Corona Discharge & the Van de Graaff

On a charged conductor, charge piles up where the surface is most sharply curved. Near sharp points the field gets strong enough to tear air apart, and the same physics sets the limit of a van de Graaff generator.

Two spheres joined by a wire#

Spheres A (radius r₁) and B (radius r₂ < r₁) are connected by a wire and share charge Q. Joined, they must be at the same potential:

The smaller sphere ends up with the larger charge density, and since the field just outside a conductor is σ/ε₀, with the stronger field too.

Corona discharge#

Any conductor behaves the same way. Flatter parts are like pieces of a big sphere and get little charge; sharply curved parts (small radius of curvature) get a lot. At a needle point the field can exceed the breakdown strength of air (about 3 × 10⁶ V/m). The air there conducts, charge leaks off, and often a faint glow surrounds the point. This is corona discharge.

The van de Graaff generator#

Designed in 1929 to reach potentials around 10⁷ V, it moves charge onto a large hollow metal dome:

1. An insulating belt runs over two pulleys, one inside the dome and one near the base, driven by a motor.
2. A comb of needles near the bottom is held at about +10⁴ V. Corona at its points makes the air conduct; positive charge settles on the belt.
3. The belt carries this charge up into the dome. Corona at a second comb, connected to the dome, transfers it to the dome.
4. Charge inside a hollow conductor goes straight to its outer surface, so the dome keeps accepting more, whatever its potential.

For a sphere, E just outside = Q/4πε₀R² and V = Q/4πε₀R, so V = ER. The dome stops charging when E reaches air’s dielectric strength: