Capacitance Calculator

Compute capacitance for common geometries (parallel-plate, coaxial/cylindrical, spherical), series/parallel combinations, and the energy stored in a capacitor. Supports unit conversions (pF, nF, μF, mF, F) and step-by-step derivations.

Parallel-plate: C = ε₀ ε_r A / d

Capacitance: fundamentals, formulas and practical considerations

Capacitance measures a system's ability to store electric charge. A capacitor stores equal and opposite charges on two conductors separated by a dielectric. The basic definition is C = Q / V, where Q is charge and V potential difference. The SI unit is the farad (F), but typical capacitors use microfarads (μF), nanofarads (nF) or picofarads (pF).

Common capacitor geometries

Parallel-plate capacitor: two plates of area A separated by distance d with dielectric ε = ε₀ ε_r give C = ε₀ ε_r A / d. This is the simplest and widely used for deriving scaling laws: capacitance grows with area and dielectric constant, and decreases with separation.

Coaxial (cylindrical) capacitor: two concentric cylinders of radii a (inner) and b (outer) and length L (≫ radii) have capacitance C = (2π ε₀ ε_r L) / ln(b/a). The formula is often written per unit length: C' = 2π ε₀ ε_r / ln(b/a).

Spherical capacitor: two concentric spheres with radii a and b have C = 4π ε₀ ε_r (ab)/(b - a). In the limit b → ∞ (isolated sphere), the self-capacitance tends to 4π ε₀ ε_r a.

Series and parallel combinations

Capacitors in parallel add directly: C_eq = Σ C_i. Series follows 1/C_eq = Σ 1/C_i, the opposite of resistors. This behaviour comes from how voltage and charge distribute in each configuration.

Energy storage

The energy stored in a capacitor is U = ½ C V². This formula is derived by integrating the work required to move incremental charge onto the capacitor while raising its voltage. For fixed charge, U = Q²/(2C).

Dielectrics and practical notes

Dielectrics increase capacitance by a factor ε_r. Real dielectrics also introduce losses (dielectric loss tangent), breakdown strength (maximum field before breakdown), and temperature dependence. For high-voltage or precision capacitors, geometry, surface finish, and connecting leads (fringe fields) matter.

Worked examples

Example 1 — Parallel-plate. Two square plates 10 cm × 10 cm (A=0.01 m²), separated by 1 mm (d=0.001 m) in air (ε_r ≈ 1). C = ε₀ A / d ≈ (8.854e-12 × 0.01) / 0.001 = 8.854e-13 / 0.001 = 8.85e-11 F = 88.5 pF.

Example 2 — Coaxial cable. Inner radius a = 0.5 mm, outer b = 2.5 mm, ε_r = 2.2 (typical dielectric), L = 1 m. C = 2π ε₀ ε_r L / ln(b/a). Compute ln(5)=1.609, so C ≈ (2π × 8.85e-12 × 2.2 × 1)/1.609 ≈ 7.6e-11 F ≈ 76 pF/m.

Limitations and design tips

  • Use correct units — converting mm² to m² or pF to F is a common error.
  • For small separations, edge effects become important; finite-element simulation may be required for accurate values.
  • Keep dielectric thickness uniform and avoid sharp edges which concentrate electric field and cause breakdown.

When to use this calculator

Use the formulas here for classroom problems, quick design checks, and sanity checks on measurements. For detailed engineering of high-voltage capacitors, consult specialized design procedures and standards.

Frequently Asked Questions

1. What value should I use for ε₀?
Use ε₀ = 8.8541878128×10⁻¹² F/m (commonly 8.854e-12 F/m).
2. How do I convert μF to F?
1 μF = 1×10⁻⁶ F. Similarly, 1 nF = 1×10⁻⁹ F; 1 pF = 1×10⁻¹² F.
3. Can I ignore fringing fields?
Only when plate dimensions are large compared to separation; otherwise include fringing corrections or simulate numerically.
4. Why is series capacitance smaller?
In series the same charge sits on all capacitors but voltages add, so equivalent capacitance decreases.
5. How do I get capacitance per unit length for cables?
Use the coaxial formula per unit length: C' = 2π ε₀ ε_r / ln(b/a) and multiply by length.
6. What is stray capacitance?
Unintended capacitance between components or traces; it can affect high-frequency circuits and must be minimized where necessary.
7. How to compute capacitance of nonstandard shapes?
Use numerical methods (FEM) or approximate by decomposing shapes into known geometries.
8. Does temperature affect capacitance?
Yes — dielectric constants and geometry (thermal expansion) can change capacitance slightly with temperature; special capacitors minimize this drift.
9. Are supercapacitors measured the same way?
Yes conceptually (C = Q/V), but supercapacitors use electrochemical double layers producing very high capacitance per volume and require careful handling.
10. Where is this calculator useful?
Classroom problems, rough design, cable capacitance estimates, and energy storage checks for simple capacitor geometries.