Cylindrical & Spherical Capacitor Calculator

Place a charge +Q on the inner cylinder and −Q on the outer. Because of cylindrical symmetry, the electric field only points radially and has magnitude E(r) = Q / (2π ε₀ εᵣ L r) at any radius a ≤ r ≤ b. Voltage between the two plates is V = ∫ₐᵇ E(r) dr = (Q / (2π ε₀ εᵣ L)) × ln(b/a). Rearranged, C = Q/V = 2π ε₀ εᵣ L / ln(b/a). Same formula applies to coaxial cables — every RG-type is literally a cylindrical capacitor, which is why they quote capacitance-per-meter (typically ~100 pF/m for RG-58).

Place +Q on the inner sphere and −Q on the outer. E(r) = Q / (4π ε₀ εᵣ r²) from Gauss's law. Integrate from a to b: V = Q / (4π ε₀ εᵣ) × (1/a − 1/b). Rearranged: C = 4π ε₀ εᵣ ab / (b − a). When b → ∞, this reduces to C = 4π ε₀ εᵣ a — the self-capacitance of an isolated sphere (used to model Tesla coils and Van de Graaff generators).

Also called a 'variable condenser'. Two sets of concentric copper cylinders mesh together — rotating one set changes the overlap length, changing capacitance linearly from a few pF up to ~1000 pF. Air is the dielectric. These are the workhorse of high-power RF transmitters, being able to handle kilowatts without arcing. Our cylinder formula calculates them directly: L is the mesh overlap length, εᵣ = 1.

A rotary variable capacitor used in vintage radios and crystal sets. Two interleaved sets of semi-circular plates — rotating one set changes the overlap area, changing capacitance roughly linearly (or logarithmically in 'log law' versions). Formula is a hybrid between parallel plate (area changes) and cylindrical (arc geometry), but the cylindrical approximation usually gives within 10% accuracy for quick estimates.

Aluminum electrolytics are rolled: two long foil sheets wound into a cylinder with paper separator. Use the cylindrical formula with a and b as outer and inner radius of the roll, and L as the foil width — but this only gives the geometrical capacitance. In reality each roll layer is also capacitively coupled to its neighbours, so the effective capacitance is higher than the isolated cylinder predicts. For electrolytics the value is largely determined by the anodized oxide thickness (~10 nm) rather than geometry — treat the geometry as a sanity check on capacitance density.

Capacitance scales exactly linearly with εᵣ (relative permittivity). Air/vacuum = 1, PTFE = 2.1, Polyethylene = 2.3, PVC = 3.3, Glass = 5.2, Mica = 7, Ceramic can be 10 to 15 000. A capacitor with a high-εᵣ dielectric gives the same capacitance in a smaller package — but high-εᵣ materials almost always have higher loss (higher tanδ), so there is always a trade-off.