The Van de Graaff Generator: How a Moving Belt Builds a Million Volts

The one idea that makes it work: charge flees to the outside

Everything about the Van de Graaff hangs on a single theorem of electrostatics. Consider a hollow conductor holding a net charge Q. Apply Gauss's law to a closed surface drawn just inside the conductor's material:

• ∮ E · dA = Q_enc / ε₀
• Inside a conductor in equilibrium the field E = 0, so the integral is zero, which forces Q_enc = 0.
• Therefore no net charge can live in the interior or in the bulk — it must reside entirely on the outer surface.

This is why you can keep pouring charge into the hollow terminal sphere and it never fights you: a small comb or brush deposits charge on the belt inside the sphere, and that charge is instantly repelled out to the sphere's exterior, leaving the interior field-free and ready to accept more. Faraday demonstrated the principle in 1843 with his ice-pail experiment, and Van de Graaff exploited it 86 years later. Crucially, the potential of the terminal can keep rising even when it is already at hundreds of kilovolts — the interior comb only has to work against the tiny local field there, not against the full terminal potential.

The charge conveyor: belt, combs, and corona

The machine is a mechanical current source. A motor drives an insulating belt (rubber, silk, or a modern polymer) over two pulleys — one at the base, one inside the metal terminal. Charge is sprayed onto the belt at the bottom and scraped off at the top:

• A lower comb — a row of fine metal points held at a few kV by a power supply (or, in the original, by triboelectric contact) — sits micrometers from the belt. The sharp points concentrate the field until it exceeds ≈ 3 MV/m and the air ionizes: this is corona discharge. Ions stream onto the belt, depositing a surface charge density σ.
• The belt physically carries that charge upward at speed v. The charging current is simply the flux of charge past a plane: I = σ · w · v, where w is the belt width. With σ ≈ 25 µC/m², w = 0.1 m, v = 20 m/s you get I ≈ 50 µA — a real, if tiny, current.
• An upper comb inside the terminal, connected to the sphere, likewise coronas onto the belt but in reverse: the strong field of the charged terminal pulls the charge off the belt and onto the comb, whence Gauss's law sends it to the sphere's exterior.

Because charge transport is mechanical, the current is essentially independent of the terminal voltage — the belt does not care how high V has climbed. That is what lets the voltage integrate upward over many belt cycles.

From current to a million volts: V = Q/C

The terminal is, electrically, an isolated sphere — a capacitor to infinity. Its capacitance is C = 4πε₀R, so a 0.15 m demonstrator sphere has C ≈ 4π(8.85×10⁻¹² F/m)(0.15 m) ≈ 17 pF. The terminal voltage is V = Q/C, and the charge builds according to a charging-circuit balance:

• C (dV/dt) = I_charge − I_leak(V), where I_charge is the belt current and I_leak is everything draining the terminal (corona off the sphere, resistive leakage, and — in an accelerator — the beam current).
• Early on, when leakage is negligible, V rises almost linearly: dV/dt ≈ I/C. With I = 50 µA and C = 17 pF, dV/dt ≈ 3 MV/s — the sphere would in principle reach 300 kV in about 0.1 s, and does climb that fast until leakage catches up.
• The machine reaches a steady state when I_leak(V) = I_charge. Because leakage rises steeply with V (corona onset is roughly exponential in field), the terminal settles at a well-defined maximum voltage.

The energy stored is U = ½CV². For the demonstrator at V = 0.3 MV, U = ½(17 pF)(0.3 MV)² ≈ 0.8 J — small, but delivered in a microsecond-scale spark that carries enough peak power to make your hair stand and a loud crack, though not enough charge to be lethal.

What sets the ceiling: dielectric breakdown

The voltage does not rise forever. The surface field of a charged sphere is E_surface = V/R (equivalently E = Q/4πε₀R²), and the ceiling is reached when that field ionizes the surrounding gas. In dry air at 1 atm the dielectric strength is about E_max ≈ 3 MV/m, so the maximum terminal voltage scales linearly with radius:

• V_max ≈ E_max · R. For R = 0.15 m this gives V_max ≈ 0.45 MV — right where classroom machines top out.
• To go higher you enlarge R (a 2 m sphere allows ≈ 6 MV in air) and, more powerfully, you raise E_max by changing the medium.

This is why the great research generators live inside pressure tanks. Sulfur hexafluoride (SF₆) at 5–10 atm has a dielectric strength several times that of air, and its heavy electronegative molecules mop up free electrons before avalanches form. Pressurizing to ~7 atm can multiply the breakdown field roughly with pressure (Paschen's law), letting a modest-radius terminal hold 10–25 MV. Sharp edges are the enemy everywhere: near a point of radius r the field is enhanced by roughly R/r, which is exactly why the corona combs work — and why every other surface on the terminal is rounded, polished, and hidden behind smooth toroidal shrouds.

Why it was built: splitting the atom

Van de Graaff was not making a science-museum toy — he wanted a controllable, steady source of fast ions to probe the nucleus. A charged particle of charge q falling through the full terminal potential V gains kinetic energy E = qV, which for a proton and V = 5 MV is 5 MeV — enough to overcome the Coulomb barrier of light nuclei. Because the terminal voltage is DC and extremely stable (parts in 10⁴ with feedback), Van de Graaff accelerators gave physicists an unusually monochromatic beam, ideal for measuring nuclear reaction cross-sections versus energy.

• A clever refinement is the tandem accelerator. Negative ions are accelerated from ground up to the positive terminal, stripped of electrons by a thin gas or foil there (becoming positive), then accelerated back down to ground — using the same voltage twice. A charge-state q ion emerges with E = (1 + q)eV, so a 12 MV terminal can push carbon ions past 70 MeV.
• Tandems remain workhorses of accelerator mass spectrometry — the technique behind modern radiocarbon dating, counting individual ¹⁴C atoms at abundances of 1 part in 10¹².
• The Pelletron variant replaces the belt with a chain of metal cylinders linked by insulating nylon, transporting charge more reliably and at higher currents.

The MIT machine Van de Graaff scaled up in the early 1930s reached 7 MV; the Oak Ridge 25URC tandem is still among the highest-voltage electrostatic accelerators ever run.

Intuitions, misconceptions, and where the physics bites

A few points routinely trip people up:

• Voltage is not energy or danger by itself. A demonstrator at 300 kV is far higher voltage than a wall outlet, yet the stored energy is under a joule and the current a few microamps — a shock stings but the tiny charge can't sustain a dangerous current through the body. What kills is coulombs, not volts.
• The belt speed and width, not the voltage, set the current. Doubling belt speed doubles I and so doubles dV/dt and the sustainable leakage — but it does not raise V_max, which is fixed by geometry and breakdown field.
• The interior really is field-free. You could, in principle, sit inside the terminal at a million volts and feel nothing electrically — the same reason a car or aircraft protects you from lightning (the Faraday-cage effect), and why the upper comb can add charge indefinitely.
• Humidity is the demonstrator's enemy. A film of water on the belt or supports provides a resistive leakage path; on a humid day the same machine may barely reach a third of its dry-air voltage. This is a practical instance of I_leak overwhelming I_charge at low V.

Run one in a dim room and you can watch the whole chain of physics at once: the faint blue glow of corona at the combs (ionization at 3 MV/m), the invisible mechanical current of the belt, and finally the branching spark to a grounded probe — a miniature, on-demand bolt of lightning obeying exactly the same breakdown physics as the sky.

Two ends of the Van de Graaff family: a classroom demonstrator versus a nuclear-physics accelerator.

Property	Classroom demonstrator	Research accelerator
Terminal radius R	≈ 0.15 m	≈ 1–2 m
Terminal voltage V	0.1–0.4 MV	5–25 MV
Insulating medium	Air, 1 atm	SF₆ or N₂/CO₂ at 5–10 atm
Charging current I	1–10 µA	50–500 µA
Terminal capacitance C	≈ 17 pF (isolated 0.15 m sphere)	10–100 pF
Stored energy ½CV²	≈ 0.4 J	up to ~10 kJ