{
  "id": "capacitance",
  "version": "7eb50c66310b",
  "status": "published",
  "name": "Capacitance Calculator",
  "question": "What is the capacitance?",
  "summary": "Capacitance calculator: C = Q ÷ V from charge and voltage, or the capacitance of parallel plates, a sphere or a cylinder with a dielectric, with the charge and energy at a voltage.",
  "category": "physics",
  "subcategory": "electricity",
  "url": "https://www.acalculator.org/physics/capacitance-calculator",
  "markdown": "https://www.acalculator.org/physics/capacitance-calculator.md",
  "kind": "function",
  "method": "C = Q ÷ V; parallel plates C = κε₀A ÷ d; sphere C = 4πκε₀R₁R₂ ÷ (R₂ − R₁), or 4πκε₀R₁ alone; cylinder C = 2πκε₀L ÷ ln(R₂ ÷ R₁); Q = CV; U = ½CV².",
  "assumptions": [
    "Ideal capacitors: the plates are large next to their gap (no edge field), and the cylinders are long next to their radii.",
    "A dielectric fills the whole gap and multiplies the vacuum capacitance by κ; ε₀ = 8.8541878188 × 10⁻¹² F/m (CODATA 2022)."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "by": {
        "title": "From",
        "description": "Work out C from the charge and voltage, or from the shape of the capacitor.",
        "type": "string",
        "enum": [
          "plates",
          "qv",
          "sphere",
          "cylinder"
        ]
      },
      "q": {
        "title": "Charge (Q)",
        "description": "The charge on each plate, in the unit chosen below.",
        "type": "number",
        "minimum": 1e-12,
        "maximum": 1000000000000
      },
      "qu": {
        "title": "Charge unit",
        "description": "The unit of the charge you type.",
        "type": "string",
        "enum": [
          "C",
          "mC",
          "uC",
          "nC",
          "pC"
        ]
      },
      "A": {
        "title": "Plate area (A)",
        "description": "The area of one plate that faces the other.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "area",
        "minimum": 1e-18,
        "maximum": 1000000000000
      },
      "d": {
        "title": "Plate separation (d)",
        "description": "The distance between the plates.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": 1e-12,
        "maximum": 1000000000
      },
      "r1": {
        "title": "Inner radius (R₁)",
        "description": "The radius of the inner sphere or cylinder.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": 1e-12,
        "maximum": 1000000000
      },
      "r2": {
        "title": "Outer radius (R₂)",
        "description": "The inner radius of the outer shell. Leave it empty for a sphere on its own.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": 1e-12,
        "maximum": 1000000000
      },
      "L": {
        "title": "Length (L)",
        "description": "The length of the two cylinders.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": 1e-12,
        "maximum": 1000000000
      },
      "die": {
        "title": "Between the conductors",
        "description": "The material that fills the gap, which sets the dielectric constant κ (OpenStax Table 8.1).",
        "type": "string",
        "enum": [
          "vacuum",
          "air",
          "ptfe",
          "paraffin",
          "siliconeOil",
          "polystyrene",
          "nylon",
          "paper",
          "quartz",
          "concrete",
          "diamond",
          "mica",
          "neoprene",
          "water",
          "srtio3",
          "custom"
        ]
      },
      "k": {
        "title": "Dielectric constant (κ)",
        "description": "The dielectric constant of the material in the gap; 1 for a vacuum.",
        "type": "number",
        "minimum": 1,
        "maximum": 10000000
      },
      "v": {
        "title": "Voltage (V)",
        "description": "The potential difference between the plates. Optional for a shape: gives the charge and energy.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "voltage",
        "minimum": 1e-9,
        "maximum": 1000000000
      }
    }
  },
  "outputs": {
    "capacitance": {
      "label": "Capacitance",
      "description": "C to 6 significant figures, in F, mF, µF, nF, pF or fF: the largest unit that keeps it at 1 or more.",
      "format": "text"
    },
    "farads": {
      "label": "Capacitance (F)",
      "description": "C in farads.",
      "format": "number"
    },
    "microfarads": {
      "label": "Capacitance (µF)",
      "description": "C in microfarads: 1 µF = 10⁻⁶ F.",
      "format": "number"
    },
    "picofarads": {
      "label": "Capacitance (pF)",
      "description": "C in picofarads: 1 pF = 10⁻¹² F.",
      "format": "number"
    },
    "chargeC": {
      "label": "Charge (C)",
      "description": "Q = C × V, in coulombs.",
      "format": "number"
    },
    "energy": {
      "label": "Energy stored (J)",
      "description": "U = ½ × C × V² = ½ × Q × V, in joules.",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "by": "plates",
      "q": 0.8,
      "qu": "nC",
      "A": "1 m2",
      "d": "1 mm",
      "r1": "5 cm",
      "r2": "10 cm",
      "L": "1 m",
      "die": "vacuum",
      "k": 1,
      "v": "40 V"
    },
    "outputs": {
      "capacitance": "8.85419 nF",
      "farads": 8.8541878188e-9,
      "microfarads": 0.0088541878188,
      "picofarads": 8854.1878188,
      "chargeC": 3.54167512752e-7,
      "energy": 0.00000708335025504
    },
    "text": "The capacitance is 8.85419 nF."
  },
  "examples": [
    {
      "given": {
        "by": "plates",
        "A": 1,
        "d": 0.001,
        "die": "vacuum",
        "v": 3000
      },
      "expect": {
        "capacitance": "8.85419 nF",
        "farads": 8.8541878188e-9,
        "chargeC": 0.0000265625634564,
        "energy": 0.0398438451846
      },
      "source": "OpenStax, University Physics Volume 2, §8.1 Capacitors and Capacitance (Equations 8.1, 8.3, 8.4 and 8.6; Examples 8.1 and 8.3), https://openstax.org/books/university-physics-volume-2/pages/8-1-capacitors-and-capacitance (retrieved 2026-10-18), Example 8.1: C = ε₀A ÷ d = 8.85 nF and Q = CV = 26.6 µC (with ε₀ = 8.85 × 10⁻¹²); NIST, CODATA 2022 recommended value of the vacuum electric permittivity ε₀ = 8.8541878188(14) × 10⁻¹² F/m, https://physics.nist.gov/cgi-bin/cuu/Value?ep0 (retrieved 2026-10-18); OpenStax, University Physics Volume 2, §8.3 Energy Stored in a Capacitor (Equation 8.10, U = Q² ÷ 2C = ½QV = ½CV²), https://openstax.org/books/university-physics-volume-2/pages/8-3-energy-stored-in-a-capacitor (retrieved 2026-10-18)"
    },
    {
      "given": {
        "by": "qv",
        "q": 0.8,
        "qu": "nC",
        "v": 40
      },
      "expect": {
        "capacitance": "20 pF",
        "farads": 2e-11,
        "picofarads": 20,
        "energy": 1.6e-8
      },
      "source": "OpenStax, University Physics Volume 2, §8.4 Capacitor with a Dielectric (Equation 8.11, C = κC₀; Example 8.10: 20.0 pF at 40.0 V holds 0.8 nC and 16.0 nJ), https://openstax.org/books/university-physics-volume-2/pages/8-4-capacitor-with-a-dielectric (retrieved 2026-10-18); OpenStax, University Physics Volume 2, §8.3 Energy Stored in a Capacitor (Equation 8.10, U = Q² ÷ 2C = ½QV = ½CV²), https://openstax.org/books/university-physics-volume-2/pages/8-3-energy-stored-in-a-capacitor (retrieved 2026-10-18); C = Q ÷ V = 0.8 nC ÷ 40 V = 20 pF, U = ½ × 0.8 nC × 40 V = 16 nJ"
    },
    {
      "given": {
        "by": "plates",
        "A": 1,
        "d": 0.001,
        "die": "ptfe"
      },
      "expect": {
        "capacitance": "18.5938 nF",
        "farads": 1.859379441948e-8
      },
      "source": "OpenStax, University Physics Volume 2, §8.4 Capacitor with a Dielectric (Equation 8.11, C = κC₀; Example 8.10: 20.0 pF at 40.0 V holds 0.8 nC and 16.0 nJ), https://openstax.org/books/university-physics-volume-2/pages/8-4-capacitor-with-a-dielectric (retrieved 2026-10-18) (C = κC₀) and Table 8.1 (κ = 2.1); hand calculation in content.mdx: 2.1 × 8.8541878188 nF"
    },
    {
      "given": {
        "by": "sphere",
        "r1": 0.05,
        "r2": 0.1,
        "die": "vacuum"
      },
      "expect": {
        "capacitance": "11.1265 pF",
        "farads": 1.1126500562018526e-11
      },
      "source": "OpenStax, University Physics Volume 2, §8.1 Capacitors and Capacitance (Equations 8.1, 8.3, 8.4 and 8.6; Examples 8.1 and 8.3), https://openstax.org/books/university-physics-volume-2/pages/8-1-capacitors-and-capacitance (retrieved 2026-10-18), Equation 8.4; hand calculation in content.mdx: 4π × ε₀ × 0.05 × 0.1 ÷ 0.05",
      "tolerance": 1e-12
    },
    {
      "given": {
        "by": "sphere",
        "r1": 6371000,
        "die": "vacuum"
      },
      "expect": {
        "capacitance": "708.869 µF",
        "farads": 0.0007088693508062003
      },
      "source": "OpenStax, University Physics Volume 2, §8.1 Capacitors and Capacitance (Equations 8.1, 8.3, 8.4 and 8.6; Examples 8.1 and 8.3), https://openstax.org/books/university-physics-volume-2/pages/8-1-capacitors-and-capacitance (retrieved 2026-10-18), Example 8.3 (C = 4πε₀R); hand calculation in content.mdx: 4π × ε₀ × 6,371,000 m",
      "tolerance": 1e-12
    },
    {
      "given": {
        "by": "cylinder",
        "r1": 0.0005,
        "r2": 0.002,
        "L": 1,
        "die": "vacuum"
      },
      "expect": {
        "capacitance": "40.1304 pF",
        "farads": 4.0130367958180986e-11
      },
      "source": "OpenStax, University Physics Volume 2, §8.1 Capacitors and Capacitance (Equations 8.1, 8.3, 8.4 and 8.6; Examples 8.1 and 8.3), https://openstax.org/books/university-physics-volume-2/pages/8-1-capacitors-and-capacitance (retrieved 2026-10-18), Equation 8.6; hand calculation in content.mdx: 2π × ε₀ × 1 ÷ ln(4)",
      "tolerance": 1e-12
    }
  ],
  "sources": [
    "OpenStax, University Physics Volume 2, §8.1 Capacitors and Capacitance: Equation 8.1, C = Q ÷ V; 8.3, C = ε₀A ÷ d; 8.4, spherical capacitor; 8.6, cylindrical capacitor; Example 8.1 (8.85 nF, 26.6 µC); Example 8.3 (isolated sphere, 4πε₀R), retrieved 2026-10-18. https://openstax.org/books/university-physics-volume-2/pages/8-1-capacitors-and-capacitance",
    "OpenStax, University Physics Volume 2, §8.3 Energy Stored in a Capacitor: Equation 8.10, U = Q² ÷ 2C = ½QV = ½CV², retrieved 2026-10-18. https://openstax.org/books/university-physics-volume-2/pages/8-3-energy-stored-in-a-capacitor",
    "OpenStax, University Physics Volume 2, §8.4 Capacitor with a Dielectric: Equation 8.11, C = κC₀; Example 8.10 (20.0 pF at 40.0 V, 0.8 nC, 16.0 nJ), retrieved 2026-10-18. https://openstax.org/books/university-physics-volume-2/pages/8-4-capacitor-with-a-dielectric",
    "OpenStax, University Physics Volume 2, §8.5 Molecular Model of a Dielectric: Table 8.1, dielectric constants at room temperature, retrieved 2026-10-18. https://openstax.org/books/university-physics-volume-2/pages/8-5-molecular-model-of-a-dielectric",
    "NIST, CODATA 2022 recommended values: vacuum electric permittivity ε₀ = 8.8541878188(14) × 10⁻¹² F/m, retrieved 2026-10-18. https://physics.nist.gov/cgi-bin/cuu/Value?ep0",
    "NIST Special Publication 811 (2008), Appendix B.8: 1 in = 0.0254 m, 1 ft = 0.3048 m, 1 mil = 0.001 in, retrieved 2026-10-18. https://www.nist.gov/pml/special-publication-811"
  ],
  "related": [
    "ohms-law",
    "impedance",
    "voltage-divider"
  ],
  "changelog": []
}