The physics
Cut the charge into tiny pieces. Each piece gives k dq / r² at the point. Add the pieces as arrows.
Symmetry helps. On a ring's axis, the sideways parts of opposite pieces become zero. Only the parts along the axis add.
ring axis: E = kQz / (R² + z²)^3/2 rod end-on: E = kQ / (d (d + L))
How fast a field falls depends on the shape. A point's field falls as 1/r². A long wire's falls as 1/r. A big sheet's does not fall.
wire: E = 2kλ / r sheet: E = σ / 2ε0
When the field changes along a path, use energy. The kinetic energy gained is q times the drop in V.
disc axis: V = (σ/2ε0)(√(R² + z²) − z)