How this instrument works
A spherical capacitor is two concentric conducting shells — an inner sphere of radius r₁ and an outer one of radius r₂ — with the electric field living entirely in the space between them. Put charge +Q on the inner sphere and an equal −Q gathers on the inner face of the outer one; Gauss's law says the field between the shells is purely radial and equal to Q ⁄ (4πε₀r²) at every point, and exactly zero outside r₂ or inside r₁. Integrate that field from r₁ out to r₂ and the result is the potential difference; divide the charge by that potential and the r₁r₂ ⁄ (r₂ − r₁) shape of the formula falls out directly — it is not an arbitrary combination of the two radii, it is what that one integral produces.
The formula carries its own limits built in. Push the outer sphere outward without bound and r₂ ⁄ (r₂ − r₁) tends to 1, leaving C = 4πε₀r₁ — the textbook capacitance of a single isolated sphere relative to a shell at infinity, which is the case an exposed charged dome with no surrounding enclosure actually approximates. Shrink the gap instead, holding both radii close to a common value r, and the same expression collapses toward ε₀ times area over separation, the ordinary parallel-plate formula, because a thin enough gap between two curved surfaces is locally almost flat.
Concentric spheres are also the one capacitor shape a Gauss's-law derivation gets exactly, with nothing left over at an edge, because a sphere has no edge to correct for. A parallel-plate capacitor's true capacitance always runs a little above the plain area-over-separation estimate near its rim, where field lines bow outward — designers of guard-ring and reference capacitors go to real trouble suppressing exactly that fringing. At planetary scale the same spherical shape models the fair-weather atmosphere: Earth's conducting surface and the conducting ionosphere roughly sixty kilometres up form a natural spherical capacitor of a few hundred picofarads that holds the charge behind the global electric circuit between the thunderstorms that keep recharging it.
- Enter the Inner sphere radius (r₁), the radius of the smaller conducting shell, in millimetres or centimetres.
- Enter the Outer sphere radius (r₂) — it must exceed r₁, since the gap between the two shells is where the field lives.
- If r₂ is not larger than r₁, the instrument flags it rather than returning a nonsensical negative capacitance.
- Read Capacitance, switching its unit between pF and nF to match the scale set by your two radii.
Worked example — 5 cm and 6 cm concentric spheres
Take two concentric conducting spheres, an inner one of radius 5 cm and an outer one of radius 6 cm — the scale of a lab electrostatics demonstration or a small high-voltage terminal. In metres that is r₁ = 0.05 and r₂ = 0.06, so r₁r₂ = 0.003 m² and r₂ − r₁ = 0.01 m, giving the ratio r₁r₂ ⁄ (r₂ − r₁) = 0.3 m. Multiply by 4πε₀ = 4π × 8.8541878128 × 10⁻¹² and the result is C = 3.33795016634 × 10⁻¹¹ F, about 33.38 pF — a small but entirely real capacitance produced by geometry alone, with no plate area or dielectric slab to specify.
Widen the gap while keeping the inner sphere fixed and capacitance falls: push the outer shell out to 10 cm instead of 6 (r₁ = 0.05, r₂ = 0.10) and the same formula returns about 11.13 pF, roughly a third of the original figure, because the field now has more distance to fall off across. That inverse relationship between spacing and capacitance is the same one a parallel-plate capacitor shows between its plates — this geometry just wraps it around a curve instead of leaving it flat.
Questions
How is the spherical capacitor formula actually derived?
From Gauss's law plus one integration. Charge Q on the inner sphere produces a purely radial field E = Q ⁄ (4πε₀r²) at every point between the two shells and nothing outside r₂. Integrating that field from r₁ to r₂ gives the potential difference V = (Q ⁄ 4πε₀)·(1/r₁ − 1/r₂), and since capacitance is defined as C = Q/V, that ratio rearranges directly into C = 4πε₀r₁r₂ ⁄ (r₂ − r₁) — nothing about the shape is assumed beyond the field a charged sphere produces.
What happens to the capacitance as the outer sphere grows very large?
It approaches C = 4πε₀r₁, the capacitance of a single isolated sphere relative to a shell at infinity — send r₂ toward infinity in the formula and the factor r₂ ⁄ (r₂ − r₁) tends to 1, leaving only r₁ behind. That limit is the standard textbook result for one charged sphere sitting in open space, and it is the case an exposed Van de Graaff dome with no surrounding chamber effectively approximates.
Why does capacitance rise as the gap between the spheres shrinks?
A smaller r₂ − r₁ squeezes the same field into a thinner shell of space, so a given charge produces less potential difference across the gap — and since C = Q/V, less potential difference per coulomb means more capacitance. Hold r₁ fixed at 5 cm and shrink the gap from 1 cm toward a few millimetres and the reading climbs sharply, the same trend a parallel-plate capacitor shows as its plates move closer together.
Where does a real spherical capacitor show up outside a physics classroom?
At two very different scales. A charged metal dome — a Van de Graaff terminal, a smooth high-voltage electrode — behaves like the inner sphere of this formula with the outer radius pushed toward infinity. At planetary scale, Earth's surface and the conducting ionosphere form an actual spherical capacitor of a few hundred picofarads, storing the charge behind the fair-weather electric field that the world's thunderstorms continuously discharge and recharge.
Why must the outer radius be larger than the inner one?
Because the two shells have to be nested, one fully inside the other, for the gap between them to hold a field at all — the outer sphere has to physically contain the inner one. Enter an outer radius equal to or smaller than the inner radius and the instrument rejects the input instead of returning a negative or undefined answer, since r₂ − r₁ would be zero or negative in the denominator.
Does the material filling the gap between the spheres matter?
This formula assumes vacuum or air, where relative permittivity is essentially 1. Fill the gap with a dielectric instead — oil, a ceramic sleeve, anything with permittivity εᵣ above 1 — and capacitance scales up by that same factor, since the general relation is C = 4πε₀εᵣr₁r₂ ⁄ (r₂ − r₁). The figure this instrument returns is the vacuum baseline that any dielectric multiplies from.