Charged sphere
(E3.6.1) E3.6 · Electric Potential Due to a Charged Conductor →The potential of an isolated conducting sphere of radius R, the same value everywhere on and inside it.
| potential of the conductor | V | |
| charge on it | C | |
| its radius | m | |
| Coulomb constant | 8.99×10⁹ N·m²/C² (=1/4πε₀) |
Conductor in equilibrium; the value holds on and inside it.
Holds for a conductor in electrostatic equilibrium only — a conductor carrying current has a nonzero field inside it and this uniform-potential result no longer applies.
Outside, the sphere's field is identical to a point charge's, so the potential is too:
At the surface:
Inside, E = 0, so moving around costs nothing and V cannot change:
So the whole conductor sits at the surface value:
Breakdown at 3 × 10⁶ V/m on a sphere of radius 0.30 m:
A higher voltage requires a larger radius.
Smaller radius means higher potential for the same charge, and a much higher surface field — which is why sharp points and thin wires break down first.