Spherical Capacitor Calculator

Spherical Capacitor Calculator
Isolated Sphere or Concentric Spherical Capacitor
a Sphere Radius
V Voltage (optional)
V
εr Relative Permittivity (1 = air/vacuum)
εr
Spherical Capacitance
Capacitance
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Type
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Permittivity
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Inner Radius (a)
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Outer Radius (b)
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Spherical Capacitor Geometry

An isolated sphere has capacitance proportional to its radius: C = 4πε0a. A concentric spherical capacitor has an inner sphere (radius a) inside an outer shell (radius b), with dielectric between them.

Inner Sphere (a) Outer Shell (b) a b εr (dielectric) E-field (radial) C = 4πε₀εr × ab/(b−a) Isolated: C = 4πε₀εr × a  (b → ∞)
Inner Sphere (a) — The inner conductor. Radius a. Carries charge +Q.
Outer Shell (b) — The outer conductor (dashed). Radius b. Carries charge −Q. For an isolated sphere, b = ∞.
εr — Relative permittivity of the dielectric filling the gap between the spheres.
E-field — Radial electric field lines from the inner sphere to the outer shell.

Spherical Capacitor Calculator

A conductor does not need a second plate nearby to have capacitance. An isolated sphere in free space has capacitance that depends only on its radius. The Earth has about 710 µF. A 10 mm metal ball has about 1.11 pF. This calculator handles two geometries: an isolated sphere (other plate at infinity) and a concentric spherical capacitor (inner sphere inside an outer shell). Enter the radius, optionally add voltage for charge and energy, and get the result.

Formulas

Isolated Sphere:
C = 4πε0εra

Concentric Spheres:
C = 4πε0εr(ab) / (b − a)

With voltage:
Q = CV — charge stored
E = ½CV² — energy stored

a = sphere radius (or inner radius). b = outer shell radius. ε0 = 8.854 × 10−12 F/m. εr = relative permittivity of the surrounding medium or gap dielectric. Capacitance scales linearly with radius — double the radius, double the capacitance.

The Earth (a = 6371 km)

C = 4π × 8.854×10−12 × 6371000
C = 4π × 8.854×10−12 × 6.371×106
C ≈ 709 µF

~710 µF — a famous result from electrostatics courses. The entire planet has less capacitance than a handful of electrolytic capacitors. This is the self-capacitance that determines how much charge the Earth can hold relative to infinity, and it is relevant to lightning physics and global electric circuit models.

Metal Ball (a = 10 mm, V = 1000 V)

C = 4π × 8.854×10−12 × 0.01 = 1.11 pF
Q = 1.11×10−12 × 1000 = 1.11 nC
E = 0.5 × 1.11×10−12 × 106 = 0.556 µJ

1.11 pF, 1.11 nC of charge, 0.556 µJ of energy. This is the ESD (electrostatic discharge) scenario — a small charged object at kilovolt potential. The energy is tiny but the voltage is high enough to damage sensitive CMOS electronics, which is why ESD protection is critical in semiconductor handling. For the energy stored in capacitors of all types, see the Capacitor Energy Calculator.

Van de Graaff Generator (a = 150 mm, V = 200 kV)

C = 4π × 8.854×10−12 × 0.15 = 16.7 pF
Q = 16.7×10−12 × 200000 = 3.34 µC
E = 0.5 × 16.7×10−12 × (200000)² = 0.334 J

16.7 pF capacitance, 3.34 µC charge, 0.334 J stored energy. Enough for a visible 10–15 cm spark but not dangerous to a healthy person (the energy threshold for cardiac fibrillation is roughly 1–10 J through the chest). This is why classroom Van de Graaff demonstrations are dramatic but safe. A larger dome (300 mm) doubles the capacitance to 33.4 pF and quadruples the energy to 1.34 J at the same voltage — approaching the caution threshold.

Why bigger domes matter: Capacitance scales linearly with radius. A 150 mm dome has 15× the capacitance of a 10 mm ball. More capacitance means more charge at the same voltage, longer sparks, and more impressive demonstrations. It also means the dome takes longer to charge — the belt current must supply more coulombs to reach the same potential.

Concentric Spheres — PTFE-Insulated (a = 50 mm, b = 60 mm, εr = 2.1)

C = 4π × 8.854×10−12 × 2.1 × (0.05 × 0.06) / (0.06 − 0.05)
C = 4π × 8.854×10−12 × 2.1 × 0.003 / 0.01
C = 4π × 8.854×10−12 × 0.63
C ≈ 70.0 pF

Q = 70.0×10−12 × 100 = 7.0 nC
E = 0.5 × 70.0×10−12 × 10000 = 350 nJ

70 pF — about 6× the capacitance of the isolated 50 mm sphere (which would be ~11.7 pF with εr = 2.1). The nearby outer shell dramatically increases the capacitance because the electric field is confined to the thin gap rather than spreading to infinity.

Thin-Gap and Large-Gap Limits

Thin Gap (a = 50 mm, b = 51 mm)

C = 4π × 8.854×10−12 × (0.05 × 0.051) / 0.001
C = 4π × 8.854×10−12 × 2.55 = 283.7 pF

283.7 pF — approaching parallel-plate territory. As the gap shrinks, the curved surface behaves more like two flat plates with area 4πa² and separation (b − a). The parallel-plate approximation C ≈ ε0 × 4πa² / (b − a) gives 278 pF — within 2% of the exact spherical result.

Large Gap (a = 50 mm, b = 5000 mm)

C = 4π × 8.854×10−12 × (0.05 × 5.0) / (5.0 − 0.05)
C = 4π × 8.854×10−12 × 0.05051 = 5.62 pF

5.62 pF — compared to 5.56 pF for the isolated 50 mm sphere. The outer shell at 100× the radius has almost no effect. This confirms that as b → ∞, the concentric formula reduces to C = 4πε0a.

Applications

Van de Graaff Generators

The dome’s self-capacitance determines the maximum charge, voltage, spark length, and stored energy. Use isolated sphere mode with the dome radius and target voltage.

ESD Modelling

The human body model (HBM) for ESD testing uses a capacitor (typically 100 pF) representing the body’s self-capacitance. A hand or finger tip approximated as a sphere gives a starting point. Real HBM values are higher due to the body’s complex geometry.

Lightning and Atmospheric Physics

The Earth’s self-capacitance (~710 µF) is central to the global electric circuit model. The atmospheric electric field charges the Earth-ionosphere system like a spherical capacitor, with the Earth as the inner conductor and the ionosphere as the outer shell (~60 km above the surface).

Concentric Electrode Systems

Spherical capacitors appear in high-voltage testing equipment, particle accelerator components, and some types of ionisation chambers. The concentric geometry provides a uniform radial electric field in the gap.

Physics Education

The spherical capacitor is a standard problem in electrostatics courses. Gauss’s law applied to spherical symmetry gives the electric field, and integrating the field gives the potential difference — from which the capacitance formula follows directly. For the fundamental charge-voltage relationship, see Ohm’s Law.

Frequently Asked Questions

How can a single sphere have capacitance without a second plate?
Capacitance is the ratio of charge to voltage: C = Q/V. An isolated sphere charged to voltage V (relative to infinity) holds charge Q = 4πε0aV. The ratio Q/V = 4πε0a is the self-capacitance. The “second plate” is at infinity — the charge on the sphere induces an equal and opposite charge distributed across the infinitely distant boundary.
Why is the Earth’s capacitance only ~710 µF?
Capacitance scales linearly with radius, and ε0 is a very small number (8.854 × 10−12 F/m). Even a 6371 km radius only produces ~710 µF. A single electrolytic capacitor on your desk can have the same capacitance — because its parallel-plate geometry with a micron-thin dielectric is far more efficient at storing charge than a sphere radiating into empty space.
How does the concentric formula relate to parallel plates?
When the gap (b − a) is very small compared to the radius, the spherical surface looks locally flat. The concentric formula approaches C ≈ ε0 × 4πa² / (b − a), which is the parallel-plate formula with area = 4πa² and gap = (b − a).
Can I use this for non-spherical objects?
As a rough estimate only. The self-capacitance of any convex conductor is approximately C ≈ 4πε0 × (equivalent sphere radius), where the equivalent radius gives the same surface area. For a cube of side s, the equivalent radius is s × √(6/(4π)) ≈ 0.69s, and the actual capacitance is about 0.66 × 4πε0s.
What dielectric should I use for the Earth calculation?
εr = 1 (vacuum/air). The Earth is surrounded by atmosphere and then space — both effectively εr ≈ 1. The atmosphere’s permittivity is 1.0006, negligibly different from vacuum.
Is a Van de Graaff spark dangerous?
A typical classroom generator (150 mm dome, 200 kV) stores about 0.33 J. This produces a startling but harmless spark. Larger generators (300+ mm domes, 500+ kV) can store several joules — enough to cause burns and potentially cardiac effects. Always check the stored energy (E = ½CV²) before operating at high voltages.

Last updated: March 2026