The charge density on the surface of conductors varies according to the curvature of the conductor at a given point.
Leybold manual: P3.1.5.1 Investigating the charge distribution on the surface of electrical conductors
Apparatus for measuring the amount of charge at individual points on a charged conductor.
By touching the surface of a charged conductor with a small metal plate, we transfer a charge proportional to the local surface charge density.
We discharge the metal plate into the Faraday cup, thereby measuring how much charge it carried.
A conical conductor has a higher charge density at its tip.