A point charge at distance R, a ring of radius R, and a spherical shell of radius R, each carrying charge q, all give the potential keq/R at one special point. Drag the probe to see how differently their potentials and fields behave away from it, and release a test charge to test the stability of the center.
Read the full note with derivations (PDF)Charge distribution
Probe point
V in units of keq/R, E in units of keq/R², distances in units of R.
Release a test charge
Release a charge of the opposite sign from rest and ask how long it takes to reach the ring. Exactly on the axis it never arrives. Just off the axis, the answer depends on the release height: near the center the sideways offset grows, but from high enough on the axis the charge spends most of each swing where the sideways force pulls it back, and it stays near the axis indefinitely. Everywhere else it slingshots around the wire, and whether it ever hits depends on how thick the wire is.
Try a starting point
Run time
Presets switch to the ring with an opposite-sign charge, free in 3D. The wire thickness is the capture radius around the ring, in units of R.
Path in the plane through the axis and the starting point. Dots: the ring's cross-section, drawn at the wire thickness.
■ distance from the axis ρ(t) ■ height z(t)
Stability of the axial oscillation versus release height z₀, from a linear (Floquet) analysis. Inside the shaded band a small sideways offset stays small; outside it the offset grows. Click the chart to release a charge at (0.01, 0, z₀).
For the ring, the axis and the plane behave differently. For the point charge and the shell, every direction is the same. The dashed line marks the probe's distance from the center.
Potential V versus distance from the center
Outward field component versus distance from the center
Potential is a scalar sum of kedq/s. When every charge element sits at the same distance R, only the total charge matters, so any such arrangement gives keq/R. The field is a gradient, so it depends on how V changes nearby, which one value cannot tell you.
Near the ring's center, V ≈ (keq/R)(1 + (x² + y² − 2z²)/4R²). The bold contour crosses itself at the center in the side slice. A test charge is pushed back along one set of directions and away along the other, so without a constraint it always escapes.
Inside the shell V is constant and E is zero everywhere. Patch area grows as s², exactly offsetting the 1/s² field, so opposite patches cancel. A test charge inside feels no force at all: neutral equilibrium.