Spherical Capacitor Calculator

C = 4 x pi x epsilon0 x R. A sphere of radius R = 10 cm gives C = 4 x pi x 8.8541878128 x 10^-12 x 0.1 = 1.11265 x 10^-11 F = 11.1265 pF, exactly what the Solid Sphere tab shows.

From Gauss's law, the field between the shells (a < r < b) is E = Q / (4 x pi x epsilon0 x r^2). Integrating from a to b gives the voltage V = Q x (b - a) / (4 x pi x epsilon0 x a x b). Capacitance C = Q / V then yields C = 4 x pi x epsilon0 x a x b / (b - a).

8.8541878128 x 10^-12 F/m (CODATA 2018), the value this calculator uses. It is the proportionality constant linking electric charge to the electric field in vacuum.

Three inverse formulas, all in the Inverse tab: solid sphere R = C / (4 x pi x epsilon0); outer shell b = C x a / (C - 4 x pi x epsilon0 x a); inner shell a = C x b / (C + 4 x pi x epsilon0 x b). The outer-shell solution exists only if C exceeds the isolated-sphere capacitance of the inner shell.

Q = C x V and U = 1/2 x C x V^2. Example: C = 11.1265 pF at V = 1000 V stores Q = 11.1265 nC and U = 5.56325 microJ, as the Charge and Energy tab computes.

Outside the shells (r >= b), E = Q / (4 x pi x epsilon0 x r^2), the same as a point charge Q at the center. Inside a hollow conductor E = 0. Example: Q = 11.1265 nC gives E = 400.0 V/m at r = 50 cm and 10 kV/m right at a 10 cm surface.

C = kappa x C_vacuum, where kappa is the material's relative permittivity. Built-in presets: vacuum 1, air about 1.0006, Teflon (PTFE) 2.1, mica typically 6 (real range about 3-7 by type; enter a custom kappa for your exact material).

Because a x b / (b - a) = 0.05 x 0.10 / 0.05 = 0.10 m, which equals the solid sphere's radius. Both give C = 4 x pi x epsilon0 x 0.10 = 11.1265 pF. It is a coincidence of these particular numbers, not a general rule.

The formula approaches the parallel-plate result: C approx epsilon0 x A / d with A = 4 x pi x a^2 (the shell area) and d = b - a (the gap). The calculator's Two Shells tab notes this limit.

k = 1 / (4 x pi x epsilon0) = 8.9875517923 x 10^9 N m^2/C^2, derived from the same epsilon0 so every tab stays consistent.

Capacitance: F, mF, microF, nF, pF. Radius/length: mm, cm, m, inch. Charge: C, mC, microC, nC, pC. Results auto-format to the most readable unit.

It assumes two perfectly concentric thin spherical shells (or one isolated sphere) with a linear dielectric. It does not apply to non-concentric shells, non-spherical geometries, or nonlinear/ferroelectric dielectrics.

Energy density in the field is u = 1/2 x epsilon0 x E^2. Integrating u over all space outside the shells recovers exactly U = 1/2 x C x V^2, a good consistency check on the formulas.

Yes, free and unlimited, running entirely in your browser with no server calls.