Spherical Capacitor Calculator — Capacitance, Charge & Energy
Compute the capacitance, stored charge, and energy for a spherical capacitor — two concentric conducting spheres separated by a vacuum gap. Enter the inner and outer radii, and optionally the voltage across the gap.
m
m
V
Capacitance of the spherical capacitor (vacuum/air dielectric)
11.127 pF
Capacitance- 1
Product a × b
0.05 × 0.1 = 0.005 - 2
Gap (b − a)
0.1 − 0.05 = 0.05 - 3
Capacitance (F)
4π × ε₀ × 0.005 ÷ 0.05 = 0.00000000001113ε₀ = 8.854 × 10⁻¹² F/m - 4
Capacitance (pF)
0.00000000001113 × 10¹² = 11.1265
How does this calculator work?
C = 4πε₀ab/(b − a). With ε₀ = 8.854 × 10⁻¹² F/m, a 5 cm inner sphere inside a 10 cm outer shell gives C ≈ 11.1 pF. Stored charge Q = CV and energy U = ½CV². A dielectric with permittivity εᵣ multiplies C by εᵣ.
Formula
How this is calculated
A spherical capacitor consists of an inner conducting sphere of radius a surrounded by a concentric outer conducting shell of radius b, with vacuum (or air) between them. Applying Gauss's law to the radial electric field in the gap gives the capacitance C = 4πε₀ab/(b − a), where ε₀ = 8.854 × 10⁻¹² F/m is the permittivity of free space. Shrinking the gap (b → a) drives C upward — exactly as with parallel-plate capacitors.
If a dielectric fills the gap, multiply the vacuum result by the relative permittivity εᵣ (roughly 2–5 for plastics, up to 80 for water). With an applied voltage V, the charge stored on the inner shell is Q = CV and the electrostatic energy in the field is U = ½CV² joules.
Note that as b → ∞ the formula reduces to C = 4πε₀a, the self-capacitance of an isolated sphere. The formula assumes a uniform vacuum gap, perfect conductors, and a symmetric charge distribution; real coaxial sphere systems can deviate due to connector supports or imperfect symmetry.
Frequently asked questions
As b → ∞ the formula C = 4πε₀ab/(b − a) → 4πε₀a, which is the self-capacitance of an isolated conducting sphere. For a sphere the size of Earth (a ≈ 6.4 × 10⁶ m), this gives C ≈ 711 μF.
Replace ε₀ with ε₀εᵣ in the formula, so capacitance scales linearly with relative permittivity. For a material with εᵣ = 4 (common plastic) the capacitance is four times the vacuum value.
Inside the inner shell (r < a) and outside the outer shell (r > b) the field is zero by Gauss's law, assuming equal and opposite charges on the two shells. The field exists only in the gap a < r < b and decays as 1/r².
Also known as
TG we-Calculate Editorial Team. (2026). Spherical Capacitor Calculator — Capacitance, Charge & Energy [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/spherical-capacitor-calculator
TG we-Calculate Editorial Team. "Spherical Capacitor Calculator — Capacitance, Charge & Energy." TG we-Calculate. 2026. https://we-calculate.com/calculator/spherical-capacitor-calculator.
TG we-Calculate Editorial Team, "Spherical Capacitor Calculator — Capacitance, Charge & Energy," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/spherical-capacitor-calculator
@misc{wecalculate_spherical_capacitor_calculator, title = {Spherical Capacitor Calculator — Capacitance, Charge & Energy}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/spherical-capacitor-calculator}}, year = {2026}, note = {TG we-Calculate} }
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