Spherical and Cylindrical Capacitors: Formulas Explained

Capacitors need not be flat. A spherical capacitor uses two concentric metal spheres; a cylindrical capacitor uses two coaxial cylinders, like a cable. Their formulas differ from the parallel plate case but come from the same definition, C equals Q divided by V. These shapes govern coaxial cables, high-voltage terminals and the capacitance of isolated conductors.

The spherical case hides a famous result: a single isolated sphere also has capacitance, with the second plate imagined at infinity. This tells us how much charge any lone conductor can hold at a given potential, a key idea in lightning protection and electrostatics.

Key Concepts

• Geometry: Inner radius R1, outer radius R2; charge plus Q inside, minus Q outside. The field lives only in the gap.
• Cylinder length L: Must be much larger than the radii so end effects can be ignored.
• C is geometric: It depends on sizes and the medium, never on Q or V separately.
• Isolated sphere: The limit of the spherical formula as R2 tends to infinity.

Formulas

• Spherical: C equals 4 pi epsilon0 R1 R2 divided by (R2 minus R1). Multiply by K if a dielectric fills the gap.
• Isolated sphere: C equals 4 pi epsilon0 R. For Earth (R about 6.37 x 10^6 m), C is about 711 microfarads.
• Cylindrical: C equals 2 pi epsilon0 L divided by ln(R2 divided by R1), ln being the natural logarithm. Multiply by K for a dielectric. If using log base 10, multiply by 2.303.

Worked Example

(a) Spherical: R1 equals 0.050 m, R2 equals 0.060 m, air filled. (b) Cylindrical: L equals 0.50 m, R1 equals 1 mm, R2 equals 3 mm.

1. Spherical: 4 pi epsilon0 equals 1.112 x 10^-10. R1 R2 equals 0.0030. Numerator equals 3.34 x 10^-13. Divide by (0.060 minus 0.050) equals 0.010 m. C equals 3.34 x 10^-11 F, about 33 pF.
2. Cylindrical: ln(3) equals 1.099. 2 pi epsilon0 L equals 5.56 x 10^-11 x 0.50 equals 2.78 x 10^-11. Divide by 1.099: C equals 2.53 x 10^-11 F, about 25 pF.

Common Misconceptions

• Misconception: A single sphere cannot be a capacitor. Correction: It can, with the second plate at infinity; C equals 4 pi epsilon0 R.
• Misconception: Log base 10 can be used directly in the cylinder formula. Correction: The formula needs natural log; log10 must be converted with the factor 2.303.
• Misconception: A much larger outer sphere keeps raising C steeply. Correction: C approaches the isolated-sphere value 4 pi epsilon0 R1 and cannot exceed it.

Real-World Applications

• Coaxial cables: Cylindrical capacitance per metre sets signal behaviour in TV and internet cables.
• High-voltage spheres: Smooth spherical terminals store charge and reduce sparking in generators and substations.
• Level sensors: Cylindrical capacitors measure fuel and liquid levels in tanks.
• Atmospheric electricity: Earths capacitance as an isolated sphere anchors models of thunderstorms.

Shape changes the algebra, not the idea: spherical C follows R1 R2 over the gap, cylindrical C follows length over ln(R2 divided by R1), and both are workhorses for cables, sensors and high-voltage design.