Spherical Capacitor Calculator

A spherical capacitor consists of two concentric conducting shells separated by an insulating region. The inner sphere has radius a and the inner surface of the outer shell has radius b. Applying a voltage places equal and opposite charges on the conductors and creates a radial electric field between them. Because of spherical symmetry, the field and potential can be derived directly from Gauss's law. The capacitance is C = 4πε₀εᵣab/(b−a), where ε₀ is vacuum permittivity and εᵣ is the relative permittivity of a uniform dielectric filling the gap. Geometry alone determines capacitance for a chosen dielectric. Bringing the shells closer increases capacitance, as does increasing either radius while retaining a practical gap. A dielectric with relative permittivity greater than one stores more charge at the same applied voltage. Once capacitance is known, charge magnitude follows from Q = CV and electrostatic energy from U = ½CV². The calculator displays capacitance in picofarads, charge in nanocoulombs, and energy in microjoules so small laboratory-scale values remain readable. Negative voltage reverses charge polarity but does not make stored energy negative because voltage is squared. Displayed charge is the signed result of the entered potential, while capacitance depends only on geometry and dielectric. For an air-filled capacitor with 1 cm and 2 cm radii, capacitance is about 2.23 pF. At 1,000 V it carries about 2.23 nC and stores 1.11 μJ. These ideal results assume perfectly concentric shells, negligible lead and edge effects, homogeneous dielectric material, and conductors in electrostatic equilibrium. Nearby objects and measurement cables can add capacitance comparable to such a small theoretical value. Electric field is strongest at the inner conductor because radial field magnitude decreases with the square of distance. Practical designs must keep peak field below the dielectric's breakdown strength and must account for surface finish, contamination, temperature, pressure, and geometric imperfections. High voltage can be dangerous even when calculated stored energy looks small, especially when another supply continuously replenishes charge. The spherical geometry is valuable in electrostatics education, field probes, shielding models, and calibration systems because symmetry makes the equations exact under ideal assumptions. Use this calculator for initial analysis and unit-consistent comparisons. For manufactured devices, include tolerances, dielectric loss, frequency behavior, parasitic capacitance, and breakdown margin. Confirm safety-critical or precision designs with field simulation, measured components, and the applicable electrical standards.