\sigma(b-a).$$

(b) Interpretation and further calculation

$$V(z)=2\pi k\sigma\left[(b-a)+\frac{z^2}{2}\left(\frac1b-\frac1a\right)+O(z^4)\right].$$$$V\prime\prime(0)=2\pi k\sigma(1/b-1/a)<0,$$so the center is an axial maximum.

Physical check

The result describes stability along the axis only; it does not establish a three-dimensional equilibrium classification.