Spherical Capacitor Calculator

Use this spherical capacitor calculator to determine the capacitance of a spherical capacitor filled with a dielectric.

Capacitance6.6759 pF4πε ab/(b−a). As the gap closes this grows without limit, and as b goes to infinity it approaches the isolated-sphere value of 1.1127 pF

As an isolated sphere1.1127 pFthe outer shell taken to infinity — 6× smaller than with the shell at 12 mm

Charge stored6.6759 nCat 1,000 V

Energy stored3.338 µJ

Peak field at the inner conductor600 kV/m0.2× the breakdown strength of dry air. The field is STRONGEST at the inner surface, never uniform, which is why breakdown always starts there

Permittivity used1 relativea dielectric multiplies the capacitance by exactly this

The formula

sphere C = 4πε₀ε_r ab/(b−a); cylinder C = 2πε₀ε_r L/ln(b/a)

Curved plates concentrate the field

A parallel-plate capacitor has a uniform field, so it breaks down everywhere at once. Concentric spheres and coaxial cylinders do not: the field is strongest at the inner conductor and falls off outward, so breakdown always starts at the inner surface however good the insulation elsewhere.

The coaxial case carries a logarithm, which makes capacitance remarkably insensitive to the radius ratio — doubling the outer diameter of a cable changes its capacitance per metre by well under a factor of two. An isolated sphere has capacitance all by itself, with infinity serving as the other plate; the entire Earth comes to about 709 µF, less than a small electrolytic capacitor.