A spherical capacitor is the textbook example of Gauss’s law in action. Van de Graaff generators are essentially isolated spheres; the Earth itself has a capacitance of about 710 µF.
How to use the Spherical Capacitor Calculator
- Inner sphere radius a
- Outer sphere radius b
- Dielectric
- Relative permittivity ε_r
Results update as you type, every field has its own unit menu, and everything is calculated in your browser.
How it works
Enter the inner and outer radii and choose the dielectric between them.
Inputs explained
| Field | What to enter |
|---|---|
| Inner sphere radius a | Units: cm, mm, m, in, ft |
| Outer sphere radius b | Units: cm, mm, m, in, ft |
| Dielectric | Choose one: Vacuum, Air, PTFE, Polyethylene … |
| Relative permittivity ε_r | A number |
What each result means
- Capacitance C = 4×pi×8.8541878128e-12×[er]×[Inner sphere radius a]×[Outer sphere radius b]÷([Outer sphere radius b] – [Inner sphere radius a])
- Isolated inner sphere = 4×pi×8.8541878128e-12×[er]×[Inner sphere radius a]
Example values
How the main result changes when Inner sphere radius a changes and everything else stays at the example values above:
| Inner sphere radius a (cm) | Capacitance C (pF) |
|---|---|
| 2.5 | 3.7088 |
| 3.75 | 6.6759 |
| 5 | 11.1265 |
| 6.25 | 18.5442 |
| 7.5 | 33.3795 |
Formula
C = 4π·ε₀·ε_r·a·b / (b − a) (isolated sphere: b → ∞, C = 4πε₀ε_r·a)
Worked example
Inputs — Inner sphere radius a: 5 cm; Outer sphere radius b: 10 cm; Dielectric: Vacuum.
Results — Capacitance C = 11.1265 pF; Isolated inner sphere = 5.56325 pF.
Key terms
- Concentric spheres
- Spheres sharing the same centre.
Frequently asked questions
What is the capacitance of an isolated sphere?
4πε₀a – about 111 pF per metre of radius.
What is Earth's capacitance?
About 710 µF.
Why must b be larger than a?
The outer sphere encloses the inner one.
How does the dielectric change C?
It multiplies it by ε_r.