Spherical Capacitor
C = 4πε0 · a · b / (b − a)
Quick answer
This calculator solves C = 4πε0 × a × b / (b − a) for Capacitance. With Inner Radius 10 mm, Outer Radius 2.2 cm, the result is 2.04 pF. It solves in 3 directions, so you can enter the values you have and get the one you need.
Calculator
Formula
Description
A spherical capacitor consists of two concentric conducting spheres separated by a gap. Its capacitance depends on the inner and outer radii and the permittivity between them. As the outer sphere recedes to infinity it reduces to the self-capacitance of an isolated sphere (4πε0·a). The geometry is the textbook model for screened spherical electrodes and Van de Graaff terminals.
Variables
- C — Capacitance (F)
- a — Inner sphere radius (m)
- b — Outer sphere radius (m)
- ε0 — Permittivity of free space (× εr if filled)
Practical Notes
For a dielectric-filled gap multiply by εr. With b → ∞ the formula gives an isolated sphere’s capacitance C = 4πε0·a (about 0.11 pF per cm of radius).
Symbols and units
| Symbol | Meaning | Default unit | Accepted units |
|---|---|---|---|
| C | Capacitance | pF | pF, nF, µF, mF, F |
| a | Inner Radius | mm | mm, cm, m, km |
| b | Outer Radius | mm | mm, cm, m, km |
Step by step
- 1. What you know: Inner Radius 10 mm, Outer Radius 2.2 cm
- 2. The relation: C = 4πε0 × a × b / (b − a)
- 3. Substitute: a = 10 mm, b = 2.2 cm
- 4. Result: Capacitance = 2.04 pF
Nearest standard component values
| Series | Nearest value | Error |
|---|---|---|
| E12 | 2.2 pF | 7.9% |
| E24 | 2 pF | 2% |
Worked examples
Minimum range
Given: Inner Radius 0.1 mm, Outer Radius 0.22 mm
Result: Capacitance = 0.0204 pF
Low range
Given: Inner Radius 1 mm, Outer Radius 2.2 mm
Result: Capacitance = 0.204 pF
Typical
Given: Inner Radius 10 mm, Outer Radius 2.2 cm
Result: Capacitance = 2.04 pF
High range
Given: Inner Radius 10 cm, Outer Radius 22 cm
Result: Capacitance = 20.4 pF
Maximum range
Given: Inner Radius 1 m, Outer Radius 2.2 m
Result: Capacitance = 0.204 nF
Solving for each variable
Solve for Capacitance
Inner Radius 0.1 mm, Outer Radius 0.22 mm → 0.0204 pF
Solve for Inner Radius
Capacitance 1 pF, Outer Radius 0.22 mm → 0.2147 mm
Solve for Outer Radius
Capacitance 1 pF, Inner Radius 0.1 mm → 0.1011 mm
How Capacitance changes with Inner Radius
| Inner Radius (input) | Capacitance (result) |
|---|---|
| 1 mm | 0.1166 pF |
| 2.5 mm | 0.3138 pF |
| 5 mm | 0.72 pF |
| 7.5 mm | 1.266 pF |
| 10 mm | 2.04 pF |
| 15 mm | 5.245 pF |
| 2 cm | 24.48 pF |
| 4 cm | -5.44 pF |
| 7 cm | -3.57 pF |
| 10 cm | -3.138 pF |
The same result in other units
| Unit | Value |
|---|---|
| pF | 2.04 |
| nF | 0.00204 |
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<p><a href="https://vbatpower.com/formulas/spherical-capacitor/">Spherical Capacitor</a> Calculator by vbatpower.com</p>Frequently asked questions
How do you calculate Spherical Capacitor?
Use C = 4πε0 × a × b / (b − a). With Inner Radius 10 mm, Outer Radius 2.2 cm, Capacitance works out to 2.04 pF.
How do you calculate spherical capacitor formula concentric spheres?
Rearrange for Capacitance: with Inner Radius 1 mm, Outer Radius 2.2 mm, Capacitance is 0.204 pF.
What is the formula for Spherical Capacitor?
The formula is C = 4πε0 × a × b / (b − a). A spherical capacitor consists of two concentric conducting spheres separated by a gap.
How do you find Inner Radius when you know the other values?
Rearrange for Inner Radius: with Capacitance 1 pF, Outer Radius 0.22 mm, Inner Radius is 0.2147 mm.
How do you find Outer Radius when you know the other values?
Rearrange for Outer Radius: with Capacitance 1 pF, Inner Radius 0.1 mm, Outer Radius is 0.1011 mm.
What units does Spherical Capacitor use?
Capacitance is shown in pF, nF. Values are held in SI base units internally, so you can mix prefixes without converting by hand.
Related Concepts
All Capacitors formulas →Need more features?
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